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Available Online Through ISSN 2229 – 5046 International Journal of Mathematical Archive-6(2), 2015, 73-78 Available online through www.ijma.info ISSN 2229 – 5046 MHD BOUNDARY LAYER FLOW AND HEAT TRANSFER OF AN UPPER-CONVECTED MAXWELL FLUID Dr. Anuj Kumar Jhankal1 and Manoj Kumar*2 1Army cadet College wing, Indian Military Academy, Dehradun, India. 2Arya Institute of Engineering Technology & Mgt., Omaxe City, Jaipur-302026, India. (Received On: 09-02-15; Revised & Accepted On: 28-02-15) ABSTRACT An analysis is made to study boundary layer flow and heat transfer of an upper-convected Maxwell fluid in presence of transverse magnetic field. The upper-convected Maxwell model is used to characterize the non-Newtonian fluid behavior. Using similarity transformation, the governing boundary layer equations are transformed into self-similar nonlinear ordinary differential equations, which are then solved numerically using a very efficient RKF45 (Runge- Kutta-Fehlberg forth-fifth) method. The effects of various parameters like magnetic parameter, Deborah number and Prandtl number on the velocity and temperature profiles as well as on the local Skin-friction coefficient and the local Nusselt number are presented and discussed through graphs and tables. Mathematics Subject Classification [MSC]: 76A05, 76D10, 76M20, 76W05, 80A20. Keywords: MHD, boundary layer flow, heat transfer, upper-convected Maxwell fluid, numerical study. NOMENCLATURE B0 Constant applied magnetic field Cp Specific heat at constant pressure f Dimensionless stream function M Magnetic field parameter Pr Prandtl number T Temperature of the fluid U Free stream velocity u, v Velocity component of the fluid along the x and y directions, respectively x, y Cartesian coordinates along the surface and normal to it, respectively Greek symbols β Deborah number η Dimensionless similarity variable ρ Density of the fluid μ Viscosity of the fluid ν Kinematic viscosity Ψ Stream function λ Retardation time σe Electrical conductivity Thermal conductivity Dimensionless temperature κ Superscriptθ ′ Derivative with respect to η Subscripts w Properties at the plane ∞ Free stream condition Corresponding Author: Manoj Kumar*2 1Army cadet College wing, Indian Military Academy, Dehradun, India. International Journal of Mathematical Archive- 6(2), Feb. – 2015 73 Anuj Kumar Jhankal1 and Manoj Kumar*2/ MHD Boundary Layer Flow and Heat Transfer of an Upper-Convected Maxwell Fluid / IJMA- 6(2), Feb.-2015. INTRODUCTION The subject of MHD is largely perceived to have been initiated by Swedish electrical engineer Hannes Alfv n [1] in 1942. If an electrically conducting fluid is placed in a constant magnetic field, the motion of the fluid induces currents which create forces on the fluid. The production of these currents has led to the design of among other deviceś the MHD generators for electricity production. The equations which describe MHD flow are a combination of continuity equation and Navier-Stokes equations of fluid dynamics and Maxwell’s equations of electromagnetism. The governing equations are differential equations that have to be solved either analytically or numerically. In recent years, the studies of boundary layer flows of non-Newtonian fluid have received considerable attention due to their numerous industrial and engineering applications. The non-Newtonian fluids in view of their diverse rheological properties cannot be examined through one constitutive relationship between shear stress and rate of strain. Many models of non-Newtonian fluids exist. Maxwell model has become the most popular as it can predict stress relaxation and also excludes the complicating effects of shear dependent viscosity Mukhopadhyay and Gorla [7] (2012). In view of its simplicity, this fluid model has acquired special status amongst model has acquired special status amongst the recent workers in the field. This kind of fluid has significant application in viscoelastic problems which have small dimensionless relaxation time. Sarpakaya [12] (1961) was the first researcher to study the MHD flow a of non- Newtonian fluid. Prandtl’s boundary layer theory proved to be of great use in Newtonian fluids as Navier-Stokes equations can be converted into much simplified boundary layer equation which is easier to handle. Wang and Tan [14] (2011) discussed the flow of Maxwell fluid in a porous medium. Zierep and Fetecau [16] (2007) studied Rayleigh- Stokes problem using Maxwell fluid and find exact solution. Sadeghy et al.[10] (2006) did a comparative study for Sakiadis flow of an upper-convected Maxwell fluid on a rigid plate. In view of the Maxwell model, several researchers have analyzed the MHD flow of a Maxwell fluid under various aspects of rotation, thermal radiation, heat and mass transfer, chemical reaction, suction/injection, thermophoresis and heat source/sink (Bataller [4] (2011); Zheng et al.