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The Effect of Prandtl Number and Magnetic Parameter on Forced Convection Unsteady Magnetohydrodynamic Boundary Layer Flow of a Viscous Fluid Past a Sphere

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The Effect of Prandtl Number and Magnetic Parameter on Forced Convection Unsteady Magnetohydrodynamic Boundary Layer Flow of a Viscous Fluid Past a Sphere International Journal of Advances in Science Engineering and Technology, ISSN: 2321-9009 Volume- 4, Issue-1, Jan.-2016 THE EFFECT OF PRANDTL NUMBER AND MAGNETIC PARAMETER ON FORCED CONVECTION UNSTEADY MAGNETOHYDRODYNAMIC BOUNDARY LAYER FLOW OF A VISCOUS FLUID PAST A SPHERE 1BASUKI WIDODO, 2DWI ARIYANI KHALIMAH, 3FIRDHA DWISHAFARINA ZAINAL, 4CHAIRUL IMRON 1,2,3,4Mathematics Department, Faculty of Mathematics and Natural Sciences, nstitutTeknologiSepuluhNopember Surabaya Indonesia E-mail: [email protected] Abstract- Magnetohydrodynamic (MHD) is a study the movement of fluid flow that can conduct electricity and are affected by magnetic fields. Many researchers have conducted research and developed on MHD flow until this day. This paper considers the unsteady MHD forced convective boundary layer flow and heat transfer of viscous fluid past a sphere. The heat transfer comes from the surface is proportional to the both of temperature on surrounding sphere surface and the velocity of the fluid. The governing equations are developed from continuity, momentum, and energy conservation. Those equations are transformed into boundary layer equations. The boundary equations further are transformed into non-dimensional form. The similarity equations is applied to solve the non-dimensional form easily. We further solve the equations numerically by using the Keller Box method. The numerical results to be obtained are used for analysize. We obtain that the variation of the magnetic variable increase, the temperature distributions decrease and the velocity distributions increase when the magnetic variableincreases. Keywords- boundary layer, forced convective, Magnetohydrodynamic, viscous fluid. 2010 Mathematics Subject Clasification : 76-XX Fluid Mechanics {} for General Continuum Mechanics. I. INTRODUCTION II. PROBLEM FORMULATION Magnetohydrodynamics (MHD) is a study the We consider unsteady MHD forced convection movement of fluid flow tha can conduct electricity and boundary layer flow past a solid sphere in a viscous are affected by magnetic fields. Many researchers has fluid. Fig. 1 illustrates the physical model and the conducted and developed research on MHD flow until coordinate system of the solid sphere. Consider a this day. One of them is forced convection flow laminar flow starts impulsively at rest in an magnetohydrodynamic. incompressible, electrically-conducting, viscous fluid Convection is heat transfer from one place which is past a non conducting solid sphere of radius a with caused by the movement of fluid. In generally, ambient velocity of fluid (1/2)U∞ and uniform convection is divided into three types: free convection temperature T∞ . It is assumed that uniform (natural), forced convection and mix convection. Free temperature is suddenly changed at the time t̅ = 0 to convection occurs when the flow of fluid influenced T where T > T∞ for heated sphere and T < T∞ for by differences in temperature or commonly called cooled sphere. The governing equations are developed power effect bouyancy, while forced convection is from mass, momentum, and energy conservation, as heat transfer fluid which is heavily influenced by follows [3]: external forces. Continuity equation: ∂(r̅u) ∂(r̅u) + Analytical solution of the convection equation is ∂x ∂y developed from nonlinear of Navier-Stokes and = 0 (1) energy equation. Boundary layer equation is an initial solution to calculate that problems. Boundary layer is Momentum equaiton: a thin layer on a solid surface where the fluid flow is ∂u ∂u ∂u influenced by viscosity and inertia force of the sphere. ρ + u + v ∂t̅ ∂x ∂y The governing equations are developed from ∂p ∂u ∂u continuity, momentum, and energy = − + µ + − σBu conservation.There are many researchers whose ∂x ∂x ∂y ∂v ∂v ∂v research about Newtonian and non-Newtonian Fluid. ρ + u + v Viscous fluid is type of Newtonian fluid. This paper ∂t̅ ∂x ∂y considers the unsteady MHD forced convection ∂p ∂v ∂v = − + µ + boundary layer flow and heat transfer of viscous fluid ∂y ∂x ∂y past a solid sphere. − σBv (2) The Effect Of Prandtl Number And Magnetic Parameter On Forced Convection Unsteady Magnetohydrodynamic Boundary Layer Flow Of A Viscous Fluid Past A Sphere 75 International Journal of Advances in Science Engineering and Technology, ISSN: 2321-9009 Volume- 4, Issue-1, Jan.