This study examines the problem of an MHD free convection flow in the presence of a temperaturedependent heat source in a viscous incompressible fluid between a long vertical wavy wall and a parallel flat wall with constant heatflux and slip flow boundary condition. A uniform magnetic field is assumed to be applied perpendicular to the walls. It is assumed that the flow consists of two parts; amean part and aperturbedpart. Expressions for the zeroth-order and first-order velocity, temperature, skin friction, and Nusselt number at the wal lr ;ire obtained. The effects of different parameters entering into the problem, viz., free convection parameter, magnetic parameter, and heat source parameter on the zeroth-order and first-order velocity fields, temperature field, skin friction, and Nusselt number at the walls are shown graphically and discussed numerically

Jain, N.C.

Taneja, Rajeev

English

Defence Science Journal

Defence Science Journal, Vol. 54, No. 1, January 22004, pp. 21-29 O 2004, DESlDOC MHD Flow with Slip Effects and Temperature-dependent Heat Source in a Viscous Incompressible Fluid Confined between a Long Vertical Wavy Wall and a Parallel Flat Wall Rajeev Taneja and N.C. Jain University of Rajasthan, Jaipur-302 004 ABSTRACT This study examines the problem of an MHD free convection flow in the presence of a temperature- dependent heat source in a viscous incompressible fluid between a long vertical wavy wall and a parallel flat wall with constant heatflux and slip flow boundary condition. A uniform magnetic field is assumed to be applied perpendicular to the walls. It is assumed that the flow consists of two parts; amean part and aperturbedpart. Expressions for the zeroth-order and first-order velocity, temperature, skin friction, and Nusselt number at the wal lr ;ire obtained. The effects of different parameters entering into the problem, viz., free convection parameter, magnetic parameter, and heat source parameter on the zeroth-order and first-order velocity fields, temperature field, skin friction, and Nusselt number at the walls are shown graphically and discussed numerically, Keywords: MHD convection flow, porous medium, skin friction, Nusselt number, slip flow, free convection flow, finite differance technique 1. INTRODUCTION The problem of free convective heat transfer Viscous fluid over a wavy wall has attracted the attention of relatively few researchers although the analysis of such flows finds applicatidns in different areas, such as transpiration cooling of reentry vehicles and rocket boosters, cross-hatching on ablative surfaces, and film vaporisation in combustion chambers. In view of these applications, Lekoudis', et al. presented a linear analysis of compressible boundary layer flows over a wavy wall. Shankar and SinhaZ studied the Rayleigh problem for a wavy wall. Lessen and Gangwani3 studied the effect of small amplitude wall waviness upon the stability of the laminar boundary layer. In all these problems, the . . authors have taken the wavy walls to be horizontal. Revised 10 September 2002 in a viscous incompressible fluid confined between a long vertical wavy wall and a parallel flat wall was considered by Vajravelu and Sastri4, and Das and Ahmed5. Patidar and Purohit6 studied free convection flow of a viscous incompressible fluid in porous medium between two long vertical wavy walls. Rao7, et al. have made an interesting analysis of an MBD convection flow in a vertical wavy channel with temperature-dependent heat source. The authors have studied the MHD free convection flow in the presence of a temperature-dependent heat source in a viscous incompressible fluid confined between a long vertical wavy wall and a parallel flat wall in slip flow regime with constant heat flux at the flat wall. The nonlinear equations governing DEP SCI I, VOL 54, NO 1, JANUARY 2W4 the flow have been solved numerically using finite = 0, v = 0, T  = T ,  at y  =~cos(hx) difference technique. Expressions for the zeroth- order and first-order velocity fields, temperature aT -q field, skin friction, and Nusselt number at the walls u = L l ( $ ) 9 v  = 0, - ay k a t y = d  have been obtained