Finite difference solution of the mixed convection flo of MHD micropolar fluid past a moving surface ith radiation effect LOKENDRA KUMAR, G. SWAPNA, BANI SINGH Department of Mathematics Jaypee Institute of Information Technology A-0, Sector-6, Noida 0307, Uttar Pradesh INDIA lokendma@gmail.com, gsapna_m@rediffmail.com, bani.singh@jiit.ac.in Abstract: - In this paper, e have studied the effect of radiation, magnetic field on the mixed convection flo of a magneto-micropolar fluid past a continuously moving surface ith suction and bloing. The governing equations have been reduced to a system of nonlinear ordinary differential equations by similarity transformations, hich have been solved numerically using finite difference method. The effect of important parameters namely, radiation parameter, magnetic parameter and Grashof number on the velocity, microrotation and temperature functions across the boundary layer are shon graphically. Numerical results are also obtained for the skin friction and for the rate of heat transfer, hich are given in tables. It is found that skin-friction coefficient increases hile the rate of heat transfer decreases as the magnetic parameter increases. The heat transfer coefficient increases ith an increase in the radiation parameter. Key-Words: - Boundary layer, Finite difference method, Mixed Convection, Radiation, MHD Introduction There are a large number of fluids existing in nature that contain suspension of small particles. The properties of such type of fluids are different from those of Netonian fluid due to fluid particle interaction and rotation. Eringen [] introduced the theory of microfluids to describe the mechanics of such complex fluids. This theory deals ith a class of fluids, hich exhibit certain microscopic effects arising from the local structure and micromotions of the fluid elements. Later, Eringen [] formulated the theory of micropolar fluids, endoed ith microinertia. The theory satisfactorily describes the characteristics of liquid crystals, polymeric fluids, animal blood, colloidal suspensions, lubricants etc. Eringen [3] extended the theory of simple microfluids to include the heat conduction and heat dissipation effects. A thorough and an excellent revie of the subject along ith applications have been presented by Ariman et al. [4-5]. The boundary layer theory for a micropolar fluid as studied by Peddieson and McNitt [6]. They derived the boundary layer equations for a micropolar fluid and applied these to the problems of steady stagnation point flo, steady flo past a semi-infinite flat plate and impulsively started flo past an infinite plate. Ebert [7] obtained a similarity solution for the boundary layer flo of a polar fluid. The flo and heat transfer from a continuous surface in a parallel free stream of micropolar fluid is studied by Gorla et al. [8]. The effect of radiation on heat transfer over a stretching surface is important in the context of space technology and processes involving high temperature. Raptis [9] studied the flo of a micropolar fluid past a continuously moving plate by the presence of radiation. Seddeek [0] investigated the flo of a magnetomicropolar fluid past a continuously moving plate. Such a fluid can be opted as a cooling liquid as its flo can be regulated by external magnetic field, hich regulates heat transfer to some extent. The thermal boundary layer flo over a stretching sheet in a micropolar fluid ith radiation is investigated by Ishak []. It is knon that constant physical properties of the fluid may change ith temperature and fluid viscosity. Elbarbary et al. [] studied the effect of variable viscosity on magneto-micropolar fluid flo in the presence of radiation. The mixed convection flo of a micropolar fluid over a stretching sheet is studied by Takhar et al. [3]. Bhargava et al. [4] studied the mixed convection from a continuous surface in a parallel moving stream of a micropolar fluid. Mahmoud et al. [5] analyzed the effects of slip and heat generation/absorption on MHD mixed convection flo of a micropolar fluid over a heated stretching ISBN: 978-960-474-98- 4
