Effect of sudden suction on the hydromagnetic natural convection from a vertical plate

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1 Proc. Indian Acad. Sci., Vol. 85 A, No. 5, 1977, pp Effect of sudden suction on the hydromagnetic natural convection from a vertical plate V. V. RAMANA RAO AND V. BALA PRASAD Department of Applied Mathematics, Andhra University, Waltair MS received 12 July 1976 ABSTRACT The effects of step function change in suction velocity are investigated for a natural convection flow from a vertical porous flat plate of infinite length in the presence of transverse magnetic field for two cases namely (i) when the plate is suddenly raised to a uniform higher temperature, (ii) when the plate suddenly begins to generate a uniform beat flux at its surface. In (i) the coefficient of heat transfer becomes independent of the Hartmann number M. In either case for a fixed time 7 except for correct steady state t ^ oo, the effect of the Hartmann number is to decrease both the velocity of the fluid and the skin-friction at the plate. In the correct steady state the velocity field and the skin-friction become independent of the Hartmann number. 1. INTRODUCTION FLOWS which are generated solely by density gradients due to large temperature diffe_ences are referred to as natural convection flows. Recently such flows have attracted wide attention in view of their importance in the fields of aeronautics, atomic power, electronics and chemical engineering. Ostrach' has studied laminar natural convection flow and heat transfer about a flat plate parallel to the direction of the generating body force. Het' ' has also studied natural convection flow in a channel, when the temperature of the walls is kept constant or vary linearly along the plate length. Sparrow and Gregg4 studied laminar natural convection flow from a vertical plate with uniform heat flux. In all these investigations it has been assumed that the flow is steady and the fluid is incompressible. The unsteady natural convection flows which are of much importance have received considerably less attention. Illingworth 5 presented an analysis of one-dimensional 280

SUCTION ON THE HYDROIYIAGNETIC NATURAL CONVECTION 281 free convection about an infinite flat plate undergoing a step function change in temperature.

2 SUCTION ON THE HYDROIYIAGNETIC NATURAL CONVECTION 281 free convection about an infinite flat plate undergoing a step function change in temperature. Kameswara Rao 6 has obtained exact solutions of natural convection flow from a vertical porous flat plate of infinite length in the two cases (i) when the plate is suddenly raised to a uniform higher temperature, (ii) when the plate suddenly begins to generate a uniform heat flux at its surface and the suction velocity is further assumed to vary as t Gupta? and Jagadeesan et a18 studied the transient free convection of an electrically conducting fluid from a vertical flat plate in the case when the induced magnetic field is negligible compared to the imposed magnetic field, i.e., when the magnetic Prandtl number Pm Q 0. Ramana Rao 9 has studied the same problem without neglecting the transient term in the magnetic field but assuming both Prandtl number and magnetic Prandtl number to be unity. A similar problem but for any Prandtl number has been recently studied by Mishra and Mohapatra 1 In this paper the effects of step function change in suction velocity are investigated for a hydromagnetic natural convection flow from a vertical porous flat plate of infinite length for the two cases discussed by Kameswara Rao, 6 using the appropriate initial conditions. 2. GOVERNING EQUATIONS AND SOLUTIONS Consider a vertical porous flat plate of infinite length in an electrically conducting viscous incompressible fluid at rest. Take the x-axis along the plate and the y-axis perpendicular to it., Let B o be the intensity of the magnetic field acting perpendicular to the plate and fixed relative to it. If the electro-magnetic force terms are combined with the Naviei^Stokes equationsy the equations expressing the conservation of mass, momentum and energy for unsteady laminar flow in a boundary layer on a vertical porous flat plate are respectively as follows: ^y 0, (1) u +vjy=gfl(t T) -I-v Z _ o P u, (2) v t + P P = 0, (3) bt zt ^ 2T (4) ^t + v ^y a ^y 2 ' where g is the acceleration due to gravity, P is the coefficient of volume expansion, a is the diffusivity and a is the electrical conductivity. In eq. (2), the

3 282 V. V. RAMANA RAO AND V. BALA PRASAD secondary effects of the magnetic induction are ignored as in the work due to Rossow.11 We shall assume that the electric field is zero; this seems to be a reasonable assumption since no external electric field is applied and the effect of polarization of the ionized fluid may be expected to be small if two, dimensional conditions occur in the ionized layer. 12 In view of these approximations the Maxwell's equations become redundant. Eq. (1) on integration leads to v = constant, (5) denoting the normal velocity at the plate; v < 0 represents suction. Eqs (2, 4) then reduce to: ^t+v^y=sp(t T )+v^yz ob ^^ u, (6) at at _ z 2T at +" ay a by p ' (7) There is a constant suction velocity ( vl, vl >0) normal to the plate for time t < 0. At t = 0, the normal velocity is doubled and is maintained for all time t > 0. The boundary conditions become: v--vi, t<0, (8) = av1, (A > 0), t >0, where we may take A = 2 for simplicity. CASE 1 t PLATE AT UNIFORM TEMPERATURE In this case the plate is maintained at a uniform temperature Tw for all time. The boundary conditions are: t>0:u=0, T=Tw :y=0, u -^ 0, T --^ T,o : y -^ oo. (9) The initial conditions are: 6=6u,a e, u= v, 2 M2 {e-e exp[ (M2 + 1 b 2)]e}, (10) 1 where we have introduced the variables, = yv1, B = T T., Ow = Tw T,,, M2 (Hartmann number) ob02v = pyl ^ (11)