for different values of the parameters involved in the solution. where 2. FORMULATION & SOLUTION OF THE PROBLEM ( 5 )  The two-dimensional steady laminar free convective hydromagnetic along a vertical has L being the mean free path and m, the Maxwell's been considered. The x-axis is taken parallel to the coefficient. flat wall and the y-axis perpendicular to it. The wavy and the flat walls are represented by y = E cos (hr) The following nondimensional quantities have and y = d ,  respectively. The flow takes place now been introduced: under buoyancy in the presence of temperature- dependent heat source. The equations governing * x r y r u d r v d  the steady two-dimensional flow and heat transfer X =-, y  =--, u =-, v  =- d v v are: av av 1 + u-+v-----+" ax ay P ay 2 aBOZd2 M =- Magnetic parameter Pv au av -+-=(I ay (3) PCP <=- Prandtl number k (4) edz  a=- Heat source parameter k where ( u ,  v )  is the velocity field, p is the pressure, a is the electrical conductivity, B, is the uniform h' = hd Nondimensional frequency magnetic field and the other symbols have their usual meanings. The last term in RHS of Eqn (4) E E =- denotes the heat generation varying directly with d  Nondimensional amplitude ratio the temperature difference. The boundary conditions relevant to the problem K K =- d Z  Permeability parameter are taken as TANEIA & IAIN: MHD FLOW WITH SLIP EFFECTS AND TEMPERATURE L h =A Rarefaction parameter d The Eqns (1) to (4) can be expressed in the nondimensional form after dropping the asterisks over them as ae ae a2e a2e ae p u--+v- =-+-- r [ ax a,] axz y (9) with corresponding boundary conditions: It has been assumed that the solution consists of a mean part and a perturbed part so that the velocity field and temperature field are: where the perturbed quantities u,, v,, 0, and p,  are small compared with the mean quantities. In view of the form Eqn ( l l ) ,  the governing Eqns (6) to (9) assume the form: to the zeroth-order and I aeI (17) to the first-order, where The boundary conditions [Eqn (lo)] reduce to: u = h u  ', I 1 1  v, = 0, 8' = 0 1 a t y = l  (19) where prime denotes differentiation wrt y. DEF SCI 1, VOL 54, NO 1, JANUARY 2004 Introducing the stream function v defined by Eliminating p, from Eqns (14) and (15) In view of Eqn (19) ,  it has been assumed that the general solution for v and 0 ,  is: Substituting these results into Eqns (20) and (21),  one obtains the following sets of ordinary differential equations to the order of hZ: a 2 yo - N  y " = G t r  O r O  (24) e "  t  "-at  = O  0 0 (25) yr - N ~ W :  + i(u;yo - uoyi ) = G r l  t' (26) f -  at, = iP (uoto + yOOi) (27) The boundary conditions [Eqn (19)l reduce to: The differential Eqns (12)  and (13) are solved with the boundary conditions [Eqn ( I S ) ]  to obtain the mean velocity (u,) and temperature (0,). 1 u = Acosh(Ny) t Bsinh(Ny) - ' O + - 0 (a -  N 2 )  NZ where The differential Eqns (24) to (29) with conditions [Eqns (30) and (31)J are solved numerically by finite difference technique. TANElA & JAIN: MHD FLOW WITH SLIP EFFECTS AND TEMPERATURE The perturbed velocity components u,, v,, and The nondimensional skin friction 'cXy is given temperature 8, are given by by i At the wavy wall, y = E cos(hx) and the flat sin(L) t c o s ( ~ ) ]  (32) wall, y =1, zq becomes: - VELOCITY FIELD (a = 5.0, h, = 0.5, K = 1.0) O.." 1 --- TEMPERATURE FIELD (FOR DIFFERENT VALUES OF a) Figure 1. The zeroth-order velocity field (u,) for different values of G, and M and zeroth-order temperature field (0,) for different values of a plotted against y. Figure 2. DEP SCI I. VOL 54, NO 1 ,  JANUARY 2004 y for different where At the wavy wall, y = E cos(hr) and the flat wall y =1, Nu takes the form: 2; = % (o), Zp = u; (I) NU =[NU: - E ( ~ s ( L ) $ ( o ) ~  s i n ( m  (o))] -I are the zeroth-order skin frictions at the walls. w (37) The nondimensional Nusselt number (Nu) is given by Nu, =[NU: t ~{cos(hr)t,(l) - sin(hr)f(l)]] -I (38) TANEJA Br JAIN: MHD FLOW WITH SLlP EFFECTS AND TEMPERATURE 8 - Pr = 0.7, 1 = 0.01, x = - SKIN FRICTION (7 )  --- NUSSELT NUMBER (Nu) -8-  F'igure 3. Skin friction (T) and Nusselt number (Nu) plotted against a for different where are the zeroth-order Nusselt numbers at the walls. 