surface. Recently, Seddeek et al. [6] investigated the analytical solution for the effect of radiation on flo of a magneto-micropolar fluid past a continuously moving plate ith suction and bloing. Hence, the purpose of the present paper is to study the effect of important parameters on the mixed convection flo of a magneto-micropolar fluid past a continuously moving surface ith suction and bloing. Certain transformations are employed to transform the governing partial differential equations to dimensionless nonlinear ordinary differential equations. The transformed equations are numerically solved by using finite difference method. Numerical results are shon in tabular form for the skin friction and Nusselt number. The effect of radiation parameter, magnetic parameter and Grashof number have been studied on velocity, microrotation and temperature functions, hich are shon graphically. Mathematical formulation Consider a steady to-dimensional flo of a laminar, viscous, incompressible micropolar and electrically conducting fluid floing past a continuously moving sheet ith uniform velocity and maintained at temperature T. The flo is assumed to be in the x-direction, hich is taken along the plate in the upard direction and y-axis is normal to. A uniform strong magnetic field B 0 is assumed to be applied in the y-direction. Equation of continuity: u v + = 0 () x Momentum Equation: u u u N σb0 u + v = + k u + gβ( T T ) x ρ () Angular Momentum Equation: N u G N = 0 (3) Energy equation: T T k f T q r u u x y c p y c p y c p y + = + (4) ρ ρ Here u, v are the velocity components in the x and y direction respectively, ρ is the density of the fluid, N is the microrotation component hose direction of rotation is in the xy plane, is the kinematic viscosity, G is the microrotation constant, T is the temperature of the fluid in the boundary layer, T is the temperature of the fluid far aay from the plate, c p is the specific heat at constant pressure, k f is the thermal conductivity, k =s/ρ is the coupling constant, s is a constant characteristic of the fluid, and σ is the electrical conductivity. The radiative heat flux is given as 4 4σ T q r = (5) 3k here σ * is the Stefan-Boltzman constant, k* is the Rosseland mean absorption coefficient. Assuming that the temperature differences ithin the flo are sufficiently small such that T 4 may be expressed as a linear function of temperature 4 3 4 T 4T T 3T (6) here the higher-order terms of the expansion are neglected. The boundary conditions for the present problem are u y = 0: u = Bx,v = v,n =,T = T y :u 0, N 0, T T (7) The governing equations [-4] subject to the boundary conditions [7] can be expressed in a simpler form by introducing the transformations B η = y, ψ = Bxf ( η), 3 B T T N = xg( η), θ = (8) T T here η bis the similarity variable and ψ is the ψ ψ stream function defined as u = and v = x satisfying equation () to obtain the ordinary η g η differential equations for the functions f ( ), ( ) and θ ( η). ( f ) Mf + Kg + Grθ = 0 ( g + f ) = 0 θ + fθ + Ec( f ) = 0 f + ff (9) Gg (0) + R () R Pr In the above equations prime denotes differentiation σb0 ith respect to η, M = is the magnetic field ρb k parameter, K = is the coupling constant gβ( T T ) parameter, Gr = is the Grashof xb GB number, G = is the microrotation parameter, ISBN: 978-960-474-98- 4
3k f k R = is the radiation parameter, 6σ T 3 the Prandtl number and Ec = c p u ( T T ) μ Pr = is k f is the Eckert number. The transformed boundary conditions are given by f = f, f =, = f, θ = () f ( ) = 0, g( ) = 0, θ( ) = 0 (3) The quantities of physical interest are the values of f and θ hich represent the skin friction coefficient and the heat transfer rate at the surface respectively. Thus, our task is to investigate ho the governing parameters, namely, M, R and Gr influence these quantities. 3 Method of solution The differential equations (9), (0), and () subject to the boundary conditions () and (3), are solved numerically using the finite difference method. The finite difference method is an efficient numerical method for solving differential equations. For the present problem, Suppose f = z (4) Therefore equations (9), (0) and () can be ritten as z + fz () z Mz + Kg + Grθ = 0 (5) Gg ( g + z ) = 0 (6) + R θ + fθ + Ec( z ) = 0 (7) R Pr ith corresponding boundary conditions f = f, z =, g = z, θ = z ( ) = 0, g( ) = 0, θ( ) = 0 (8) Using central difference formulae dy yi+ yi = dη h i d y y i+ y i + y i = dη (9) h i the differential equations are