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Gomes, Anik; Alam, Jahangir; Murtaza, Ghulam; Sultana, Tahmina; Tzirtzilakis, Efstratios E.; Ferdows, Mohammad

AppliedMath
, Volume 1 (1) – Dec 14, 2021

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Article Aligned Magnetic Field and Radiation Effects on Biomagnetic Fluid over an Unsteady Stretching Sheet with Various Slip Conditions 1 1 1 2 3 , Anik Gomes , Jahangir Alam , Ghulam Murtaza , Tahmina Sultana , Efstratios E. Tzirtzilakis * and Mohammad Ferdows Department of Mathematics, Comilla University, Comilla 3506, Bangladesh; thomasanikgomes@gmail.com (A.G.); jahangircu1994@gmail.com (J.A.); limonn@yahoo.com (G.M.) Department of Science and Humanities, Military Institute of Science and Technology, Dhaka 1216, Bangladesh; tahmina2011@yahoo.com Fluid Mechanics & Turbomachinery Laboratory, Department of Mechanical Engineering, University of the Peloponnese, 26334 Patras, Greece Research Group of Fluid Flow Modeling and Simulation, Department of Applied Mathematics, University of Dhaka, Dhaka 1000, Bangladesh; ferdows@du.ac.bd * Correspondence: etzirtzilakis@uop.gr Abstract: The aim of the present study is to analyze the effects of aligned magnetic ﬁeld and radiation on biomagnetic ﬂuid ﬂow and heat transfer over an unsteady stretching sheet with various slip conditions. The magnetic ﬁeld is assumed to be sufﬁciently strong enough to saturate the ferroﬂuid, and the variation of magnetization is approximated by a linear function of temperature difference. The governing boundary layer equations with boundary conditions are simpliﬁed by Citation: Gomes, A.; Alam, J.; suitable transformations. Numerical solution is obtained by using the bvp4c function technique in Murtaza, G.; Sultana, T.; Tzirtzilakis, MATLAB software. The numerical results are derived for the velocity, temperature, the skin friction E.E.; Ferdows, M. Aligned Magnetic coefﬁcient, and the rate of heat transfer. The evaluated results are compared with analytical study Field and Radiation Effects on documented in scientiﬁc literature. The present investigation illustrates that the ﬂuid velocity is Biomagnetic Fluid over an Unsteady Stretching Sheet with Various Slip decreased with the increasing values of radiation parameter, magnetic parameter, and ferromagnetic Conditions. AppliedMath 2021, 1, interaction parameter, though is increased as the Prandtl number, Grashof number, permeable 37–62. https://doi.org/10.3390/ parameter and thermal slip parameter are increased. In this investigation, the suction/injection appliedmath1010004 parameter had a good impact on the skin friction coefﬁcient and the rate of heat transfer. Academic Editor: Takayuki Hibi Keywords: biomagnetic ﬂuid; thermal radiation; heat ﬂux; magnetic ﬁeld; unsteady; stretching sheet; skin friction; rate of heat transfer Received: 27 October 2021 Accepted: 10 December 2021 Published: 14 December 2021 1. Introduction Publisher’s Note: MDPI stays neutral During the last few decades, due to its application in several areas in science and engi- with regard to jurisdictional claims in neering, the study of ﬂow and heat transfer over an unsteady stretching sheet has drawn published maps and institutional afﬁl- signiﬁcant attention to researchers. The study of rotating ﬂow and heat transfer has re- iations. ceived fervent interest in modern ﬂuid dynamics research, with applications in geophysics, biomedical engineering, medical science, planetary science, thermal insulations, etc. Biomagnetic Fluid Dynamics (BFD) is the study of the effects of an applied magnetic ﬁeld on biological ﬂuid ﬂow [1]. The most characteristic biomagnetic ﬂuid is blood. Blood Copyright: © 2021 by the authors. is a suspension of numerous cells such as red blood cells, white blood cells, and platelets in Licensee MDPI, Basel, Switzerland. a liquid electrolyte solution called plasma. Plasma contains 7% of principal proteins and This article is an open access article 90% of water, along with considerable concentration of ions. Blood as a whole is considered distributed under the terms and as a non-Newtonian ﬂuid predominantly when the characteristic dimension of the ﬂow is conditions of the Creative Commons nearby the cell dimension. As far as the stretching sheet ﬂows are concerned, Crane [2] Attribution (CC BY) license (https:// computed an exact similarity solution for the boundary layer ﬂow of a Newtonian ﬂuid creativecommons.org/licenses/by/ towards an elastic sheet. The sheet was stretched with the velocity proportional to the 4.0/). AppliedMath 2021, 1, 37–62. https://doi.org/10.3390/appliedmath1010004 https://www.mdpi.com/journal/appliedmath AppliedMath 2021, 1 38 distance from the origin. Barozzi and Dumas [3] numerically studied the convective heat transfer in blood vessels of the circulatory system. They observed that the rheological behavior of blood does not signiﬁcantly affect the heat transfer rate in small blood vessels. Pennes [4] studied the effects of blood perfusion and metabolic heat generation in living tissues using a simpliﬁed bio-heat transfer model. Although this model bears the potential to describe the effect of blood ﬂow on tissue temperature, it has some considerable short comings. This is because uniform perfusion rate was assumed, and the direction of blood ﬂow was not accounted for. Moreover, in his model, only the stream of venous blood as the ﬂuid stream equilibrated with tissue was considered. In recent years, the study of the magneto hydrodynamic (MHD) ﬂow of blood through the arteries has gained considerable interest because of its important applications in physi- ology. Theoretical estimates of blood ﬂow in arteries during the therapeutic procedure of electromagnetic hyperthermia used for cancer treatment were reported by Misra et al. [5]. A few important discussions were also available in that paper. The effects of electromagnetic radiation/ultrasonic radiation on blood ﬂow were studied by other investigations such as those of Inoue et al. [6], Nishimoto et al. [7], Bidin and Nazar [8], Irfan et al. [9], and Ishak [10]. The effect of viscous dissipation and radiation on the unsteady ﬂow of electri- cally conducting ﬂuid passed over a stretching surface was considered by Brickman [11] and Chand et al. [12]. The variable viscosity and thermal conductivity effects of combined heat and mass transfer in mixed convection over a UHF/UMF wedge in porous media were analyzed by Hassanien et al. [13] and Khan et al. [14]. Moreover, the effects of variable vis- cosity and thermal conductivity on a thin ﬁlm ﬂow over a shrinking/stretching sheet were also studied. Pal and Mondal [15] studied the effects of temperature-dependent viscosity and variable thermal conductivity on an MHD, non-Darcy mixed convection diffusion of species over a stretching sheet. The effects of thermal radiation over a stretching sheet under several ﬂow conditions have been also studied by several researchers [16–20]. Furthermore, a study of microploar ﬂuid under the inﬂuence of a magnetic ﬁeld through a stretched curved surface, using the Cattaneo–Christov heat model investigated by Khan et al. [21], had found that, in ﬂuid velocity, the magnetic ﬁeld parameter plays a signiﬁcant role. The movement of peristaltic ﬂow of a dusty ﬂuid with elastic properties in a curved conﬁguration was analyzed by Khan et al. [22]. A fully developed model of non-Newtonian ﬂuid through a 2D stretching sheet in the presence of Lorentz force and internal heat was presented by Vijaya et al. [23]. The BFD model [1], which involves both ferrohydrodynamics (FHD) and magnetohydrodynamics (MHD) principles, was utilized for the study of the effect of thermal radiation through a two-dimensional unsteady stretch- ing sheet by Alam et al. [24]. Finally, the impact of MHD on non-Newtonian mass and heat transfer along a curved stretched sheet was numerically studied by Yasim et al. [25]. The aim of the present investigation is to study the ﬂow and heat transfer in a stretch- ing sheet with an angle a to the vertical plane in the presence of a non-uniform source/sink. The mathematical formulation of the effect of the magnetic ﬁeld is that of BFD, involving both principles of ferrohydrodynamics (FHD) and MHD [1]. The governing partial dif- ferential equations have been transformed by similarity transformations into a coupled system of nonlinear ordinary differential equations. The solution was attained by using a MATLAB package. The effects of various parameters on the momentum and heat transfer characteristics have been studied, and the numerical results are presented graphically for the various values of the parameters entering the problem into consideration. 