3. DISCUSSION & CONCLUSION To understand the physical solution, the numerical values have been calculated for the zeroth-order velocity and temperature fields (Fig. l), the first- order velocity field and temperature field (Fig. 2), skin friction, and Nusselt number (Fig. 3) for different values of Gr (free convection parameter), M (magnetic parameter), and a (heat source parameter). values of G, and M In the Fig. 1, the zeroth-order velocity field (u,) is plotted against,y for fixed values of a = 5.0, h ,  = 0.5, K = 1.0 and different values of GI and M. It has been observed that for G,2 0, the velocity u, decreases throughout the channel when M increases. For G ,  = 0, velocity u, becomes negative near thk flat wall while it remains positive near the wavy wall. Further, it is seen from the graph numbers 6, 7 and 8 (in Fig. 1)  that for G, < 0, the velocity u, becomes negative throughout the channel. In this case when M is increased, velocity u, is increased. In this figure the zeroth-order temperature field (0,) is also plotted DEF SCI I, VOL 54, NO 1,  JANUARY 2004 against y for different values of a. It  is being observed that when a is increased, temperature 8 ,  is increased. In the Fig. 2, the first-order velocity and temperature fields (u , ,  v , ,  8 , )  are plotted against y for fixed values of P , =  0.7, h = 0.01, x = 1.0, h,  = 0.5, K = 1.0 and different values of Gr , M and a. It is being observed that when M and a are increased, velocity u ,  and temperature 8 ,  are increased but the phenomena reverses for the case of G,. Further, it is seen that when G, and M are increased, velocity v ,  is decreased but the phenomena reverses for the case of a. In the Fig. 3 the skin friction (7) and Nusselt number (Nu) are plotted against a for fixed values of P,=  0.7, h = 0.01, x = 1.0, h, = 0.5, K = 1.0 and different values of G, and M. It is being observed that when M is increased, skin friction (zw ) and Nusselt number (Nuw) at the wavy wall are decreased but the phenomena reverses for the case G,. Further, it is seen that when M is increased, skin friction (2,) and Nusselt number (Nu,)  at the flat wall are increased but the phenomena reverses for the case of Gr. ACKNOWLEDGEMENTS The authors are thankful to the learned referee for their valuable suggestions for the improvement of the paper. Dr N. C. Jain is thankful to University Grants Commission and the University of Rajasthan for providing financial assistance. REFERENCES 1. Lekoudis, S.G.; Nayfeh, A.H. & Saric, W.S. Compressible boundary layers over wavy walls. Physics of Fluids, 1976, 19, 514-19. 2. Shankar, P.N. & Sinha, U.N. The Rayleigh problem for a wavy wall. J.  Fluid Mech., 1976, 77, 243-56. 3. Lessen, M. & Gangwani, S.T. Effects of small amplitude wall waviness upon the stability of the laminar boundary layer. Physics of Fluids, 1976, 19, 510-513. 4. Vajravelu, K. & Sastri, K.S. Free convective heat transfer in a viscous incompressible fluid confined between a long vertical wavy wall and a parallel flat wall. J. Fluid Mech., 1978, 86, 365-83. 5. Das, U.N. & Ahmed, N. Free convective MHD flow and heat transfer in a viscous incompressible fluid confined between a long vertical wavy wall and a parallel flat wall. lndian J.  Pure Appl. Math., 1992, 23, 295-04. 6. Patidar, R.P. & Purohit, G.N. Free convection flow of a viscous incompressible fluid in a porous medium between two long vertical wavy walls. lndian J.  Math., 1998, 40, 76-86. 7. Rao, D.R.V.P.; Krishna, D.V. & Sivaprasad, R. MHD convection flow in a vertical wavy channel with temperature-dependent heat sources. Proc. lndian Nat. Sci. Acad. ,  1987, 53, 63-74. 

MHD Flow with Slip Effects and Temperature-dependent Heat Source in a Viscous Incompressible Fluid Confined between a Long Vertical Wavy Wall and a Parallel Flat Wall

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MHD Flow with Slip Effects and Temperature-dependent Heat Source in a Viscous Incompressible Fluid Confined between a Long Vertical Wavy Wall and a Parallel Flat Wall

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