transformed to difference equations for each i. As the equations are nonlinear, they cannot be solved directly. Therefore an iterative scheme is used. Re-riting the equations in the form i ( f, f,... f ) x = F (0) n here each f i is a function of the variable fi, z i, gi, θ i. Initial guess values are assigned and starting ith these guess values, ne iterate values are calculated. The process is repeated until the correct value of the functions is determined. Equations (5) - (7) are thus transformed to 3f i + 4f i+ f i+ zi = h () z i+ z i + z i z i+ z i + f i h h g i+ g i ( z i ) Mz i + K + Grθ i = 0 h () g i+ g i + g i z i+ z i G g i + = 0 h h (3) + R θi+ θ i + θi θ i+ θi + f i + R.Pr h h z i+ z i Ec = 0 h (4) here h is the step length. No, riting Eqs. () - (4) in the form of Eq. (0) f i = ( 4f i+ f i+ hzi ) (5) 3 ( z i+ + z i ) + f ih( z i+ z i ) + z i = ( ) + (6) 4 + h yi + Mh K g i+ g i h Grθ i G( g i+ + g i ) ( ) g i = (7) 4G + 4h h z i+ z i ( + R) ( θ + θ ) 4 i+ i + R.Pr R.Pr θ ( ) i = hf i θ i+ θi + (8) 8( + R) Ec( z i+ z i ) The boundary conditions are ritten as f = f, z =, g = z, θ = (9) z ( ) = 0, g( ) = 0, θ( ) = 0 (30) The all shear stress is given by τ = μ u y y= 0 τ = μ u u f x (3) Hence, the skin friction coefficient is found using τ c ( Re) f = = f ρu (3) c f ISBN: 978-960-474-98- 43
u x here Re = is the Reynolds number. The heat flux at the all is given by q q T u = k = k( T T ) θ y (33) x y= 0 hich is used to compute the Nusselt number xq Nu = = ( Re) θ () 0 k T T ( ) f and θ ( 0 ) The results of, Table and Table 3. (34) are given in Table Fig.. Velocity distribution for different Gr. 4 Results and Discussion f ( η) The behavior of the velocity, angular velocity g ( η) and temperature profiles θ( η) for different values of the parameter that describe the flo are displayed through graphs. Fig. display results for the velocity distribution for various values of Gr. It is seen that f ( η) increases ith increasing Gr, thereby, increasing the boundary layer. Fig. shos the angular velocity distribution. Angular velocity g ( η) decreases ith increasing Gr, thus creating a reverse rotation of micro-constituents for large values of Gr. Fig. 3 shos the temperature distribution. The temperature distribution θ ( η) decreases continuously ith increasing Gr. It is evident that ithin the boundary layer the cooling of fluid takes place ith an increase in the parameter Gr. Fig. 4, 5 and 6 represents the effect of radiation parameter R on velocity f ( η) g ( η) θ( η), angular velocity and temperature distribution, respectively. It is seen that velocity and temperature distribution decreases ith increasing R and angular velocity distribution increases ith an increase in R. A decrease in the radiation parameter brings in a decrease in the Rosseland radiation absorptivity, thereby increasing the radiative heat flux. Thus, the rate of radiative heat transferred to the fluid increases and consequently the fluid temperature and the velocity of the particles increases. It is also seen that a decrease in temperature ith increasing radiation parameter results in decreasing the thermal boundary layer thickness. Fig.. Angular distribution for different Gr. Fig. 3. Temperature distribution for different Gr. Fig. 4. Velocity distribution for different R. ISBN: 978-960-474-98- 44
Proceedings of the 4th WSEAS International Conference on Finite Differences - Finite Elements - Finite Volumes - Boundary Elements Fig. 5. Angular distribution for different R. Fig. 9. Temperature distribution for different M. Fig. 7, 8 and 9 depict the effect of Magnetic parameter M on velocity, angular velocity and temperature distribution. Fig. 7 depicts the variation of velocity ith M. It is observed that velocity decreases ith the increase in M along the surface. It can also be seen from the figure that the momentum boundary layer thickness decreases as M increases, and hence induces an increase in the absolute value of the velocity gradient at the surface. Fig. 8 shos the profile of the angular velocity ith the variation of M. It is clear from the figure that angular velocity increases ith an increase in M near the surface and the reverse is true, aay from the surface. Fig. 9 shos the resulting temperature profile for various values of M. It is noted that an increase of M leads