2. Model Description The unsteady two-dimensional BFD ﬂow of a viscous incompressible ﬂuid past a stretching sheet with an acute angle a to the vertical is considered. Where u and v are the velocity components along X-direction and Y-direction, re- spectively. The X-axis is considered along the plate and Y-axis is taken normal to it (see ax Figure 1). Initially (t = 0), the sheet is stretched with velocity U (x, t) = along (1ct) the X-axis, whereas the origin is kept ﬁxed in the ﬂuid medium of ambient temperature AppliedMath 2021, 1, FOR PEER REVIEW 3 of 26 2. Model Description The unsteady two-dimensional BFD flow of a viscous incompressible fluid past a stretching sheet with an acute angle to the vertical is considered. X Y Where u and v are the velocity components along -direction and -direc- tion, respectively. The X -axis is considered along the plate and Y -axis is taken normal to it (see Figure 1). Initially , the sheet is stretched with velocity (t = 0) ax U x,t = ( ) along the X -axis, whereas the origin is kept fixed in the fluid medium AppliedMath 2021, 1 1 − ct 39 ( ) of ambient temperature , and is the stretching surface temperature. A magnetic field of uniform strength B(t ) is acting normal to the direction of the flow, with an acute T , and T is the stretching surface temperature. A magnetic ﬁeld of uniform strength ¥ w angle . The magnetic Reynold’s number is assumed as much less than unity, and the flow B(t) is acting normal to the direction of the ﬂow, with an acute angle x. The magnetic is considered two-dimensional, therefore the induced magnetic field can be neglected in Reynold’s number is assumed as much less than unity, and the ﬂow is considered two- comparison to the applied magnetic field. Moreover, the fluids exhibit polarization due to dimensional, therefore the induced magnetic ﬁeld can be neglected in comparison to the the incorporated principles of FHD, and the applied magnetic field is assumed to be applied magnetic ﬁeld. Moreover, the ﬂuids exhibit polarization due to the incorporated strong enough to attain equilibrium magnetization. The velocity slip, thermal slip, viscous principles of FHD, and the applied magnetic ﬁeld is assumed to be strong enough to attain dissipation parameter, and ferromagnetic interaction parameter have been taken into ac- equilibrium magnetization. The velocity slip, thermal slip, viscous dissipation parameter, count. The boundary layer equations of the fluid and energy equation for the problem can and ferromagnetic interaction parameter have been taken into account. The boundary layer be written as [1,26,27]. equations of the ﬂuid and energy equation for the problem can be written as [1,26,27]. Figure 1. Physical sketch of the problem. Figure 1. Physical sketch of the problem. C Conservation onservation of of m mass: ass: ¶u ¶v + = 0 (1) u v ¶x ¶y + = 0 (1) Momentum conservation: x y 2 2 ¶u ¶u ¶u ¶ u sB (t) J m ¶H Momentum conservation: 0 + u + v = J sin xu u + M + gb (T T ) cos a (2) 1 ¥ ¶t ¶x ¶y ¶y r k (t) r ¶x 2 2 u u u u B (t) 2 0 * + u + v = − sin u − u + + g (T − T )cos (2) 1 Energy (Heat) conservation: t x y k (t ) x y ¶T ¶T ¶T ¶M ¶H ¶H ¶ T ¶q ¶u rC + u + v + m T u + v = K + q + rJ (3) p 0 ¶t ¶x ¶y ¶T ¶x ¶y ¶y ¶y ¶y Here, r is the biomagnetic ﬂuid density, B(t) = B (1 ct) , where B is a constant 0 0 representing the magnetic ﬁeld strength at t = 0, k (t) = k (1 ct) is the time dependent 1 2 permeability parameter, k is the constant permeability of the medium, g is the acceleration due to gravity, b is the coefﬁcient of thermal expansion, K is the thermal conductivity, C is the speciﬁc heat at constant pressure, J is the kinematic coefﬁcient of viscosity, s is the electrical conductivity, q is the radiative heat ﬂux, M is the magnetization, H is the r 1 magnetic ﬁeld of the ﬂuid, and T is the temperature of the ﬁeld. AppliedMath 2021, 1 40 The boundary conditions for the problem can be written as: [27,28] ¶u ¶T y = 0 : u = U + Nm , v = V , T = T + K w w w ¶y ¶y (4) y ! ¥ : u ! 0, T ! T ax bx Here, U (x, t) = is the stretching velocity, T (x, t) = T + is the surface w w ¥ (1ct) (1ct) temperature. Where a, b, c are the constants such that a > 0, b > 0, c 0 and ct < 1. In Equation (4), V represents the blood velocity at the wall and is equal to injec- tion/suction velocity given by JU V = f (0) (5) As it is implied by Equation (5), the mass transfer at the sheet of the wall takes place with a velocity V . For the case of injection, it is considered that V > 0, whereas V < 0 w w w is considered for the case of suction. N = N (1 ct) is the velocity slip factor, m is the coefﬁcient of viscosity, K = K (1 ct) is the thermal slip factor. The no-slip conditions hold when N = K = 0. Using the Rosseland approximation [27], the radiation heat ﬂux q is simpliﬁed as 4s ¶T q = (6) 3k ¶y where s and k are the Stefan–Boltzman constant and the mean absorption co-efﬁcient, respectively. Considering that the temperature differences within the ﬂow are such that the term 4 4 T may be expressed as a linear function of the temperature, T is expanded in a Taylor series about T . By neglecting the higher order terms beyond the ﬁrst degree in (T T ), ¥ ¥ it is obtained that 4 3 4 T =e 4T T 3T (7) ¥ ¥ The non-uniform heat source/sink q is deﬁned as KU m 0 q = A (T T ) f + B (T T ) w ¥ ¥ xJ where A and B are the coefﬁcient of a space- and temperature-dependent heat source/sink, respectively. The case of A > 0, B > 0 corresponds to internal heat generation, and that of A < 0, B < 0 corresponds to internal heat absorption. Physically, the role of a heat source in a ﬂuid transport is to enhance its thermal conductivity, which consequently results in increased ﬂuid temperature. On the other hand, heat sink decreases the thermal conductivity, which results in a decrease in the temperature of the ﬂuid. By substituting (5) and (6) into (3), the energy equation is reduced to 2 3 2 ¶T ¶T ¶T ¶M ¶H ¶H ¶ T 16s T ¶ T ¶u ¥ m rC + u + v + m T u + v = K + q + rJ (8) p 0 2 2 ¶t ¶x ¶y ¶T ¶x ¶y ¶y 3k ¶y ¶y ¶H The term m M in Equation (2) denotes the component of magnetic force per unit ¶x volume. This term is heavily dependent on the presence of a magnetic gradient, and, when the magnetic gradient is absent, this force vanishes. The heating due to adiabatic magnetization is represented by the second term, on the left hand side of the thermal energy Equation (8). The components H and H of the magnetic ﬁeld H = H , H , x y x y which are due to a magnetic dipole, are given by [29,30] ¶V g x (y + d) H (x, y) = = ¶x 2p 2 [x + (y + d) ] AppliedMath 2021, 1 41 ¶V g 2x(y + d) H (x, y) = = ¶y 2p 2 [x + (y + d) ] a x where V = is a scalar potential of the magnetic dipole, g = a and a is a 2 1 1 2p 2 x +(y+d) dimensionless distance deﬁned as a = d. xJ Thus, the magnetic ﬁeld strength intensity kHk = H is given by " # g 1 x 2 2 2 H(x, y) = [H + H ] = x y 2 4 2p (y + d) (y + d) The corresponding gradients are given by ¶H g 2x ¶x 2p (y + d) " # ¶H g 2 4x = + 3 5 ¶y 2p y + d y + d ( ) ( ) The magnetization M is generally determined by the ﬂuid temperature provided that the applied magnetic ﬁeld H is sufﬁciently strong enough to saturate the biomagnetic ﬂuid. Anderson and Valnes [29] considered that the variation of magnetization M with temperature T can be approximated by the linear equation M = k(T T ), 1 ¥ where k is a constant. To transform the momentum and energy equations, the following transformations are deﬁned: U T T w ¥ h = y;y = xJU f (h); q(h) = (9) xJ T T w ¥ Here, h is the similarity variable, y is the stream function, f and q are dimension- less quantities. The continuity Equation (1) is satisﬁed by the stream function y as ¶y u = ¶y and ¶y v = ¶x Making use of Equation (9), Equations (2) and (8) can be written as 1 1 2bq 000 02 00 0 00 0 2 0 f f + f f A f + h f M f sin x f + Grq cos a = 0 (10) 2 k (h + a ) (1 + R) 1 2bl(# + q) (A f + B q) 00 0 0 0 002 q A q + hq f q + f q f + + Ec f = 0 (11) Pr 2 Pr Pr(h + a ) The boundary conditions are transformed to: 0 00 0 h = 0 : f = S, f = 1 + S f , q = 1 + S q (12) h ! ¥ : f ! 0, q ! 0 AppliedMath 2021, 1 42 In Equation (12), S < 0 and S > 0 correspond to injection and suction, respectively. In the equation written above, primes denote derivatives with respect to h. S = Nr aJ and S = K are the non-dimensional velocity slip factor and thermal slip factor, respectively. mC c p Furthermore, A = is the unsteadiness parameter, Pr = is the Prandtl number, a K am l = is the viscous dissipation parameter, # = is the dimensionless T T rK(1ct)(T T ) w ¥ w ¥ g m rk(T T ) ak 0 w ¥ curie temperature, b = is the ferromagnetic interaction parameter, k = 2 3 2p J sB 16s T 0 ¥ is the permeability parameter,M = is the magnetic ﬁeld parameter, R = is ar 3Kk xgb (T T ) U w ¥ the radiation parameter, Gr = is the Grashof number, Ec = is the C (T T ) U p w ¥ U xU w w Eckert number, a = d is the dimensionless distance, and Re = is the local xJ J Reynolds number. The skin friction coefﬁcient and the Nusselt number constitute important characteris- tics of the ﬂow, deﬁned as: t xq w w C = ; Nu = f 2 ¥ K(T T ) w ¥ where, the wall stress t and the heat transfer q from the sheet are given by w w ¶u ¶T t = m ; q = K . w w ¶y ¶y y=0 y=0 Using the similarity variables (9), it is obtained that: 1 1 00 0 2 2 C = 2Re f (0); Nu = Re q (0) 3. Numerical Method The numerical solution of the set of nonlinear ordinary differential Equations (10) and (11) subject to boundary conditions (12) is utilized by using bvp4c function technique MATLAB 0 00 0 package. We consider f = y , f = y , f = y , q = y , q = y . Then, the governing equations 1 2 3 4 5 are transformed into the following system of first order ordinary differential equations: f = y 00 0 f = y = y 3 > h y 2by 000 0 2 2 2 4 f = y = A(y + y ) + y y y + My sin x + + Gry cos a 2 3 1 3 2 4 3 2 2 k (13) 3 (h+a ) 0 > q = y > h 2bly (#+y ) Pr 1 PrEc 00 0 1 4 2 ; q = y = A(y + y ) + y y y y + [A y + B y ] y 4 5 2 4 1 5 3 2 4 5 3 (1+R) 2 (1+R) (1+R) (1+R) (h+a) Along with the initial boundary conditions: y (0) = S, y (0) = 1 + S y (0), y (0) = 1 + S y (0), y (¥) = 0, y (¥) = 0 (14) 1 2 3 4 t 1 2 4 Equations (13) and (14) are integrated numerically as an initial value problem to a given terminal point. All the calculations are made by using bvp4c function available in MATLAB software. 