[15] (2013); Noor [8] 2012; Vajravelu [13] (2012); Eosboee et al.[5] (2010)). In this study we aim at investigating boundary layer flow and heat transfer of an upper-convected Maxwell fluid in presence of transverse magnetic field. The governing partial differential equation is transformed into ordinary differential equation by means of similarity transformations. This equation is solved numerically by RKF45 (Runge- Kutta-Fehlberg forth-fifth) method using symbolic algebra software Maple. The effects of various parameters are presented with the aid of graphs & tables and discussed. FORMULATION OF THE PROBLEM Let us consider the steady and incompressible MHD boundary layer flow an electrically conducting fluid obeying UCM model. The x- and y-axes are taken along and perpendicular to the flow, respectively. A uniform magnetic field of strength B0 is imposed along the y-axis. It is assumed that external field is zero, the electric field owing to polarization of charges and Hall effect are neglected. Under the usual boundary layer approximations, the governing equation of continuity, momentum and energy [Pai [9] (1956), Schlichting [11] (1964), Bansal [2] (1977)] under the influence of externally imposed transverse magnetic field [Jeffery [7] (1966), Bansal [3] (1994)] are: + = 0 (1) 2 2 2 2 2 2 2 0 + + 2 + 2 + 2 = 2 (2) 2 + �= 2 � − (3) � � Along with the boundary conditions for the problem are given by ( , 0) = , ( , 0) = 0, ( , 0) = ( ), ( , ∞) 0, ( , ∞) (4) ∞ → → Ψ Ψ The continuity equation (1) is satisfied by introducing a stream function Ψ such that u = and v = y x ∂ ∂ ∂ − ∂ The momentum equation can be transformed into the corresponding ordinary nonlinear differential equation by the following transformation: = , = ( ) , � √ ∞ ( ) = (5) − ∞ − © 2015, IJMA. All Rights Reserved 74 Anuj Kumar Jhankal1 and Manoj Kumar*2/ MHD Boundary Layer Flow and Heat Transfer of an Upper-Convected Maxwell Fluid / IJMA- 6(2), Feb.-2015. Where η is the independent similarity variable. The transformed nonlinear ordinary equation is 2 ′′′ 2 ′ ′′ + 2 ′′′ + ′2 ′′ + ′′ 2 ′ = 0 (6) (2 ) = 0 (7) ′′ − � ′ ′ � − The − transformed − boundary conditions are: = 0: = 0, = 1, = 1 : = 0,′ = 0 (8) 2 Where prime′ denotes differentiation with respect to , = 0 is the magnetic parameter, = is the Deborah 2 → ∞ number and = is the Prandtl number. RESULTS AND DISCUSSIONS The system of governing equations (6)-(7) together with the boundary condition (8) is non-linear ordinary differential equations depending on the various values of magnetic parameter M, Deborah number β and Prandtl number Pr. The non-linear differential equations (6) and (7) subject to the boundary conditions (8) are solved numerically using Runge- Kutta-Fehlberg Forth-Fifth order method. To solve this equation we adopted symbolic algebra software Maple. Maple uses the well known Runge-Kutta-Feulberg Forth-Fifth (RKF45) order method to generate the numerical solution of boundary value problem. In order to verify the accuracy of our present method, we have compared our results with HPM and BVP Eosboee et al. [5]. Table 1 compares the values of velocity component ( ) for M=1.0 and β=0.4. The comparisons are found to be in excellent agreement. Table-1: Numerical values of f( ) compared with the result obtained by Eosboee et al. [5], when M=1.0 and β=0.4. Eosboee et al. [5] ( ) η Present paper results ( ) η HPM BVP 0 0 0 0 0.5 0.38662 0.386325 0.386128 1.0 0.60613 0.605299 0.604469 1.5 0.72485 0.724257 0.722237 2.0 0.78562 0.786745 0.782737 2.5 0.81461 0.818662 0.811468 3.0 0.82698 0.834332 0.822121 3.5 0.83119 0.841253 0.832975 4.0 0.83192 0.843124 0.843124 Table 2 represents the Skin friction parameter against magnetic parameter M for various values of Deborah number β. It is noted that the effect of increasing M is to decrease the skin friction at the surface. The positive values of f ′′(0) imply that fluid exerts the drag force on the surface, and the negative values f ′′ (0) imply the opposite meaning. Thus as M becomes very large, there will be a decrease in skin friction f ′′ (0), and then a decrease in the drag force at the surface. Results also shows that an increase in the Deborah number β leads to increases the skin friction f ′′(0) and the drag force at the surface. Table-2: Numerical values of Skin friction coefficient f''(0) M β=0 β=0.1 β=0.2 β=0.3 β=0.4 β=0.5 β=0.6 β=0.7 0 -0.49664 -0.48124 -0.46522 -0.44854 -0.43117 -0.41316 -0.39465 -0.37598 0.5 -0.83234 -0.82348 -0.81449 -0.80536 -0.79608 -0.78665 -0.77706 -0.76731 1 -1.08534 -1.07869 -1.07198 -1.06521 -1.05836 -1.05144 -1.04445 -1.03738 1.5 -1.29338 -1.28783 -1.28223 -1.27660 -1.27092 -1.26520 -1.2594 -1.25362 2 -1.4733 -1.46844 -1.46354 -1.45860 -1.45364 -1.44864 -1.44361 -1.43855 Table 3 represents the Nusselt number against magnetic parameter M for various values of Deborah number β.
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