-2016 Energy equaiton: where, , and are dimensionless parameter. ∂T ∂T ∂T Those parameters are defined as follows: ρC + u + v ∂t̅ ∂x ∂y = (Magnetic Parameter) ∞ ∂ T = c = (Convection Parameter) ∂x ( ) Gr = ∞ (Grashof Number) ∂ T + (3) ∂y Pr = (Prandtl Number) Using the boundary layer approximation where the Reynold number Re → ∞ which imply that → 0, we may discard related in the governing equations as below: Continuity equation: ∂(ru) ∂(rv) + ∂x ∂y = 0 (9) Momentum equation: + + Fig. 1. Physical model and coordinate system These dimensinless boundary layer equation are = + − ( − ) transformed into non-dimensional governing equation + sin x (10) by substituting non-dimensional variables. The Energy equation: non-dimensional variables as follows: ∂T ∂T ∂T + u + v x y U t̅ u ∂t ∂x ∂y x = , y = Re/ , t = ∞ , u = , 1 ∂T a a a U∞ = (11) Pr ∂y / u p T − T∞ v = Re , p = , T = , In order to connect the flow velocity u in x − U∞ ρU∞ T − T∞ direction, and the flow velocity v in y − direction, the r̅(x) r(x) = (4) stream function is introduced. The stream function is a introduced as follow: ∞ u = andv = − where Re = . By substituting equation (4) into Continuity equation: dimensional governing equations (1), (2), and (3), we obtain the non-dimensional equation i.e.: ∂ψ ∂x ∂y ∂(ru) ∂(rv) ∂ψ + = (12) ∂x ∂y ∂y ∂x = 0 (5) ∂u ∂u ∂u Momentum equation: + u + v ∂t ∂x ∂y + − − = ∂p 1 ∂ u ∂ u = − + + − Mu u + + αT sin x + ∂x Re ∂x ∂y + αT sin x (6) M u − (13)Energy 1 ∂v ∂v ∂v + u + v equation: Re ∂t ∂x ∂y ∂p 1 ∂v 1 ∂v α + − = = − + + − T cos x / ∂y Re ∂x Re ∂y Re (14)The similarity M − v (7) Re variables for small time case are introduced as bellow: ∂T ∂T ∂T 1 1 ∂ T ψ = tu (x)r(x)f(x, η, t), T = s(x, η, t), η = + u + v = ∂t ∂x ∂y Pr Re ∂x y/t/ (15)meanwhile the similarity variables for the ∂T large time are + (8) ( ) ( ) ( ) ( ) ∂y ψ = u x r x F x, Y, t , T = S x, Y, t , Y = y (16)By applying (15) into equation (12), (13), and (14), the following governing equations for small times (t < t∗) are obtained The Effect Of Prandtl Number And Magnetic Parameter On Forced Convection Unsteady Magnetohydrodynamic Boundary Layer Flow Of A Viscous Fluid Past A Sphere 76 International Journal of Advances in Science Engineering and Technology, ISSN: 2321-9009 Volume- 4, Issue-1, Jan.-2016 ∂f η ∂f du ∂f ∂f + + t 1 − + f ∂η 2 ∂η dx ∂η ∂η At the forward and rear stagnation points, the ∂f governing equations for the large times case are + Mt 1 − ∂F 3 ∂F ∂F ∂F ∂η + λ 1 − + F + M 1 − ∂f ∂Y 2 ∂Y ∂Y ∂Y = t ∂F ∂η ∂t = (23) ∂f ∂f ∂f ∂f ∂Y ∂t + tu − ∂η ∂η ∂x ∂η ∂η ∂S 3 ∂S 1 dr ∂f sin x + λ Pr F − f – tα s (17) ∂Y 2 ∂Y r dx ∂η u ∂S = Pr (24) ∂t ∂s Pr η ∂s du ∂s + + Pr t f ∂η 2 ∂η dx ∂η Subjected to the following conditions: ∂s F = = 0, S = 1atY = 0 = Pr t ∂t = 1, S = 0asY → ∞ ∂f ∂s ∂s ∂f + u − ∂η ∂η ∂η ∂x 1 dr ∂s III. NUMERICAL PROCEDURES − f (18) r dx ∂η This problem is then solved numerically by using Keller-Box method. The method has the following At the forward and rear stagnation points the four main steps: governing equations for small time case are reduced to (i) Reduce (19) and (20) to a first order equations (ii) Write the difference equations using central ∂f η ∂f 3 ∂f ∂f + + λt 1 − + f differences ∂η 2 ∂η 2 ∂η ∂η (iii) Linearize the resulting algebraic equation by ∂f ∂f Newton’s Method and write in matrix-vector form + Mt 1 − = t (19) ∂η ∂η ∂t (iv) Use the block tridiagonal elimination technique to solve the linear system [4]. ∂s Pr η ∂s 3 ∂s + + λ Pr t f IV. RESULT AND DISCUSSION ∂η 2 ∂η 2 ∂η ∂s = Pr t (20) The system of equation (19)-(20) and (23)-(24) are ∂t solved numerically for some values of the Prandtl numbers ( and magnetic parameter ( ) using Subjected to boundary conditions: Pr) M Keller-Box method. The variation on velocity and temperature profile at front stagnation point (x = 0°) t < 0 ∶ f = = s = 0for any x, η with various value of magnetic parameter are t ≥ ∶ f = = 0, s = 1atη = 0 illustrated in Fig. 2 and Fig. 3 respectively. These numerical results have been made at fixed values of = 1, s = 0asη → ∞ Pr = 0.7. The results show that velocity profiles in For the large time ( > ∗) case, the following Fig. 2 increase when the magnetic parameter increase, governing equations are there is no values of velocity in negative. This clarify ther is no reversal of flow occurred at least up ∂F du ∂F ∂F ∂F until t = 1.5 s . The temperature profiles in Fig. 3 decrease when magnetic parameter increase. + 1 − + F + M 1 − ∂Y dx ∂Y ∂Y ∂Y The influence of Prandtl number on the temperature ∂ F profiles can be seen in Fig. 4. These numerical results = ∂Y ∂t have been made at fixed values of M = 1.5. The ∂F ∂F ∂F ∂F results show that the temperature in Fig. 4 decrease + u − ∂Y ∂Y ∂x ∂x ∂Y when the Prandtl number increase. / 1 dr ∂ F The skin friction coefficients CRe is essential in − F (21) r dx ∂Y understanding what is happening between the surface of the sphere and viscous fluid and defined as u ∂ f / + Pr F = Pr + u − − CRe = ∂η t F (22) The results show that the skin friction in Fig.
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