4. Parameter Estimation In this study, the unsteady biomagnetic ﬂuid ﬂow along a two-dimensional stretch- ing/shrinking sheet under the action of a magnetic ﬁeld is investigated numerically. In order to achieve the numerical solution, it is necessary to determine some speciﬁc values for the dimensionless parameters, such as the Prandtl number, the unsteadiness parameter, the magnetic ﬁeld parameter, the permeability parameter, the radiation parameter, the ferromagnetic interaction parameter, the Grashof number, the Eckert number, the suc- AppliedMath 2021, 1 43 tion/injection parameter, the non-dimensional velocity-slip factor, the non-dimensional thermal slip factor, the acute angle of magnetic ﬁeld, the inclination angle, the co-efﬁcient of space and temperature. Many researchers have, in scientiﬁc literatures, reported various values of the above- mentioned dimensionless parameters. It is understood that, for human blood, the following data are considered: 3 1 1 1 1 3 m = 3.2 10 kgm s , C = 14.65 jkg K , k = 2.2 10 j(msK) in [31–33], where, human body temperature is considered to be T = 37 C and the body Curie temperature is considered to be T = 41 C. For this value of temperature, the dimensionless temperature is # = 78.5. mC Using these values, we have Pr = = 21. That is, for human blood ﬂow, the Prandtl number is 21. For the results presented in the following Figures 2–39, we consider the values of the dimensionless parameters entering into the problem under consideration as follows: (1) Unsteadiness parameter A = 0.0, 0.5, 0.9 as in [34] (2) Radiation parameter R = 1, 2, 3 as in [24,28,35,36] (3) Prandtl number Pr = 17, 21, 25 as in [29,37,38] (4) Ferromagnetic interaction parameter b = 1, 4, 7 as in [30,37,38] (5) Dimensionless distance a = 1 as in [34,36,38] (6) Viscous dissipation parameter l = 1.6 10 as in [24,37,38] (7) Dimensionless curie temperature # = 78.5 as in [29,37,38] (8) Suction/injection parameter S = 0.1, 0.5, 1.0, 1.5 as in [20,27] (9) Magnetic ﬁeld parameter M = 1, 2, 3 as in [24,31,38] (10) Permeability parameter k = 0.1, 0.3, 0.5 as in [27,31] (11) Eckert number Ec = 1, 2, 3 as in [27,31,37,38] (12) Grashof number Gr = 1, 3, 5 as in [27,31] (13) Non-dimensional velocity-slip factor S = 0.5, 1.0, 1.5 as in [27,31] (14) Non-dimensional thermal slip factor S = 0.5, 1.0, 1.5 as in [27,31] (15) Acute angle of magnetic ﬁeld x = 0, p/4, p/2 as in [27,31,34] (16) Inclination angle a = 0, p/4, p/2 as in [27,31] (17) Co-efﬁcient of space A = 0, 2, 4 as in [27,31] (18) Co-efﬁcient of temperature B = 0, 2, 4 as in [27,31] 5. Results and Discussion In order to assess the validity of the numerical results, the values of local Nusselt number q (0) have been compared with the existing works of Magyari and Keller [35], El-Aziz [36], Bidin and Nazar [8], and Anwar Ishak [10] by setting S = 0, S = 0, S = 0, b = 0, l = 0, A = 0, k ! ¥, Gr = 0, Ec = 0, A = 0, B = 0, x = p/2. It is apparent from the Table 1 that the numerical scheme and the coding used give results in good agreement with the abovementioned, previously published studies. A comparison of the local Nusselt number q (0) for various values R, M, Pr. Figures 2 and 3 show the velocity and temperature distributions with various values of the unsteadiness parameter A. From Figure 2, it is observed that the velocity proﬁles are decreased as the unsteadiness parameter is increasing. This is justiﬁed because the ac- companying reduction in the thickness of the momentum in the boundary layer. Moreover, from Figure 3, it is obtained that the temperature proﬁles are decreased signiﬁcantly as the unsteadiness parameter is increased. The fact is that, when the unsteadiness parameter is increased, less heat is transferred from the sheet to the ﬂuid. AppliedMath 2021, 1, FOR PEER REVIEW 9 of 26 coding used give results in good agreement with the abovementioned, previously pub- lished studies. A comparison of the local Nusselt number for various values R, M , Pr . − ' (0) Table 1. Comparisons with previous studies. R M Pr Magyari and Keller [35] El-Aziz [36] Bidin and Nazar [8] Anwar Ishak [10] Present Results 0 1 1 −0.954782 −0.954785 −0.9548 −0.9548 −0.954806 2 −1.4714 −1.4715 −1.471442 3 −1.869075 −1.869074 −1.8691 −1.8691 −1.869057 5 −2.500135 −2.500132 −2.5001 −2.50018 10 −3.660379 −3.660372 −3.6604 −3.660369 1 1 −0.8611 −0.861094 1 0 −0.5315 −0.5312 −0.531162 1 −0.4505 −0.450620 Figures 2 and 3 show the velocity and temperature distributions with various values AppliedMath 2021, 1 44 of the unsteadiness parameter A . From Figure 2, it is observed that the velocity profiles are decreased as the unsteadiness parameter is increasing. This is justified because the accompanying reduction in the thickness of the momentum in the boundary layer. More- Table 1. Comparisons with previous studies. over, from Figure 3, it is obtained that the temperature profiles are decreased significantly as the unsteadiness parameter is increased. The fact is that, when the unsteadiness param- R M Pr Magyari and Keller [35] El-Aziz [36] Bidin and Nazar [8] Anwar Ishak [10] Present Results eter is increased, less heat is transferred from the sheet to the fluid. 0 1 1 0.954782 0.954785 0.9548 0.9548 0.954806 Figures 4 and 5 show the velocity and temperature profiles for various values of the 2 1.4714 1.4715 1.471442 3 1.869075 1.869074 1.8691 1.8691 1.869057 radiation parameter R . From Figures 4 and 5, it is observed that an increment in radia- 5 2.500135 2.500132 2.5001 2.50018 tion parameter R results in a decrement in the fluid velocity profile, whereas the tem- 10 3.660379 3.660372 3.6604 3.660369 perature profile increases. The temperature profile is increased because the effect of the 1 1 0.8611 0.861094 radiation parameter is to enhance heat transfer. The thermal boundary layer thickness is 1 0 0.5315 0.5312 0.531162 increased with the increment of the thermal radiation. 1 0.4505 0.450620 0.7 -14 Pr=21,=1.610 , =1,=1,=78.5,M=3,R=3,k =0.5, 0.6 1 3 Gr=5,=/6,Ec=3,A*=2,B*=1,=/6,S =0.5,S=1,S =0.5 f t 0.5 0.4 A=0.0,0.5,0.9 f'() 0.3 0.2 0.1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 AppliedMath 2021, 1, FOR PEER REVIEW 10 of 26 Figure 2. Velocity profile Figure ( 2. ) for Velocity vario pr us oﬁle vaflues of (h) for various . values of A. f ' 0.8 -14 Pr=21,=1.610 , =1,=1,=78.5,M=3,R=3,k =0.5, 0.7 1 3 Gr=5,=/6,Ec=3,A*=2,B*=1,=/6,S =0.5,S=1,S =0.5 f t 0.6 0.5 A=0.0,0.5,0.9 () 0.4 0.3 0.2 0.1 0 0.5 1 1.5 Figure 3. Temperature proﬁle q(h) for various values of A. Figure 3. Temperature profile ( ) for various values of . Figures 4 and 5 show the velocity and temperature proﬁles for various values of the 0.35 radiation parameter R. From Figures 4 and 5, it is observed that an increment in radiation -14 Pr=25,=1.610 , =1,=5,A=0.4,=78.5,M=1,k =0.5, 1 3 0.3 Gr=3,=/4,=/4,Ec=1,A*=1,B*=2,S=0.5,S =1,S =0.5 f t 0.25 0.2 f'() R=1,2,3 0.15 0.1 0.05 0 0.5 1 1.5 2 2.5 Figure 4. Velocity profile for various values of R f ' ( ) 0.4 -14 Pr=25,=1.610 , =1,=1,A=0.8,=78.5,M=3,k =0.2,Gr=2, 1 3 0.35 =/4,=/4,Ec=1,A*=1,B*=2,S=0.5,S =0.5,S =1.5 f t 0.3 0.25 () 0.2 R=1,2,3 0.15 0.1 0.05 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 Figure 5. Temperature profile for various values of R . ( ) Pr The velocity and temperature profiles with various values of Prandtl number are shown in Figures 6 and 7. It is observed that an increment in Pr causes an increment in the velocity profile, whereas the temperature profile is decreased. This occurs because an increment in the Prandtl number means a decrement in the thermal diffusivity, and this AppliedMath 2021, 1, FOR PEER REVIEW 10 of 26 AppliedMath 2021, 1, FOR PEER REVIEW 10 of 26 0.8 0.8 -14 Pr=21,=1.610 , =1,=1,=78.5,M=3,R=3,k =0.5, -14 0.7 1 3 Pr=21,=1.610 , =1,=1,=78.5,M=3,R=3,k =0.5, 0.7 1 3 Gr=5,=/6,Ec=3,A*=2,B*=1,=/6,S =0.5,S=1,S =0.5 f t Gr=5,=/6,Ec=3,A*=2,B*=1,=/6,S =0.5,S=1,S =0.5 0.6 f t 0.6 0.5 0.5 A=0.0,0.5,0.9 AppliedMath 2021, 1 45 A=0.0,0.5,0.9 () 0.4 () 0.4 0.3 0.3 parameter R results in a decrement in the ﬂuid velocity proﬁle, whereas the temperature proﬁle increases. The temperature proﬁle is increased because the effect of the radiation 0.2 0.2 parameter is to enhance heat transfer. The thermal boundary layer thickness is increased 0.1 with the increment of the thermal radiation. 0.1 The velocity and temperature proﬁles with various values of Prandtl number Pr are shown in Figures 6 and 7. It is observed that an increment in Pr causes an increment in 0 0.5 1 1.5 the velocity proﬁle, whereas the temperature proﬁle is decreased. This occurs because an 0 0.5 1 1.5 increment in the Prandtl number means a decrement in the thermal diffusivity, and this phenomenon leads to the decreasing of energy ability that ﬁnally results in the reduction Figure 3. Temperature profile for various values of A . ( ) Figure 3. Temperature profile for various values of A . ( ) of the thermal boundary layer thickness. 0.35 0.35 -14 Pr=25,=1.610 , =1,=5,A=0.4,=78.5,M=1,k =0.5, -14 1 3 0.3 Pr=25,=1.610 , =1,=5,A=0.4,=78.5,M=1,k =0.5, 1 3 0.3 Gr=3,=/4,=/4,Ec=1,A*=1,B*=2,S=0.5,S =1,S =0.5 f t Gr=3,=/4,=/4,Ec=1,A*=1,B*=2,S=0.5,S =1,S =0.5 0.25 f t 0.25 0.2 0.2 f'() R=1,2,3 f'() R=1,2,3 0.15 0.15 0.1 0.1 0.05 0.05 00 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5 Figure 4. Velocity profile for various values of R f ' ( ) Figure 4. Velocity profile for various values of R Figure f ' ( ) 4. Velocity proﬁle f (h) for various values of R. 0.4 -14 0.4 Pr=25,=1.610 , =1,=1,A=0.8,=78.5,M=3,k =0.2,Gr=2, -14 1 3 0.35 Pr=25,=1.610 , =1,=1,A=0.8,=78.5,M=3,k =0.2,Gr=2, 1 3 0.35 =/4,=/4,Ec=1,A*=1,B*=2,S=0.5,S =0.5,S =1.5 f t =/4,=/4,Ec=1,A*=1,B*=2,S=0.5,S =0.5,S =1.5 0.3 f t 0.3 0.25 0.25 () 0.2 () R=1,2,3 0.2 R=1,2,3 0.15 0.15 0.1 0.1 0.05 0.05 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 Figure 5. Temperature proﬁle q(h) for various values of R. Figure 5. Temperature profile ( ) for various values of . Figure 5. Temperature profile for various values of . ( ) The velocity and temperature profiles with various values of Prandtl number Pr are Pr The velocity and temperature profiles with various values of Prandtl number are shown in Figures 6 and 7. It is observed that an increment in Pr causes an increment in shown in Figures 6 and 7. It is observed that an increment in Pr causes an increment in the velocity profile, whereas the temperature profile is decreased. This occurs because an the velocity profile, whereas the temperature profile is decreased. This occurs because an increment in the Prandtl number means a decrement in the thermal diffusivity, and this increment in the Prandtl number means a decrement in the thermal diffusivity, and this AppliedMath 2021, 1, FOR PEER REVIEW 11 of 26 AppliedMath 2021, 1, FOR PEER REVIEW 11 of 26 phenomenon leads to the decreasing of energy ability that finally results in the reduction of the thermal boundary layer thickness. phenomenon leads to the decreasing of energy ability that finally results in the reduction Figures 8 and 9 show the effect of the Grashof number Gr on the profiles of velocity of the thermal boundary layer thickness. and temperature. It was found that, with an increment in the Grashof number, which in- Figures 8 and 9 show the effect of the Grashof number Gr on the profiles of velocity creases the velocity profile, the opposite is true for the temperature profile. This is due to and temperature. It was found that, with an increment in the Grashof number, which in- the fact that an increase in the Grashof number means increment of the buoyancy forces creases the velocity profile, the opposite is true for the temperature profile. This is due to which finally reduce the thermal boundary layer thickness. AppliedMath 2021, 1 46 the fact that an increase in the Grashof number means increment of the buoyancy forces which finally reduce the thermal boundary layer thickness. 