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MHD Casson Fluid Flow Over a Stretching Sheet with Entropy Generation Analysis and Hall Influence

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MHD Casson Fluid Flow Over a Stretching Sheet with Entropy Generation Analysis and Hall Influence entropy Article MHD Casson Fluid Flow over a Stretching Sheet with Entropy Generation Analysis and Hall Influence Mohamed Abd El-Aziz 1 and Ahmed A. Afify 2,* 1 Department of Mathematics, Faculty of Science, King Khalid University, Abha 9004, Saudi Arabia; [email protected] 2 Department of Mathematics, Deanship of Educational Services, Qassim University, P.O. Box 6595, Buraidah 51452, Saudi Arabia * Correspondence: afi[email protected] Received: 3 May 2019; Accepted: 12 June 2019; Published: 14 June 2019 Abstract: The impacts of entropy generation and Hall current on MHD Casson fluid over a stretching surface with velocity slip factor have been numerically analyzed. Numerical work for the governing equations is established by using a shooting method with a fourth-order Runge–Kutta integration scheme. The outcomes show that the entropy generation is enhanced with a magnetic parameter, Reynolds number and group parameter. Further, the reverse behavior is observed with the Hall parameter, Eckert number, Casson parameter and slip factor. Also, it is viewed that Bejan number reduces with a group parameter. Keywords: MHD Casson fluid; slip factor; Hall current; entropy generation; Bejan number 1. Introduction The study of magnetohydrodynamic flows with Hall currents has evinced interest attributable to its numerous applications in industries, such as MHD power generators, Hall current accelerators, Hall current sensors, and planetary fluid dynamics. Sato [1] was the first author who investigated the impact of Hall current on the flow of ionized gas between two parallel plates. The influence of Hall current on the efficiency of an MHD generator was investigated by Sherman and Sutton [2]. Several authors [3–9] discussed the influence of Hall current on hydromagnetic flow problems for various aspects. Recently, Abdel-Wahed [10] examined the impacts of hall current on the MHD boundary layer flow and heat transfer of Ferro nanofluid in a curved tube. Some different fluids are termed as non-Newtonian fluids such as Jeffrey fluid, viscoelastic fluid, power-law flow, Williamson fluid, micropolar fluid, and Casson fluid. Casson [11] was the first investigator who introduced the Casson fluid model. Reviews of Casson fluid over different geometries have been presented in Refs. [12–17]. Recently, Ramana Reddy et al. [18] numerically analyzed the combined influences of thermal radiation and viscous dissipation of a paraboloid along an upper convective surface. All the above previous researchers have employed the first law of thermodynamics only. On the other hand, the second law of thermodynamics is utilized to minimize the entropy generation in thermal engineering systems. Entropy generation analysis in applied thermal engineering was proposed by Bejan [19–21]. Later on, many researchers [22–31] have effectively applied his approach to calculating the entropy generation analysis for different geometrical configurations. Recently, Reddy et al. [32] investigated the entropy generation for MHD Casson fluid flow with thermal radiation influence. Very recently, Afridi et al. [33] discussed the second law analysis for MHD flow and heat transfer past a slender stretching surface by taking Joule heating and variable thickness. To our knowledge, no document has yet been initiated for the MHD boundary layer of a Casson fluid owing to a stretching surface considering Hall effect, slip phenomenon and entropy generation. Entropy 2019, 21, 592; doi:10.3390/e21060592 www.mdpi.com/journal/entropy Entropy 2019, 21, x 2 of 22 Entropy 2019, 21, 592 2 of 20 phenomenon and entropy generation. The objective of the present document is to discuss the second law of thermodynamics for a Casson fluid flow along a stretching surface taking the Hall The objective of the present document is to discuss the second law of thermodynamics for a Casson current, velocity slip, and viscous dissipation influences. The impact of physical parameters is fluid flow along a stretching surface taking the Hall current, velocity slip, and viscous dissipation analyzed with the help of graphs and tables. influences. The impact of physical parameters is analyzed with the help of graphs and tables. 