to an increase of temperature. Thus, the heat transfer rate at the surface decreases ith increasing M. The local skin-friction coefficient in terms of f (0) Fig. 6. Temperature distribution for different R. and the local Nusselt number in terms of θ (0 ) for different values of Gr, M and R are tabulated in Table, Table and Table 3. It is obvious from the table that local skin-friction coefficient increases ith an increase in M and R, but decreases ith an increase in Gr. The Nusselt number increases ith the increase of the Gr and R, but decreases ith and increase in M. Fig. 7. Velocity distribution for different M. Table Gr f (0) 0.7699.33 4 0.037 7 0.7533 0.638 Fig. 8. Angular distribution for different M. ISBN: 978-960-474-98- 45 θ (0 ) 0.9706.77.335.499.496
Table M f θ ( 0 ) 0 0.884.383.33.77 4.59 0.895 7.8896 0.7506 0 3.3878 0.655 Table 3 R f θ ( 0 ) 0..374 0.604.33.77.5.37.3753 3.5.3845.4409 5.395.4959 5 Conclusions The problem considered in this ork is the steady, laminar, MHD mixed convection in the micropolar boundary layer flo over a continuously moving surface. The effects of radiation, magnetic field and Grashof number are examined. The governing equations are transformed to a system of nonlinear ordinary differential equations by similarity transformations and are solved numerically. The shear stress and Nusselt number as ell as the details of velocity, angular velocity and temperature fields are presented for various values of the parameters for the problem, namely magnetic field parameter (M), radiation parameter(r) and Grashof number (Gr). References: [] A. C. Eringen, Simple Microfluids, International Journal of Engineering Science, Vol., No., 964, pp. 05-7. [] A. C. Eringen, Theory of Micropolar Fluids, Journal of Mathematics and Mechanics, Vol.6, 966, pp. -8. [3] A. C. Eringen, Theory of Thermomicrofluids, Journal of Mathematical Analysis and Applications, Vol.38, No., 97, pp. 480-496. [4] T. Ariman, M. A. Turk, N. D. Sylvester, Microcontinuum Fluid Mechanics A Revie, International Journal of Engineering Science, Vol., No.8, 973, pp. 905-930. [5] T. Ariman, M. A. Turk, N. D. Sylvester, Applications of Microcontinuum Fluid Mechanics, International Journal of Engineering Science, Vol., No.4, 974, pp. 73-93. [6] J. Peddieson, R. P. McNiit, Boundary-layer Theory for a Micropolar Fluid, Recent Advances in Engineering Science, Vol.5, 968, pp. 405-46. [7] F. Ebert, A similarity Solution for the Boundary Layer Flo of a Polar Fluid, Chemical Engineering Journal, Vol.5, No., 973, pp. 85-9. [8] R. S. R. Gorla, P. V. Reddy, Flo and Heat Transfer from a Continuous Surface in a Parallel Free Stream of Micropolar Fluid, International Journal of Engineering Science, Vol.5, No.0, 987, pp. 43-49. [9] A. Raptis, Flo of a Micropolar Fluid past a Continuously Moving Plate by the Presence of Radiation, International Journal of Heat and Mass Transfer, Vol.4, No.8, 998, pp. 865-866. [0] M. A. Seddeek, Flo of a Magneto-micropolar Fluid Past a Continuously Moving Plate, Physics Letters A, Vol.306, No.4, 003, pp. 55-57. [] Ishak, Thermal Boundary Layer Flo over a Stretching Sheet in a Micropolar Fluid ith Radiation Effect, Meccanica, Vol.45, No.3, 00, pp. 367-373. [] E. M. E. Elbarbary, N. S. Elgazery, Chebyshev Finite Difference Method for the Effect of Variable Viscosity on Magneto-Micropolar Fluid Flo ith Radiation, International Communications in Heat and Mass Transfer, Vol.3, No.3, 004, pp. 409-49. [3] H. S. Takhar, R. S. Agaral, R. Bhargava, S. Jain, Mixed Convection Flo of a Micropolar Fluid over a Stretching Sheet, Heat and Mass Transfer, Vol.34, No.-3, 998, pp. 3-9. [4] R. Bhargava, L. Kumar, H. S. Takhar, Mixed Convection from a Continuous Surface in a Parallel Moving Stream of a Micropolar Fluid, Heat and Mass Transfer, Vol.39, No.5-6, 003, pp. 407-43. [5] M. Mahmoud, S. Waheed, Effects of Slip and Heat Generation/Absorption on MHD Mixed Convection Flo of a Micropolar Fluid over a Heated Stretching Surface, Mathematical Problems in Engineering, Vol.00, 00, pp. 0 [6] M. A. Seddeek, S. N. Odda, M. Y. Akl, M. S. Abdelmeguid, Analytical solution for the effect of radiation on flo of a magneto-micropolar fluid past a continuously moving plate ith suction and bloing, Computational Materials Science, Vol.45, No., pp. 43-4. ISBN: 978-960-474-98- 46