0.25 -14 =1.610 , =1,=5,A=0.4,=78.5,M=1,R=3,k =0.3, 0.25 1 3 -14 0.2 Gr=3,=/4,=/4,Ec=1,A*=1,B*=2,S=1,S =1,S =0.5 =1.610 , =1,=5,A=0.4,=78.5,M=1,R=3,k =0.3, f t 1 3 0.2 Gr=3,=/4,=/4,Ec=1,A*=1,B*=2,S=1,S =1,S =0.5 f t 0.15 Pr=17,21,25 f'() 0.15 Pr=17,21,25 0.1 f'() 0.1 0.05 0.05 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 Pr Figure 6. Velocity profile for various values of . f ' ( ) Figure 6. Velocity proﬁle f (h) for various values of Pr. Figure 6. Velocity profile for various values of Pr . f ' ( ) 0.5 -14 =1.610 , =1,=1,A=0.5,=78.5,M=1,R=1,k =0.5,Gr=3, 0.5 1 3 -14 0.4 =/2,=/4,Ec=3,A*=1,B*=4,S=1.5,S =0.5,S =1.5 =1.610 , =1,=1,A=0.5,=78.5,M=1,R=1,k =0.5,Gr=3, f t 1 3 0.4 =/2,=/4,Ec=3,A*=1,B*=4,S=1.5,S =0.5,S =1.5 f t 0.3 Pr=17,21,25 () 0.3 Pr=17,21,25 0.2 () 0.2 0.1 0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Figure 7. Temperature proﬁle q(h) for various values of Pr. Pr Figure 7. Temperature profile ( ) for various values of . Pr Figure 7. Temperature profil Figur e es ( 8)and for v 9 show arious val the efues fectof of the Grashof . number Gr on the proﬁles of velocity and temperature. It was found that, with an increment in the Grashof number, which increases the velocity proﬁle, the opposite is true for the temperature proﬁle. This is due to the fact that an increase in the Grashof number means increment of the buoyancy forces which ﬁnally reduce the thermal boundary layer thickness. The velocity and temperature proﬁles for various values of the Eckert number Ec are shown in Figures 10 and 11. The relationship between the kinetic energy in the ﬂow and the enthalpy is expressed by the Eckert number. It assimilates the conversion of kinetic energy into internal energy by the work done against the viscous ﬂuid stresses. The positive Eckert number implies cooling of the sheet. Hence, greater viscous dissipative heat causes a rise in temperature as well as the velocity, both of which are evident in Figures 10 and 11. AppliedMath 2021, 1, FOR PEER REVIEW 12 of 26 AppliedMath 2021, 1, FOR PEER REVIEW 12 of 26 AppliedMath 2021, 1 47 0.35 0.35 -14 Pr=25,=1.610 , =1,=5,A=0.4,=78.5,M=1,R=1,k =0.5, -14 1 3 0.3 Pr=25,=1.610 , =1,=5,A=0.4,=78.5,M=1,R=1,k =0.5, 1 3 0.3 =/4,=/4,Ec=1,A*=1,B*=2,S=0.5.S =1,S =0.5 f t =/4,=/4,Ec=1,A*=1,B*=2,S=0.5.S =1,S =0.5 0.25 f t 0.25 0.2 f'() 0.2 Gr=1,3,5 f'() Gr=1,3,5 0.15 0.15 0.1 0.1 0.05 0.05 0 0.5 1 1.5 2 0 0.5 1 1.5 2 Figure 8. Velocity profile for various values of Gr . f ' ( ) Figure 8. Velocity proﬁle f (h) for various values of Gr. Figure 8. Velocity profile for various values of Gr . f ' ( ) 0.7 0.7 -14 Pr=21,=1.610 , =1,=1,A=0.5,=78.5,M=1,R=1,k =0.5, -14 1 3 0.6 Pr=21,=1.610 , =1,=1,A=0.5,=78.5,M=1,R=1,k =0.5, 1 3 0.6 Ec=1,A*=1,B*=1,S =0.5,S =1.5,S=-0.1,=/4,=/4 f t Ec=1,A*=1,B*=1,S =0.5,S =1.5,S=-0.1,=/4,=/4 0.5 f t 0.5 0.4 0.4 Gr=1,3,5 () Gr=1,3,5 () 0.3 0.3 0.2 0.2 0.1 0.1 0 0.5 1 1.5 2 2.5 3 3.5 4 AppliedMath 2021, 1, FOR PEER REVIEW 13 of 26 0 0.5 1 1.5 2 2.5 3 3.5 4 Figure 9. Temperature proﬁle q(h) for various values of Gr. Figure 9. Temperature profile for various values of Gr . ( ) Figure 9. Temperature profile for various values of Gr . ( ) 0.5 -14 Ec The velocity and temperature profiles for various values of the Eckert number are Pr=25,=1.610 , =1,=1,A=0.5,=78.5,M=1,R=1, The velocity and temperature profiles for various values of the Eckert number Ec are shown in Figures 10 and 11. The relationship between the kinetic energy in the flow and k =0.3,Gr=1,A*=1,B*=2,S=-0.1,S =0.5,S =1,=/4,=/4 shown in Figures 10 and 11. The relationship between the kinetic energy in the flow and 0.4 the enthalpy is 3 expressed by the Eckert number. Itf assimilates t the conversion of kinetic the enthalpy is expressed by the Eckert number. It assimilates the conversion of kinetic energy into internal energy by the work done against the viscous fluid stresses. The posi- energy into internal energy by the work done against the viscous fluid stresses. The posi- tive Eckert number implies cooling of the sheet. Hence, greater viscous dissipative heat 0.3 tive Eckert number implies cooling of the sheet. Hence, greater viscous dissipative heat causes a rise in temperature as well as the velocity, both of which are evident in Figures causes a rise in temperature as well as the velocity, both of which are evident in Figures Ec=1,2,3 f'() 10 and 11. 0.2 10 and 11. Figures 12 and 13 show the effect of the suction/injection parameter S on the ve- Figures 12 and 13 show the effect of the suction/injection parameter S on the ve- locity and temperature profiles. From the figures, it is observed that the momentum locity and temperature profiles. From the figures, it is observed that the momentum boundary layer thickness is decreased with increasing values of S . It is expected that 0.1 boundary layer thickness is decreased with increasing values of S . It is expected that the increment of the suction results in the decrement of the thickness of the hydrodynamic the increment of the suction results in the decrement of the thickness of the hydrodynamic boundary layer. It also illustrates that an increment in S decreases the temperature pro- boundary layer. It also illustrates that an increment in S decreases the temperature pro- files in th 0 e flow regi 0.5 on. This is1 due to the 1.fa 5 ct that, as 2 the suction 2.5is increas3 ed, more warm 3.5 files in the flow region. This is due to the fact that, as the suction is increased, more warm fluid is taken away from the fluid region, causing a reduction in the thermal boundary fluid is taken away from the fluid region, causing a reduction in the thermal boundary layer thickness. Figure 10. Velocity proﬁle f (h) for various values of Ec. Figure 10. Velocity profile for various values of Ec . f ' ( ) layer thickness. 0.7 -14 Pr=25,=1.610 , =1,=1,A=0.5,=78.5,M=1,R=1, 0.6 1 k =0.3,Gr=1,=/2,=/4,A*=1,B*=4,S=1,S =0.5,S =0.5 3 f t 0.5 0.4 () Ec=1,2,3 0.3 0.2 0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Ec Figure 11. Temperature profile ( ) for various values of . 0.2 -14 Pr=25,=1.610 , =1,=1,A=0.8,=78.5,M=2,R=3, k =0.2,Gr=3,=/4,=/4,Ec=1,A*=1,B*=2,S =1.5,S =0.5 3 f t 0.15 f'() 0.1 S=0.5,1,1.5 0.05 0 0.5 1 1.5 2 Figure 12. Velocity profile for various values of S . f ' ( ) AppliedMath 2021, 1, FOR PEER REVIEW 13 of 26 0.5 -14 Pr=25,=1.610 , =1,=1,A=0.5,=78.5,M=1,R=1, k =0.3,Gr=1,A*=1,B*=2,S=-0.1,S =0.5,S =1,=/4,=/4 0.4 3 f t 0.3 f'() Ec=1,2,3 AppliedMath 2021, 1, FOR PEER REVIEW 13 of 26 0.2 0.1 0.5 -14 Pr=25,=1.610 , =1,=1,A=0.5,=78.5,M=1,R=1, k =0.3,Gr=1,A*=1,B*=2,S=-0.1,S =0.5,S =1,=/4,=/4 0.4 0 0.5 1 1.5 2 2.5 3 3.5 3 f t AppliedMath 2021, 1 48 Ec Figure 10 0.. 3Velocity profile f ' ( ) for various values of . Ec=1,2,3 f'() 0.7 0.2 -14 Pr=25,=1.610 , =1,=1,A=0.5,=78.5,M=1,R=1, 0.6 1 k =0.3,Gr=1,=/2,=/4,A*=1,B*=4,S=1,S =0.5,S =0.5 0.1 3 f t 0.5 0.4 () 0 0.5 Ec 1 =1,2,3 1.5 2 2.5 3 3.5 0.3 Figure 10. Velocity profile for various values of Ec . f ' ( ) 0.2 0.7 0.1 -14 Pr=25,=1.610 , =1,=1,A=0.5,=78.5,M=1,R=1, 0.6 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 k =0.3,Gr=1,=/2,=/4,A*=1,B*=4,S=1,S =0.5,S =0.5 3 f t 0.5 Figure 11. Temperature proﬁle q(h) for various values of Ec. Figure 11. Temperature profile for various values of Ec . ( ) 0.4 () Figures 12Ec and= 13 1, show 2,3the effect of the suction/injection parameter S on the velocity 0.2 and temperature proﬁles. From the ﬁgures, it is observed that the momentum boundary 0.3 -14 layer thickness is decreased with increasing values of S. It is expected that the increment Pr=25,=1.610 , =1,=1,A=0.8,=78.5,M=2,R=3, of the suction results in the decrement of the thickness of the hydrodynamic boundary 0.2 k =0.2,Gr=3,=/4,=/4,Ec=1,A*=1,B*=2,S =1.5,S =0.5 layer. It also illustrates that an increment in S decreases the temperature proﬁles in the ﬂow 3 f t 0.15 region. This is due to the fact that, as the suction is increased, more warm ﬂuid is taken 0.1 away from the ﬂuid region, causing a reduction in the thermal boundary layer thickness. Figures 14 and 15 show the effect of velocity slip parameter S on the velocity and f'() temperature proﬁles. From the Figure 14, it is observed that the presence of velocity slip 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.1 S=0.5,1,1.5 within the boundary layer causes the velocity level along the sheet to decrease. This is happening because the quantity 1 f (0) increases monotonically with S . We also observe Figure 11. Temperature profile for various values of Ec . ( ) that the temperature proﬁle decreases, as shown in Figure 15. 