2.2. Mathematical Mathematical Formulation Formulation InIn this this paper paper,, the the magnetohydrodynamic magnetohydrodynamic flow flow of of incompressible incompressible Casson Casson fluid fluid is is considered. considered. The flow is generated owing to the stretching surface with linear velocity u() x= cx , in the x- The flow is generated owing to the stretching surface with linear velocity uw(x) =wcx, in the x-direction. direction.Hall current Hall is producedcurrent is dueproduced to the strongdue to magnetic the strong field magnetic which is verticalfield which to the is stretching vertical surfaceto the stretchingin the y-direction, surface in as the shown y-direction, in Figure as1. shown The induced in Fig magneticure 1. The field induced is ignored magnetic with field respect is ignored to small withmagnetic respect Reynolds to small number. magnetic The Reynolds heat transfer number. characteristic The heat is transfer examined characteristic via viscous dissipation.is examined Bejan via viscousnumber dissipation. and entropy Bejan generation number are and utilized entropy to evaluategeneration the are loss utilized of energy to evaluate for the existing the los flows of energy regime. forFurther, the existing it is assumed flow regime. that the Further, Joule heatingit is assumed is neglected that the in Joule this study.heating The is neglected generalized in this Ohm’s study. law, Theincluding generalized Hall current,Ohm’s law, is stated including in the Hall form current, Sutton andis stated Sherman in the [2 form]: Sutton and Sherman [2]: ee !JJBEVB+!eτe (! !) = ( ! +! )! (1) J + B0 J B = σ E + V B (1) B0 × × en2 e2n τ ee where !JJJJ= ,, !, V!, !E , BB= (0,0 ,0) , , , e=e , e , n and m are the current where J =( Jxx, Jy , Jz z ), V, E, B = (0, B0, 0), τe, !e, σ e= m, e, ne and me aree the currente density vector, e me the velocity vector, the electric field vector, the magnetic induction vector, the electron collision time, density vector, the velocity vector, the electric field vector, the magnetic induction vector, the the cyclotron frequency of electron, the electrical conductivity of the fluid, the charge of electron, the electron collision time, the cyclotron frequency of electron, the electrical conductivity of the fluid, number density of electrons, and the mass of the electron, respectively. In this work, an electric field is the charge of electron, the number density of electrons, and the mass of the electron, respectively. In neglected, thus Equation (1) becomes: this work, an electric field is neglected, thus Equation (1) becomes: σBB0 J = 0 (mu w) (2) Jxx =−2( mu w) (2) 11++mm2 − σB0 Jz = 0 (mw + u) (3) Jz =++ 2( mw u) (3) 1+ mm2 where, m = !eτe is the Hall parameter. where, m = eeis the Hall parameter. w y u z Uw x Force B0 Stretching surface Slot Figure 1. Physical model and coordinate system. Figure 1. Physical model and coordinate system. AccordingAccording to to Refs Refs [12] [12,–13[13],] the, the rheological rheological equation equation of of the the Casson Casson fluid fluid is is given given by: by: 8 py > 2 µB + eij, π > πc <> p2π τij = (4) > py :> 2 µB + eij, π < πc p2πc 2 Entropy 2019, 21, 592 3 of 20 th where π = eijeij with eij being the (i, j) component of the deformation rate, π depicts the product of the component of the deformation rate with itself, πc denotes a critical value of this product based on the non-Newtonian model, µB indicates the plastic dynamic viscosity of non-Newtonian fluids, and py 1 is the yield stress of the fluid. When π < πc, Equation (4) can be expressed as: τij = µB 1 + γ 2eij . µ p2π Here γ = B c is the Casson parameter. py Due to the above-mentioned assumptions and the boundary layer approximations, the governing equations of Casson fluid and generalized Ohm’s law with Hall current influence are given by: @u @v + = 0 (5) @x @y ! 2 @u @u 1 @2u σB u + v = υ 1 + 0 (u + mw) (6) @x @y γ @y2 − ρ(1 + m2) ! 2 @w @w 1 @2w σB u + v = υ 1 + + 0 (mu w) (7) @x @y γ @y2 ρ(1 + m2) − 8 9 2 !> !2 !2> @T @T @ T υ 1 <> @u @w => u + v = α + 1 + > + > (8) @x @y @y2 cp γ :> @y @y ;> Subject to the boundary conditions: 1 @u 1 @w 2 u = uw + 1 + L , v = 0, w = 1 + L , T = Tw(x) = T + bx at y= 0, γ @y γ @y 1 (9) u = w = 0, T = T as y 1 ! 1 The following non-dimensional variables are defined as: 1/2 c 1/2 T T η = y, u = c x f 0(η), v = (c υ) f (η), w = c x h(η), θ(η) = − 1 (10) υ − Tw T − 1 where υ = µB/ρ, α = k/ρcp, k, ρ, cp, Tw, T , b, c, L, f, and h, are the kinematic viscosity,thermal diffusivity, 1 thermal conductivity, fluid density, specific heat, temperature at the surface, ambient temperature, positive constants, characteristic length, dimensionless stream function, and dimensionless transverse velocity, respectively. By invoking Equation (10), Equation (5) is automatically satisfied whereas the other equations and the boundary condition take the following form: ! 1 2 M 1 + f 000 + f f 00 f 0 ( f 0 + mh) = 0 (11) γ − − 1 + m2 ! 1 M 1 + h00 + f h0 f 0h + (m f 0 h) = 0 (12) γ − 1 + m2 − ! 1 1 2 2 θ00 + f θ0 2 f 0θ + Ec 1 + f 00 + h0 = 0 (13) Pr − γ 1 1 f (0) = 0, f (0) = 1 + χ 1 + f 00 (0), h(0) = χ 1 + h (0), θ(0)= 1, 0 γ γ 0 (14) f ( ) = 0, h( ) = 0, θ( ) = 0 0 1 1 1 Here, prime denotes differentiation with respect to η, f is a dimensionless stream function, h is υ the dimensionless transverse velocity, θ is the dimensionless temperature, Pr = α is Prandtl number, σB2 1/2 2 M = 0 is a magnetic parameter, χ = L c is the slip parameter, Ec = c is the Eckert number, and ρc υ bcp m is the Hall parameter.
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