0.05 0.2 -14 Pr=25,=1.610 , =1,=1,A=0.8,=78.5,M=2,R=3, 0 k =0.2,G 0.5 r=3,=/4,=1/4,Ec=1,A*=1,B* 1.5=2,S =1.5,S =0 2.5 3 f t 0.15 Figure 12. Velocity profile for various values of S . f ' ( ) f'() 0.1 S=0.5,1,1.5 0.05 0 0.5 1 1.5 2 Figure 12. Velocity proﬁle f (h) for various values of S. Figure 12. Velocity profile ( ) for various values of S . f ' AppliedMath 2021, 1, FOR PEER REVIEW 14 of 26 0.2 -14 Pr=21,=1.610 , =1,=1,A=0.8,=78.5,,M=2,R=1, k =0.4,Gr=2,Ec=1,A*=0.5,B*=0.5,=/4,=/4,S =1.5,S =1 0.15 3 f t () 0.1 S=0.5,1,1.5 0.05 0 0.5 1 1.5 AppliedMath 2021, 1, FOR PEER REVIEW 14 of 26 AppliedMath 2021, 1 49 Figure 13. Temperature profile for various values of S . ( ) 0.2 -14 Figures 14 and 15 show the effect of velocity slip parameter S on the velocity and Pr=21,=1.610 , =1,=1,A=0.8,=78.5,,M=2,R=1, temperature profiles. From the Figure 14, it is observed that the presence of velocity slip k =0.4,Gr=2,Ec=1,A*=0.5,B*=0.5,=/4,=/4,S =1.5,S =1 0.15 3 f t within the boundary layer causes the velocity level along the sheet to decrease. This is happening because the quantity 1 − 𝑓 (0) increases monotonically with S . We also ob- () serve that the temperature profile decreases, as shown in Figure 15. 0.1 S=0.5,1,1.5 The effect of the thermal slip parameter on the velocity and temperature profiles is shown in Figures 16 and 17. In Figure 16, it can be observed that the presence of the 0.05 thermal slip factor on the temperature profiles has a significant effect. It is clear that the temperature near the surface is decreased as the values of are increased. This is hap- pening because the increment in the thermal slip parameter results in the increment of the 0 0.5 1 1.5 thermal coefficient, and the thermal diffusion towards the blood flow is reduced. The re- verse behavior takes place for the velocity boundary layer, as shown in Figure 16. Figure 13. Temperature proﬁle q(h) for various values of S. Figure 13. Temperature profile ( ) for various values of S . 0.5 -14 Figures 14 and 15 show the effect of velocity slip parameter S on the velocity and Pr=25,=1.610 , =1,=1,A=0.8,=78.5,M=2,R=3, temperature 0.4 profiles. From the Figure 14, it is observed that the presence of velocity slip k =0.2,Gr=3,=/4,=/4,Ec=1,A*=1,B*=2,S=0.5,S =0.5 3 t within the boundary layer causes the velocity level along the sheet to decrease. This is happening because the quantity 1 − 𝑓 (0) increases monotonically with S . We also ob- 0.3 f'() serve that the temperature profile decreases, as shown in Figure 15. S =0.5,1,1.5 The effect of the thermal slip parameter S on the velocity and temperature profiles 0.2 f is shown in Figures 16 and 17. In Figure 16, it can be observed that the presence of the thermal slip factor on the temperature profiles has a significant effect. It is clear that the 0.1 temperature near the surface is decreased as the values of are increased. This is hap- pening because the increment in the thermal slip parameter results in the increment of the 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 thermal coefficient, and the thermal diffusion towards the blood flow is reduced. The re- verse behavior takes place for the velocity boundary layer, as shown in Figure 16. Figure 14. Velocity proﬁle f (h) for various values of S . Figure 14. Velocity profile for various values of . f ' ( ) 0.5 The effect of the thermal slip parameter S on the velocity and temperature proﬁles -14 is shown in Figures 16 and 17. In Figure 16, it can be observed that the presence of the Pr=25,=1.610 , =1,=1,A=0.8,=78.5,M=2,R=3, thermal slip factor on the 1 temperature proﬁles has a signiﬁcant effect. It is clear that 0.4 k =0 the.temperatur 2,Gr=3, e= near /4the ,= surface /4,Ec=1 is decr ,A eased *=1as ,B* the =2 values ,S=0. of 5,SS ar =0 e incr .5 eased. This is 3 t happening because the increment in the thermal slip parameter results in the increment of the thermal coefﬁcient, and the thermal diffusion towards the blood ﬂow is reduced. The 0.3 reverse behavior takes place for the velocity boundary layer, as shown in Figure 16. f'() Figures 18 and 19 show the effects of the permeability parameter k on the velocity and temperatur S =0.e5, pr1, oﬁles. 1.5It is observed from Figure 18 that the presence of permeability 0.2 parameter k on the velocity proﬁles has a signiﬁcant incremental effect. This is happening because the ﬂow increases over the sheet as the permeability parameter is increased. The resistance on the ﬂow above the sheet is decreased as the permeability of the sheet 0.1 increases. From Figure 19, it can be noticed that the temperature proﬁle declines when the permeability parameter k enhances. 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 Figure 14. Velocity profile for various values of . f ' ( ) AppliedMath 2021, 1, FOR PEER REVIEW 15 of 26 AppliedMath 2021, 1, FOR PEER REVIEW 15 of 26 AppliedMath 2021, 1 50 AppliedMath 2021, 1, FOR PEER REVIEW 15 of 26 0.7 -14 0.7 Pr=21,=1.610 , =1,=1,A=0.8,=78.5,M=2,R=1, -14 1 0.6 Pr=21,=1.610 , =1,=1,A=0.8,=78.5,M=2,R=1, 0.7 k =0.4,Gr=2,Ec=2,A*=0.5,B*=0.5,S=0.5,S =1.5,=/2,=/4 -14 3 t 0.6 Pr=21,=1.610 , =1,=1,A=0.8,=78.5,M=2,R=1, k =0.4,Gr=2,Ec=2,A*=0.5,B*=0.5,S=0.5,S =1.5,=/2,=/4 0.5 3 t 0.6 k =0.4,Gr=2,Ec=2,A*=0.5,B*=0.5,S=0.5,S =1.5,=/2,=/4 0.5 3 t 0.4 0.5 S =0.5,1,1.5 0.4 () S =0.5,1,1.5 (0 ) .3 0.4 S =0.5,1,1.5 0.3 () 0.2 0.3 0.2 0.1 0.2 0.1 0.1 0 0.5 1 1.5 0 0.5 1 1.5 0 0.5 1 1.5 Figure 15. Temperature profile ( ) for various values of . Figure 15. Temperature profile for various values of . ( ) Figure 15. Temperature proﬁle q h for various values of S . ( ) Figure 15. Temperature profile for various values of . ( ) 0.14 -14 0.14 Pr=25,=1.610 , =1,=5,A=0.8,=78.5,M=2,R=3,k =0.2, 0.12 -14 1 3 0.14 Pr=25,=1.610 , =1,=5,A=0.8,=78.5,M=2,R=3,k =0.2, Gr=3,=/4,=/4,Ec=2,A*=1,B*=2,S=0.5,S =1.5 0.12 1 3 -14 Pr=25,=1.610 , =1,=5,A=0.8,=78.5,M=2,R=3,k =0.2, 0.1 Gr=3,=/4,=/4,Ec=2,A*=1,B*=2,S=0.5,S =1.5 0.12 1 3 0.1 Gr=3,=/4,=/4,Ec=2,A*=1,B*=2,S=0.5,S =1.5 0.08 f'() 0.1 S =0.5,1,1.5 0.08 f'() 0.06 S =0.5,1,1.5 0.08 f'() 0.06 S =0.5,1,1.5 0.04 0.06 0.04 0.02 0.04 0.02 0.02 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Figure 16. Velocity profile for various values of . f ' ( ) Figure 16. Velocity proﬁle f (h) for various values of S . Figure 16. Velocity profile for various values of . f ' ( ) Figure 16. Velocity profile for various values of . f ' ( ) 0.35 -14 0.35 Pr=21,=1.610 , =1,=1,A=0.8,=78.5,M=2,R=1, 0.3 -14 0.35 Pr=21,=1.610 , =1,=1,A=0.8,=78.5,M=2,R=1, k =0.4,Gr=2,Ec=1,A*=0.5,B*=0.5,S =1.5,S=0.5,=/4,=/4 0.3 1 -14 3 f Pr=21,=1.610 , =1,=1,A=0.8,=78.5,M=2,R=1, 0.25 k =0.4,Gr=2,Ec=1,A*=0.5,B*=0.5,S =1.5,S=0.5,=/4,=/4 0.3 3 f 0.25 k =0.4,Gr=2,Ec=1,A*=0.5,B*=0.5,S =1.5,S=0.5,=/4,=/4 3 f 0.2 S =0.5,1,1.5 0.25 () 0.2 S =0.5,1,1.5 () 0.15 0.2 S =0.5,1,1.5 () 0.15 0.1 0.15 0.1 0.05 0.1 0.05 0.05 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.2 0 0 0.2 0.4 0.6 0.8 1 1.2 Figure 17. Temperature proﬁle q(h) for various values of S . Figure 17. Temperature profile ( ) for various val ues of . Figure 17. Temperature profile for various values of . ( ) Figure 17. Temperature profile ( ) for various values of . Figures 18 and 19 show the effects of the permeability parameter on the velocity Figures 18 and 19 show the effects of the permeability parameter k on the velocity and temperature profiles. It is observed from Figure 18 that the presence of permeability Figures 18 and 19 show the effects of the permeability parameter on the velocity and temperature profiles. It is observed from Figure 18 that the presence 3 of permeability and temperature profiles. It is observed from Figure 18 that the presence of permeability AppliedMath 2021, 1, FOR PEER REVIEW 16 of 26 AppliedMath 2021, 1, FOR PEER REVIEW 16 of 26 parameter on the velocity profiles has a significant incremental effect. This is happen- para ing met becer ause th o e n fth low e velocity increasepro s ov files er th he as sheet a sign as ifica the nt p in ermeabi crement lity al p ef aramet fect. Th er isis is in hap creased. pen- The resistance on the flow above the sheet is decreased as the permeability of the sheet ing because the flow increases over the sheet as the permeability parameter is increased. increases. From Figure 19, it can be noticed that the temperature profile declines when the The resistance on the flow above the sheet is decreased as the permeability of the sheet increases. From Figure 19, it can be noticed that the temperature profile declines when the permeability parameter enhances. permea Fibi gures lity para 20 an met d er 21 show enhances the effect . of inclination parameter on the velocity and temperature profiles. From Figure 20, we observe that the velocity profile decreases with Figures 20 and 21 show the effect of inclination parameter on the velocity and an increment in the inclination parameter . It seems that the angle of inclination de- temperature profiles. From Figure 20, we observe that the velocity profile decreases with creases the effect of the buoyancy force due to thermal diffusion by a factor of cos . an increment in the inclination parameter . It seems that the angle of inclination de- Therefore, the driving force to the fluid decreases, and, as a result, the velocity is finally creases the effect of the buoyancy force due to thermal diffusion by a factor of cos . decreased. The reverse is happening in the temperature profile, which is shown in Figure Therefore, the driving force to the fluid decreases, and, as a result, the velocity is finally AppliedMath 2021, 1 51 decreased. The reverse is happening in the temperature profile, which is shown in Figure 0.25 -14 0.25 Pr=25,=1.610 , =1,=5,A=0.4,=78.5,M=1,R=1, -14 0.2 Pr=2 Gr=1 5,, =1 =./6 4 ,10 =/,4 ,Ec=1 =1,,=5 A*=1 ,A=0 ,B* .4 =2 ,,=7 S=0 8.5 .5 ,M ,S=1 =1 ,R ,S =1 =0 , .5 f t 0.2 Gr=1,=/4,=/4,Ec=1,A*=1,B*=2,S=0.5,S =1,S =0.5 f t 0.15 f'() 0.15 f'() k =0.1,0.3,0.5 0.1 k =0.1,0.3,0.5 0.1 0.05 0.05 0 0.5 1 1.5 2 0 0.5 1 1.5 2 Figure 18. Velocity profile for various values of . f ' ( ) Figure 18. Velocity proﬁle f (h) for various values of k . Figure 18. Velocity profile for various values of . f ' ( ) 0.7 -14 Pr=21,=1.610 , =1,=1,A=0.5,=78.5,M=1,R=1,Gr=1, 0.7 0.6 -14 Pr=2 Ec=2 1,, A=1 *=1 .6 , B* 10 =1,S , =1 =1 .5,, S=1 =0 ,A .5 =0 ,S=-0 .5,.=7 1, 8= .5 ,M /4=1 ,,= R =1 /4,Gr=1, f 1 t 0.6 0.5 Ec=2,A*=1,B*=1,S =1.5,S =0.5,S=-0.1,=/4,=/4 f t 0.5 0.4 () 0.4 0.3 () k =0.1,0.3,0.5 0.3 0.2 k =0.1,0.3,0.5 0.2 0.1 0.1 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 Figure 19. Temperature proﬁle q(h) for various values of k . Figure 19. Temperature profile for various values of . ( ) Figures 20 and 21 show the effect of inclination parameter a on the velocity and Figure 19. Temperature profile for various values of . ( ) temperature proﬁles. From Figure 20, we observe that the velocity proﬁle decreases with an increment in the inclination parameter a. It seems that the angle of inclination decreases the effect of the buoyancy force due to thermal diffusion by a factor of cos a. Therefore, the driving force to the ﬂuid decreases, and, as a result, the velocity is ﬁnally decreased. The reverse is happening in the temperature proﬁle, which is shown in Figure 21. Figures 22 and 23 show the effect of the inclination angle of the magnetic ﬁeld x on the velocity and temperature proﬁles. It is noticed that the velocity proﬁle is reduced and that the temperature proﬁle is enhanced by the increment in the inclination angle. This may be due to the fact that a rise in the aligned angle makes the applied magnetic ﬁeld stronger. Figures 24 and 25 depict the effects of the magnetic parameter M on the velocity and temperature proﬁles. It is observed that the velocity proﬁle is decreased as the magnetic parameter is increased. The increment of the magnetic parameter increases the introduced Lorentz force in the boundary layer, and, hence, the velocity proﬁle in the boundary layer is decreased. An increment in the magnetic parameter would enhance the Lorentz force AppliedMath 2021, 1 52 and, consequently, an augmentation of the Lorentz force opposes the ﬂow, and the ﬂuid motion is reduced. From Figure 25, it is noticed that the temperature proﬁles increase as the AppliedMath 2021, 1, FOR PEER REVIEW 17 of 26 magnetic parameter increases. This indicates the fact that the introduction of the transverse AppliedMath 2021, 1, FOR PEER REVIEW 17 of 26 magnetic ﬁeld to an electrically conductive ﬂuid gives rise to the Lorentz force. All these effects result in the increment of the temperature of the ﬂuid. 0.5 0.5 -14 Pr=25,=1.610 -14, =1,=1,A=0.8,=78.5,M=3,R=3,k =0.5, Pr=25,=1.610 ,1 =1,=1,A=0.8,=78.5,M=3,R=3,k3 =0.5, 1 3 0.4 Gr=3,=/4,Ec=2,A*=1,B*=1,S=1,S =0.5,S =0.5 0.4 Gr=3,=/4,Ec=2,A*=1,B*=1,S=1,Sf =0.5,St =0.5 f t 0.3 0.3 =0,/4,/2 f'() =0,/4,/2 f'() 0.2 0.2 0.1 0.1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Figure 20. Velocity profile for various values of . f ' ( ) Figure 20. Velocity profile for various values of . f ' ( ) Figure 20. Velocity proﬁle f (h) for various values of a. -14 Pr=21,=1.610 -14, =1,=1,A=0.5,=78.5,M=3,R=1,k =0.5, Pr=21,=1.610 ,1 =1,=1,A=0.5,=78.5,M=3,R=1,k3 =0.5, 1 3 0.8 Gr=1,Ec=1,A*=1,B*=1,S =0.5,S =1.5,S=-0.1,=/4 0.8 f t Gr=1,Ec=1,A*=1,B*=1,S =0.5,S =1.5,S=-0.1,=/4 f t 0.6 0.6 () =0,/4,/2 () =0,/4,/2 0.4 0.4 0.2 0.2 0 1 2 3 4 5 6 0 1 2 3 4 5 6 Figure 21. Temperature Figure profile 21. Temperatur for v earious proﬁle q val (h)ues for various of . values of a. ( ) Figure 21. Temperature profile for various values of . ( ) The velocity and temperatre proﬁles for various values of ferromagnetic interaction Figures 22 and 23 show the effect of the inclination angle of the magnetic field on parameter b are shown in Figures 26 and 27. It is observed that the velocity of the ﬂuid Figures 22 and 23 show the effect of the inclination angle of the magnetic field on decreases with an increment of ferromagnetic number, whereas the temperature proﬁle the velocity and temperature profiles. It is noticed that the velocity profile is reduced and the velocity and temperature profiles. It is noticed that the velocity profile is reduced and is increased in these cases. The region behind that ferromagnetic number is directly that the temperature profile is enhanced by the increment in the inclination angle. This that the temperature profile is enhanced by the increment in the inclination angle. This related to the celvin force, which is also known as the drug force. The results observed in may be due to the fact that a rise in the aligned angle makes the applied magnetic field Figures 24–27 are in accordance with those presented in [30,31]. may be due to the fact that a rise in the aligned angle makes the applied magnetic field stronger. stronger. Figures 24 and 25 depict the effects of the magnetic parameter M on the velocity Figures 24 and 25 depict the effects of the magnetic parameter M on the velocity and temperature profiles. It is observed that the velocity profile is decreased as the mag- and temperature profiles. It is observed that the velocity profile is decreased as the mag- netic parameter is increased. The increment of the magnetic parameter increases the in- netic parameter is increased. The increment of the magnetic parameter increases the in- troduced Lorentz force in the boundary layer, and, hence, the velocity profile in the troduced Lorentz force in the boundary layer, and, hence, the velocity profile in the boundary layer is decreased. An increment in the magnetic parameter would enhance the boundary layer is decreased. An increment in the magnetic parameter would enhance the Lorentz force and, consequently, an augmentation of the Lorentz force opposes the flow, Lorentz force and, consequently, an augmentation of the Lorentz force opposes the flow, and the fluid motion is reduced. From Figure 25, it is noticed that the temperature profiles and the fluid motion is reduced. From Figure 25, it is noticed that the temperature profiles increase as the magnetic parameter increases. This indicates the fact that the introduction increase as the magnetic parameter increases. This indicates the fact that the introduction of the transverse magnetic field to an electrically conductive fluid gives rise to the Lorentz of the transverse magnetic field to an electrically conductive fluid gives rise to the Lorentz force. All these effects result in the increment of the temperature of the fluid. force. All these effects result in the increment of the temperature of the fluid. The velocity and temperatre profiles for various values of ferromagnetic interaction The velocity and temperatre profiles for various values of ferromagnetic interaction parameter are shown in Figures 26 and 27. It is observed that the velocity of the fluid parameter are shown in Figures 26 and 27. It is observed that the velocity of the fluid decreases with an increment of ferromagnetic number, whereas the temperature profile is decreases with an increment of ferromagnetic number, whereas the temperature profile is increased in these cases. The region behind that ferromagnetic number is directly related increased in these cases. The region behind that ferromagnetic number is directly related AppliedMath 2021, 1, FOR PEER REVIEW 18 of 26 AppliedMath 2021, 1, FOR PEER REVIEW 18 of 26 AppliedMath 2021, 1, FOR PEER REVIEW 18 of 26 to the celvin force, which is also known as the drug force. The results observed in Figures AppliedMath 2021 to ,th 1 e celvin force, which is also known as the drug force. The results observed in Figures 53 to the celvin force, which is also known as the drug force. The results observed in Figures 24–27 are in accordance with those presented in [30,31] 24–27 are in accordance with those presented in [30,31] 24–27 are in accordance with those presented in [30,31] 0.5 0.5 0.5 -14 -14 Pr=25,=1.610 , =1,=1,A=0.8,=78.5,M=3,R=3,k =0.5, -14 Pr=25,=1.610 , =1,=1,A=0.8,=78.5,M=3,R=3,k =0.5, 1 3 Pr=25,=1.610 ,1=1,=1,A=0.8,=78.5,M=3,R=3,k3=0.5, 0.4 1 3 Gr=3,=/4,Ec=2,A*=1,B*=1,S=1,S =0.5,S =0.5 0.4 Gr=3,=/4,Ec=2,A*=1,B*=1,S=1,S =0.5,S =0.5 0.4 f t Gr=3,=/4,Ec=2,A*=1,B*=1,S=1,S =0.5,S =0.5 f t f t 0.3 0.3 0.3 f'() f'() f'() =0, /4, /2 =0, /4, /2 0.2 0.2 =0, /4, /2 0.2 0.1 0.1 0.1 0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5 Figure 22. Velocity profile for various values of . f ' ( ) Figure 22. Velocity profile for various values of . f ' ( ) Figure 22. Velocity profile Figuref '22. ( ) V elocity for various proﬁleva f lue (h)s for of various . values of x . 0.5 0.5 0.5 -14 -14 Pr=21,=1.610 , =1,=1,A=0.5,=78.5,M=3,R=1,k =0.5, -14 Pr=21,=1.610 , =1,=1,A=0.5,=78.5,M=3,R=1,k =0.5, 1 3 Pr=21,=1.610 , =1,=1,A=0.5,=78.5,M=3,R=1,k =0.5, 1 3 1 3 0.4 Gr=1,Ec=1,A*=1,B*=1,S =1.5,S =1.5,S=-0.1,=/4 0.4 Gr=1,Ec=1,A*=1,B*=1,S =1.5,S =1.5,S=-0.1,=/4 f t 0.4 Gr=1,Ec=1,A*=1,B*=1,Sf=1.5,St=1.5,S=-0.1,=/4 f t 0.3 0.3 0.3 () () () =0,/4,/2 =0,/4,/2 0.2 0.2 =0,/4,/2 0.2 0.1 0.1 0.1 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 Figure 23. Temperature profile for various values of . ( ) Figure 23. Temperature profile for various values of . ( ) Figure 23. Temperature proﬁle q(h) for various values of x. Figure 23. Temperature profile for various values of . ( ) 0.4 0.4 0.4 -14 -14 Pr=25,=1.610 , =1,=5,A=0.5,R=1,k =0.2,Gr=1,Ec=1, -14 Pr=25,=1.610 , =1,=5,A=0.5,R=1,k =0.2,Gr=1,Ec=1, 0.35 1 3 0.35 Pr=25,=1.610 , =1,=5,A=0.5,R=1,k =0.2,Gr=1,Ec=1, 1 3 0.35 1 3 =78.5,A*=0.5,B*=2,S=1,S =0.5,S =0.5,=/4,=/4 =78.5,A*=0.5,B*=2,S=1,S =0.5,S =0.5,=/4,=/4 f t 0.3 =78.5,A*=0.5,B*=2,S=1,S =0.5,S =0.5,=/4,=/4 f t 0.3 f t 0.3 0.25 0.25 0.25 f'() f'() M=1,2,3 M=1,2,3 f'() 0.2 0.2 M=1,2,3 0.2 0.15 0.15 0.15 0.1 0.1 0.1 0.05 0.05 0.05 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.2 Figure 24. Velocity proﬁle f (h) for various values of M. Figure 24. Velocity profile f ' ( ) for various values of . Figure 24. Velocity profile f ' ( ) for various values of . Figure 24. Velocity profile for various values of M . f ' ( ) AppliedMath 2021, 1, FOR PEER REVIEW 19 of 26 AppliedMath 2021, 1, FOR PEER REVIEW 19 of 26 AppliedMath 2021, 1, FOR PEER REVIEW 19 of 26 AppliedMath 2021, 1 54 0.8 0.8 -14 0.8 -14 Pr=21,=1.610 , =1,=1,A=0.5,=78.5,R=1,k =0.5,Gr=1, Pr=21,=1.610 -14, =1,=1,A=0.5,=78.5,R=1,k =0.5,Gr=1, 0.7 1 3 0.7 Pr=21,=1.610 , =1,=1,A=0.5,=78.5,R=1,k =0.5,Gr=1, 1 3 0.7 1 3 Ec=2,A*=1,B*=1,S =1,S =1,S=-0.1,=/2,=/4 Ec=2,A*=1,B*=1,S =1,S =1,S=-0.1,=/2,=/4 f t 0.6 Ec=2,A*=1,B*=1,Sf =1,St=1,S=-0.1,=/2,=/4 0.6 f t 0.6 0.5 0.5 0.5 () () 0.4 () 0.4 M=1,2,3 0.4 M=1,2,3 M=1,2,3 0.3 0.3 0.3 0.2 0.2 0.2 0.1 0.1 0.1 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 Figure 25. Temperature profile ( ) for various values of . Figure 25. Temperature profile for various values of M . ( ) Figure 25. Temperatur Figure e profil 25. e Temperatur for v earious proﬁle val q(hues ) forof various M . values of M. ( ) 0.2 0.2 0.2 -14 -14 Pr=25,=1.610 , =1,A=0.8,=78.5,M=1,R=1,k =0.3, Pr=25,=1.610 -14, =1,A=0.8,=78.5,M=1,R=1,k =0.3, 1 3 Pr=25,=1.610 , =1,A=0.8,=78.5,M=1,R=1,k =0.3, 1 3 1 3 Gr=3,Ec=2,A*=1.5,B*=1,S =1.5,S=1,S =1,=/2,=/4 0.15 Gr=3,Ec=2,A*=1.5,B*=1,S =1.5,S=1,S =1,=/2,=/4 f t 0.15 Gr=3,Ec=2,A*=1.5,B*=1,Sf =1.5,S=1,St=1,=/2,=/4 0.15 f t =1,4,7 =1,4,7 f'() 0.1 =1,4,7 f'() 0.1 f'() 0.1 0.05 0.05 0.05 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 Figure 26. Velocity profile for various values of . f ' ( ) Figure 26. Velocity profile f ' ( ) for various values of . Figure 26. Velocity proﬁle f (h) for various values of b. Figure 26. Velocity profile ( ) for various values of . f ' 0.7 0.7 -14 0.7 -14 Pr=21,=1.610 , =1,A=0.8,=78.5,M=1,R=2,k =0.3,Gr=3, -14 Pr=21,=1.610 , =1,A=0.8,=78.5,M=1,R=2,k =0.3,Gr=3, 1 3 0.6 Pr=21,=1.610 , =1,A=0.8,=78.5,M=1,R=2,k =0.3,Gr=3, 1 3 0.6 1 3 Ec=1,A*=1,B*=2,S =1.5,S =0.5,S=-0.1,=/4,=/4 0.6 Ec=1,A*=1,B*=2,S =1.5,S =0.5,S=-0.1,=/4,=/4 f t Ec=1,A*=1,B*=2,Sf =1.5,St=0.5,S=-0.1,=/4,=/4 0.5 f t 0.5 0.5 0.4 0.4 0.4 () () () =1,4,7 0.3 =1,4,7 0.3 =1,4,7 0.3 0.2 0.2 0.2 0.1 0.1 0.1 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8 Figure 27. Temperature proﬁle q(h) for various values of b. Figure 27. Temperature profile for various values of . ( ) Figure 27. Temperature profile for various values of . ( ) Figure 27. Temperature profile for various values of . ( ) Figures 28–39 depict the skin friction coefficient and the rate of wall heat transfer Figures 28–39 depict the skin friction coefficient and the rate of wall heat transfer Figures 28–39 depict the skin friction coefficient and the rate of wall heat transfer with regard to the magnetic parameter for various values of the inclination angle of the with regard to the magnetic parameter for various values of the inclination angle of the with regard to the magnetic parameter for various values of the inclination angle of the AppliedMath 2021, 1, FOR PEER REVIEW 20 of 26 AppliedMath 2021, 1, FOR PEER REVIEW 20 of 26 AppliedMath 2021, 1 55 sheet, angle of the magnetic field, radiation parameter, ferromagnetic parameter, unstead- sheet, angle of the magnetic field, radiation parameter, ferromagnetic parameter, unstead- iness parameter, and Eckert number. From the figures, it can be observed that skin friction Figures 28–39 depict the skin friction coefﬁcient and the rate of wall heat transfer with iness parameter, and Eckert number. From the figures, it can be observed that skin friction decreases with increasing values of the inclination angle of the sheet and the acute angle regard to the magnetic parameter for various values of the inclination angle of the sheet, decreases with increasing values of the inclination angle of the sheet and the acute angle angle of the magnetic ﬁeld, radiation parameter, ferromagnetic parameter, unsteadiness of magnetic field, whereas the rate of the wall heat transfer is increased in these cases. parameter, and Eckert number. From the ﬁgures, it can be observed that skin friction of magnetic field, whereas the rate of the wall heat transfer is increased in these cases. Moreover, both skin friction and the rate of wall heat transfer are decreased with increas- decreases with increasing values of the inclination angle of the sheet and the acute angle Moreover, both skin friction and the rate of wall heat transfer are decreased with increas- ing values of the radiation parameter, ferromagnetic parameter, and unsteadiness param- of magnetic ﬁeld, whereas the rate of the wall heat transfer is increased in these cases. ing values of the radiation parameter, ferromagnetic parameter, and unsteadiness param- Moreover, both skin friction and the rate of wall heat transfer are decreased with increasing eter. Moreover, both skin friction and the rate of wall heat transfer are increased with values of the radiation parameter, ferromagnetic parameter, and unsteadiness parameter. eter. Moreover, both skin friction and the rate of wall heat transfer are increased with increasing values of the Eckert number. Moreover, both skin friction and the rate of wall heat transfer are increased with increasing increasing values of the Eckert number. values of the Eckert number. -0.72 = -0.72 = = -0.73 = = = -0.73 = = -0.74 -0.74 -0.75 -0.75 -0.76 -0.76 -0.77 Pr=21,A=0.9,R=2,k =0.3,Gr=3, -0.77 Pr=21,A=0.9,R=2,k =0.3,Gr=3, Ec=1,A*=2,B*=4,S=1,S =1,S =1.5, f t -0.78 Ec=1,A*=2,B*=4,S=1,S =1,S =1.5, − f t -0.78 = ==== − = ==== -0.79 -0.791.0 1.5 2.0 2.5 3.0 1.0 1.5 2.0 2.5 3.0 Figure 28. Skin friction coefficient with M for different values of . f '' (0) Figure 28. Skin friction coefﬁcient f 0 with M for different values of a. ( ) Figure 28. Skin friction coefficient with M for different values of . f '' (0) -0.16 -0.16 -0.18 -0.18 -0.20 -0.20 -0.22 Pr=21,R=1,k =0.3,A=0.5,Gr=3,Ec=2 - -0.2 0.22 4 Pr=21,R=1,k =0.3,A=0.5,Gr=3,Ec=2 -0.24 -0.26 A*=2,B*=4,S=-0.1,S =1,S =0.5, f t -0.26 -0.28 A*=2,B*=4,S=-0.1,S =1,S =0.5, − f t - -0.2 0.38 0 = ==== − -0.30 -0.32 = ==== = -0.32 -0.34 = - -0.3 0.34 6 = = -0.36 = -0.38 = = -0.38 -0.40 = - -0.4 0.40 2 - -0.4 0.42 4 -0.44 -0.46 - -0.4 0.46 8 - -0.4 0.58 0 1.0 1.5 2.0 2.5 3.0 -0.50 1.0 1.5 2.0 2.5 3.0 Figure 29. Local Nusselt number q (0) with M for different values of a. Figure 29. Local Nusselt number with M for different values of . − ' (0) Figure 29. Local Nusselt number with for different values of . − ' (0) −− ()() f''(0) f''(0) AppliedMath 2021, 1, FOR PEER REVIEW 21 of 26 AppliedMath 2021, 1, FOR PEER REVIEW 21 of 26 AppliedMath 2021, 1 56 -0.4 -0.4 Pr=21,A=0.9,R=2,k =0.3,Gr=3, Pr=21,A=0.9,R=2,k =0.3,Gr=3, Ec=1,A*=2,B*=4,S=1,S 3 =1,S =1.5, f t Ec=1,A −* =2,B*=4,S=1,S =1,S =1.5, f t = ==== -0.6 − = ==== -0.6 -0.8 -0.8 = = = = = = = -1.0 = 1.0 1.5 2.0 2.5 3.0 -1.0 1.0 1.5 2.0 2.5 3.0 Figure 30. Skin friction coefficient with for different values of . f '' (0) Figure 30. Skin friction coefﬁcient f (0) with M for different values of x. Figure 30. Skin friction coefficient with for different values of . f '' (0) -0.2 Pr=21,R=1,k =0.3,A=0.5,Gr=3, -0.2 Ec=2,A*=2,B*=4,S=-0.1,S =1,S =0.5, Pr=21,R=1,k =0.3,A=0.5f,Gr=3, t − = ==== Ec=2,A*=2,B*=4,S=-0.1,S =1,S =0.5, f t − = ==== -0.4 -0.4 = = = = = = = -0.6 1.0 1.5 2.0 2.5 3.0 = -0.6 1.0 1.5 0 2.0 2.5 3.0 Figure 31. Local Nusselt number q (0) with M for different values of x . Figure 31. Local Nusselt number with for different values of . − ' (0) Figure 31. Local Nusselt number with M for different values of . ( ) − ' 0 -0.755 R=1 -0.760 -0.755 R=2 -0.765 R=1,2,3 R=3 R=1 -0.760 -0.770 R=2 -0.765 R=1,2,3 R=3 -0.775 -0.770 -0.780 -0.775 -0.785 -0.780 Pr=21,A=0.9,k =0.3,Gr=1,Ec=2, -0.790 3 A*=4,B*=2,S=1.5,S =1,S =1, -0.785 f t -0.795 − = ===== Pr=21,A=0.9,k =0.3,Gr=1,Ec=2, -0.790 3 -0.800 A*=4,B*=2,S=1.5,S =1,S =1, 1.0 1.5 2.0 2.5 3.0 f t -0.795 − = ===== -0.800 M R Figure 32. Skin friction coefficient with for different values of . f '' (0) 1.0 1.5 2.0 2.5 3.0 M R Figure 32. Skin friction coefficient with for different values of . f '' (0) f''(0) −() f''(0) f''(0) −() f''(0) AppliedMath 2021, 1, FOR PEER REVIEW 21 of 26 -0.4 Pr=21,A=0.9,R=2,k =0.3,Gr=3, Ec=1,A*=2,B*=4,S=1,S =1,S =1.5, f t − = ==== -0.6 -0.8 = = = = -1.0 1.0 1.5 2.0 2.5 3.0 Figure 30. Skin friction coefficient with for different values of . f '' (0) -0.2 Pr=21,R=1,k =0.3,A=0.5,Gr=3, Ec=2,A*=2,B*=4,S=-0.1,S =1,S =0.5, f t − = ==== -0.4 = = = = -0.6 1.0 1.5 2.0 2.5 3.0 AppliedMath 2021, 1 57 Figure 31. Local Nusselt number with M for different values of . ( ) − ' 0 -0.755 R=1 -0.760 R=2 -0.765 R=1,2,3 R=3 -0.770 -0.775 -0.780 -0.785 Pr=21,A=0.9,k =0.3,Gr=1,Ec=2, -0.790 3 A*=4,B*=2,S=1.5,S =1,S =1, f t -0.795 − = ===== AppliedMath 2021, 1, FOR PEER REVIEW 22 of 26 -0.800 1.0 1.5 2.0 2.5 3.0 Figure 32. Skin friction coefﬁcient f (0) with M for different values of R. M R Figure 32. Skin friction coefficient with for different values of . f '' (0) Pr=21,k =0.5,A=0.5,Gr=3,Ec=2,A*=1, -0.46 − B*=2,S=-0.1,S =1,S =0.5,= f t -0.48 ===== -0.50 -0.52 -0.54 R=1 R=2 R=1,2,3 -0.56 R=3 1.0 1.5 2.0 2.5 3.0 Figure 33. Local Nusselt number q (0) with M for different values of R. M R Figure 33. Local Nusselt number with for different values of . − ' (0) -0.755 -0.760 Pr=21,A=0.9,R=2,k =0.3,Gr=1, -0.765 Ec=2,A*=4,B*=2,S=1.5, − -0.770 = === -0.775 S =1,S =1, = f t -0.780 -0.785 -0.790 -0.795 -0.800 -0.805 -0.810 = = -0.815 = -0.820 = -0.825 -0.830 1.0 1.5 2.0 2.5 3.0 Figure 34. Skin friction coefficient with M for different values of . f '' (0) -1.576 Pr=21,R=2,A=0.5,k =0.3,Gr=3,Ec=1, -1.578 -1.580 A*=2,B*=4,S=1.5,S =1,S =0.5, f t -1.582 = − = == -1.584 = -1.586 == -1.588 = -1.590 -1.592 -1.594 -1.596 -1.598 -1.600 = -1.602 -1.604 -1.606 -1.608 -1.610 1.0 1.5 2.0 2.5 3.0 Figure 35. Local Nusselt number with for different values of . − ' (0) −() f''(0) −() f''(0) −() f''(0) AppliedMath 2021, 1, FOR PEER REVIEW 22 of 26 AppliedMath 2021, 1, FOR PEER REVIEW 22 of 26 Pr=21,k =0.5,A=0.5,Gr=3,Ec=2,A*=1, -0.46 − Pr=21,k =0.5,A=0.5,Gr=3,Ec=2,A*=1, -0.46 B*=2,S=-0.1,S =1,S =0.5,= f t − B*=2,S=-0.1,S =1,S =0.5,= -0.48 f t ===== -0.48 ===== -0.50 -0.50 -0.52 -0.52 -0.54 -0.54 R=1 R=1 R=2 R=1,2,3 R=2 R=1,2,3 -0.56 R=3 -0.56 R=3 1.0 1.5 2.0 2.5 3.0 1.0 1.5 2.0 2.5 3.0 AppliedMath 2021, 1 58 M M R R Figure Figure 33 33 . . Lo Lo ca ca l l Nus Nus sese lt lt nu nu mber mber with with for di for di fferent v fferent v aluesalues of of . . − −' (0') (0) -0.755 -0.755 Pr=21,A=0.9,R=2,k =0.3,Gr=1, -0.760 Pr=21,A=0.9,R=2,k =0.3,Gr=1, -0.760 3 -0.765 Ec=2,A*=4,B*=2,S=1.5, -0.765 Ec=2,A*=4,B*=2,S=1.5, − -0.770 − -0.770 = === = === -0.775 S =1,S =1, = -0.775 f t S =1,S =1, = -0.780 f t -0.780 -0.785 -0.785 -0.790 -0.790 -0.795 -0.795 -0.800 -0.800 -0.805 -0.805 -0.810 = = --0.815 0.810 = = = --0.820 0.815 = -0.825 = -0.820 -0.830 = -0.825 1.0 1.5 2.0 2.5 3.0 -0.830 1.0 1.5 2.0 2.5 3.0 Figure 34. Skin friction coefficient with M for different values of . f '' (0) Figure 34. Skin friction coefﬁcient f (0) with M for different values of b. Figure 34. Skin friction coefficient with for different values of . f '' (0) -1.576 Pr=21,R=2,A=0.5,k =0.3,Gr=3,Ec=1, -1.578 -1.576 -1.580 A*=2,B*=4,S=1.5,S =1,S =0.5, f t Pr=21,R=2,A=0.5,k =0.3,Gr=3,Ec=1, -1.578 -1.582 = − = == --1.5 1.5 84 80 A*=2,B*=4,S=1.5,S =1,S =0.5, = f t -1.586 -1.582 == = − -1.588 = = == -1.584 = -1.590 -1.586 == -1.592 -1.588 = -1.594 -1.590 -1.596 -1.592 -1.598 -1.594 -1.600 = -1.596 -1.602 -1.598 -1.604 -1.600 -1.606 = -1.602 -1.608 -1.604 -1.610 -1.606 1.0 1.5 2.0 2.5 3.0 -1.608 M -1.610 Figure 35. Local Nusselt number q (0) with M for different values of b. 1.0 1.5 2.0 2.5 3.0 Figure 35. Local Nusselt number with for different values of . − ' (0) Figure 35. Local Nusselt number with for different values of . − ' (0) −() −() −() f''(0) −() f''(0) AppliedMath 2021, 1, FOR PEER REVIEW 23 of 26 AppliedMath 2021, 1, FOR PEER REVIEW 23 of 26 AppliedMath 2021, 1 59 -0.750 A=0.0 -0.755 A=0.5 -0.750 A=0.0 A=0.9 A=0.0,0.5,0.9 -0.760 -0.755 A=0.5 -0.765 A=0.9 -0.760 A=0.0,0.5,0.9 -0.770 -0.765 -0.775 -0.770 -0.780 -0.775 -0.785 -0.780 Pr=21,R=2,k =0.3,Gr=1,Ec=2, -0.790 − -0.785 A*=4,B*=2,S=1.5,S =1,S =1,= Pr=21,R=2,k =0.3,Gr f =1,E t c=2, -0.795 -0.790 ===== − A*=4,B*=2,S=1.5,S =1,S =1,= -0.800 f t -0.795 1.0 1.5 2.0 2.5 3.0 ===== -0.800 1.0 1.5 2.0 2.5 3.0 M M A Figure 36. Skin friction coefficient with for different values of . f '' (0) Figure 36. Skin friction coefﬁcient f (0) with M for different values of A. M A Figure 36. Skin friction coefficient with for different values of . f '' (0) 0.0 -0.1 0.0 Pr=21,R=1,k =0.5,Gr=3,Ec=2, -0.2 -0.1 A*=1,B*=2,S=-0.1,S =1,S =0.5, f t Pr=21,R=1,k =0.5,Gr=3,Ec=2, -0.3 -0.2 3 A=0.0 − = === A*=1,B*=2,S=-0.1,S =1,S =0.5, A=0.5 f t -0.4 == -0.3 − A=0.0 A=0.9 = === A=0.5 -0.5 == -0.4 A=0.9 -0.6 -0.5 A=0.0,0.5,0.9 -0.7 -0.6 A=0.0,0.5,0.9 1.0 1.5 2.0 2.5 3.0 -0.7 1.0 1.5 2.0 2.5 3.0 Figure 37. Local Nusselt number q (0) with M for different values of A. M A Figure 37. Local Nusselt number with for different values of . − ' (0) M A Figure 37. Local Nusselt number with for different values of . − ' (0) -0.745 Ec=1 -0.750 Ec=2 -0.745 Ec=1 Ec=1,2,3 Ec=3 -0.755 -0.750 Ec=2 -0.760 Ec=1,2,3 Ec=3 -0.755 -0.765 -0.760 -0.770 -0.765 -0.775 -0.770 -0.780 Pr=21,Gr=3,k =0.3,A=0.9,R=2, -0.775 -0.785 A*=4,B*=2,S=1.5,S =1,S =1,= -0.780 f t Pr=21,Gr=3,k =0.3,A=0.9,R=2, − -0.790 = ==== A*=4,B*=2,S =1.5,S =1,S =1,= -0.785 f t − 1.0 1.5 2.0 2.5 3.0 = ==== -0.790 1.0 1.5 2.0 2.5 3.0 Figure 38. Skin friction coefficient M with for different values of Ec . f '' (0) Figure 38. Skin friction coefficient with for different values of Ec . f '' (0) −() f''(0) f''(0) −() f''(0) f''(0) AppliedMath 2021, 1, FOR PEER REVIEW 23 of 26 -0.750 A=0.0 -0.755 A=0.5 A=0.9 A=0.0,0.5,0.9 -0.760 -0.765 -0.770 -0.775 -0.780 -0.785 Pr=21,R=2,k =0.3,Gr=1,Ec=2, -0.790 − A*=4,B*=2,S=1.5,S =1,S =1,= f t -0.795 ===== -0.800 1.0 1.5 2.0 2.5 3.0 Figure 36. Skin friction coefficient with M for different values of A . f '' (0) 0.0 -0.1 Pr=21,R=1,k =0.5,Gr=3,Ec=2, -0.2 3 A*=1,B*=2,S=-0.1,S =1,S =0.5, f t -0.3 − A=0.0 = === A=0.5 -0.4 == A=0.9 -0.5 -0.6 A=0.0,0.5,0.9 -0.7 1.0 1.5 2.0 2.5 3.0 AppliedMath 2021, 1 60 Figure 37. Local Nusselt number with M for different values of A . − ' (0) -0.745 Ec=1 -0.750 Ec=2 Ec=1,2,3 Ec=3 -0.755 -0.760 -0.765 -0.770 -0.775 -0.780 Pr=21,Gr=3,k =0.3,A=0.9,R=2, A*=4,B*=2,S=1.5,S =1,S =1,= -0.785 f t − -0.790 = ==== 1.0 1.5 2.0 2.5 3.0 AppliedMath 2021, 1, FOR PEER REVIEW 24 of 26 Figure 38. Skin friction coefﬁcient f (0) with M for different values of Ec. Figure 38. Skin friction coefficient with for different values of Ec . f '' (0) -0.20 -0.25 -0.30 Ec=1 -0.35 Ec=2 -0.40 Ec=3 Ec=1,2,3 -0.45 -0.50 -0.55 -0.60 Pr=21,R=1,k =0.5,A=0.5,Gr=3,A*=1, -0.65 − B*=2,S=-0.1,S =1,S =0.5,= f t -0.70 ===== -0.75 -0.80 1.0 1.5 2.0 2.5 3.0 Figure 39. Local Nusselt number q 0 with M for different values of Ec. ( ) Figure 39. Local Nusselt number with M for different values of Ec . − ' (0) 6. Conclusions The results concern the study of the effect of the magnetic ﬁeld and radiation effects on 6. Conclusions the two-dimensional unsteady inclined stretching sheet with various slip conditions. We The results concern the study of the effect of the magnetic field and radiation effects observed that the suction/injection parameter has a good impact on the Nusselt number and skin friction coefﬁcient. Such types of biological ﬂuid ﬂow problems are interesting in on the two-dimensional unsteady inclined stretching sheet with various slip conditions. the biomedical and bioengineering sectors, especially in drug and gene delivery, cancer We observed that the suction/injection parameter has a good impact on the Nusselt num- treatment, and MRIs. The important ﬁndings are given below: ber and skin friction coefficient. Such types of biological fluid flow problems are interest- (1) Fluid velocity is reduced with ferromagnetic interaction parameter, radiation pa- ing in the biomedical and bioengineering sectors, especially in drug and gene delivery, rameter, magnetic ﬁeld parameter, inclination angle of the sheet, and acute angle of magnetic ﬁeld, whereas the temperature is increased in all cases; cancer treatment, and MRIs. The important findings are given below: (1) Fluid velocity is reduced with ferromagnetic interaction parameter, radiation param- eter, magnetic field parameter, inclination angle of the sheet, and acute angle of mag- netic field, whereas the temperature is increased in all cases; (2) Fluid velocity is enhanced with Prandtl number, Grashof number, permeability pa- rameter, non-dimensional thermal slip factor, whereas temperature is decreased in all cases; (3) Both fluid velocity and temperature are decreased with unsteadiness parameter, suc- tion/injection parameter, and non-dimensional velocity slip factor; (4) Both fluid velocity and temperature are increased with Eckert number, coefficient of space, and a temperature-dependent heat source/sink; (5) The effects of suction parameter on skin friction are enhanced with the increment of Eckert number, but decreased with an inclination angle of the sheet, acute angle of magnetic field, thermal slip factor, and unsteadiness parameter; (6) The effects of the injection parameter on the Nusselt number enhanced with the in- clination angle of the sheet, acute angle of magnetic field, and Eckert number, but decreased with an increment in the radiation parameter and unsteadiness parameter; (7) In case of suction parameter, both skin friction and Nusselt number are decreased with increasing values of the ferromagnetic interaction parameter. Author Contributions: Conceptualization, M.F.; Data curation, G.M.; Formal analysis, T.S.; Funding acquisition, G.M.; Investigation, A.G., J.A. and T.S.; Methodology, G.M.; Project administration, M.F.; Resources, A.G. and J.A.; Software, A.G., J.A. and G.M.; Supervision, E.E.T. and M.F.; Valida- tion, G.M., T.S. and M.F.; Writing—original draft, A.G., J.A., G.M. and T.S.; Writing—review & ed- iting, E.E.T. and M.F. All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Data Availability Statement: Not applicable. Conflicts of Interest: The authors declare no conflict of interest. −() −() f''(0) f''(0) AppliedMath 2021, 1 61 (2) Fluid velocity is enhanced with Prandtl number, Grashof number, permeability pa- rameter, non-dimensional thermal slip factor, whereas temperature is decreased in all cases; (3) Both ﬂuid velocity and temperature are decreased with unsteadiness parameter, suction/injection parameter, and non-dimensional velocity slip factor; (4) Both ﬂuid velocity and temperature are increased with Eckert number, coefﬁcient of space, and a temperature-dependent heat source/sink; (5) The effects of suction parameter on skin friction are enhanced with the increment of Eckert number, but decreased with an inclination angle of the sheet, acute angle of magnetic ﬁeld, thermal slip factor, and unsteadiness parameter; (6) The effects of the injection parameter on the Nusselt number enhanced with the inclination angle of the sheet, acute angle of magnetic ﬁeld, and Eckert number, but decreased with an increment in the radiation parameter and unsteadiness parameter; (7) In case of suction parameter, both skin friction and Nusselt number are decreased with increasing values of the ferromagnetic interaction parameter. Author Contributions: Conceptualization, M.F.; Data curation, G.M.; Formal analysis, T.S.; Funding acquisition, G.M.; Investigation, A.G., J.A. and T.S.; Methodology, G.M.; Project administration, M.F.; Resources, A.G. and J.A.; Software, A.G., J.A. and G.M.; Supervision, E.E.T. and M.F.; Validation, G.M., T.S. and M.F.; Writing—original draft, A.G., J.A., G.M. and T.S.; Writing—review & editing, E.E.T. and M.F. All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Data Availability Statement: Not applicable. Conﬂicts of Interest: The authors declare no conﬂict of interest. References 1. Tzirtzilakis, E.E. A mathematical model for blood ﬂow in magnetic ﬁeld. Phys. Fluids 2005, 17, 077103. [CrossRef] 2. Crane, L.J. Flow past a stretching plate. Z. Angew. Phys. 1970, 21, 645–647. [CrossRef] 3. Barozzi, G.S.; Dumas, A.J. Convective Heat Transfer Coefﬁcients in the Circulation. J. Biomech. Eng. 1991, 113, 308–310. 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AppliedMath – Multidisciplinary Digital Publishing Institute

**Published: ** Dec 14, 2021

**Keywords: **biomagnetic fluid; thermal radiation; heat flux; magnetic field; unsteady; stretching sheet; skin friction; rate of heat transfer

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