4 and SUCTION ON THE HYDROMAGNETIC NATURAL. CONVECTIO 283 b2 = M2 %/11 2+ j++. (12) Tn terms of the further variable defined' by, v^2 t (13) we solve eqs (6, 7) by Laplace transform technique, making use of eqs (8-10) and obtain for Prandtl number Pr = 1,.1= 2, B ew L 2'le -2E eifc(2^t s/t)+crfc( t +/t) + exp (E + i) erf ( 2^ 1)], (14) U V 2 exp( (E + t) )C erf ( 1) + 2 exp (b 2 i) x eb^erfc^2/t +b^/1)-i-e- b`erfc(2 y l b^/t)i exp [b2 t (M2 b2) E] t + SyI8w exp ( (E + t)) x [1, erf (? I) + q et e -; erfc (;-i. 44) ee erfc (rj + 1) ), (15) which as M -* 0, educes to: U _ g j r [(E + I) exp( (E - i +- t)) + j e_i { 1 e'"'^ x erfc (2 t V r) 2 e erfc (2 V t + v t ) j( 2 - / exp ( (t+ E 2/41)) + Es'4 erfc G^-X exp ( (E + t)){(r + i erfc ( =) E exp ( E 2/41)},, (16) The skin-friction at the plate can be calculated as: To = µ! ilul = µ8p Ow rexp L (b2 1 -t {Ma b 2 b erf (bv 1)} ( ayjv_o z I + et + jar 2 erf (^/ r) J, (17)

5 284 V. V. RAMANA RAO AND V. BALA PRASAD which as M --> 0, becomes To = MOew I e-t (1 +l + 2 erf (,/ 1)]. (18) The coefficient of heat transfer is lgiven by h = _ /c (0) xvl {l + erf (V i)}, (19) Bw ^yyau Bu,v where K is the coefficient of thermal conductivity. The coefficient of heat transfer as seen from eq. (19) is independent of the Hartmann number and from table 1, it is found to increase with increasing t. In figure 1 velocity profiles are drawn for different values of t" and for M = 0 and In figure 2 temperature profiles are drawn for different I and are not affected by the presence of magnetic field. For t = 0 and oo (correct steady state) there are steady asymptotic suction profiles for different suction velocities. It is observed that as M increases, the velocity decreases at any point of the fluid but for a fixed time t" (t'0 oo). The skin- Table 1. Calculated values of coefficient of heat transfer ' [h/( K!- )J Bw v _MO Ms0 30 Y ^.,3 ""'^^t ice. ti / t wao 0 ^ Figure 1. Volocity profiles,

6 SUCTION ON THE HYDROMAGNETIC NATURAL CONVECTION 285 it Cs0. m. S -^ E Figure 2. Temperature profiles. Table 2. Calculated values of the skin-friction fr o/( V9.1,8 )] v i M t friction at the plate has been entered in table 2 for different t and M. For a fixed t (t oo) an increase in the Hartmann number decreases the skinfriction. For correct steady state, i.e., t = oo both the velocity and the skin-friction become independent of the Hartmann number. For a fixed M, the skin friction decreases. with increase in I. CASE 2: PLATE WITH UNIFORM HEAT FLUX In this case the plate is made to generate a uniform heat flux at the surface for all time. The boundary conditions are: t>0: 0,jy = q: y=0, u -3. 0, T -^ TT : y --- oo (20) where q is the heat flux per unit area of the plate. The initial conditions are: 6 = i a-e, u = v^a Ma [e-6 exp { (M2 -I- 1 b 2) a}] (21)

7 286 V. V. RAMANA RAO AND V. BALA PRASAD We solve eqs (6, 7) by Laplace transform technique, making use of eqs (8, 20, 21) and obtain for Prandtl number Pr = 1, A = 2, 6 = UL 14 2 erfc(... Vt ) + (4 2 i ) E x erfc + Vi) +,l /t exp ( (t + E + E 2/41)) + exp ( (E + t)) erf ( it)], (22) u =Ky_SM2 exp [ ( E + t)}{ 1 exp [1,21_ (M2 bz ) E] erfc (2--1) + 2 exp (bzs) C ebq erfc G01 e + b Vi e-b^ Qrfc (2 ii b Vi)] gfly $" exp[ (E + t)] x {t erf (---) Z exp ( E 2/41) + t exp ( E 2/41)+ 2 exp (E+t) ( i- t) x erfc + Vi) $+exp[ (e t)]erfc`27 t Vt)} 1 (23) which as M --> 0, becomes u = 8pv4 3 ( exp[_ (E + t)] { t orf \ / + E + 2 exp. [ ^t E2/4!)] J + 2 ( i- t') erfc (Z I + sit) + e e' x 4 e-e erfc ( 2,VI V) /exp( (i + E 2/41))}]. (24) The skin-friction at the plate becomes: To = yg^ exp [(be 1) ij{m2 b$ b erf (b Vi)) + µgl^ iga 2 e-t {AJ + ^t I \2 4I erfc (s/1)1 (25)

8 SUCTION ON THE HYDROMAGNETIC NATURAL CONVECTION 287 which as M -- 0, becomes TO_ f.8^9a [I e--t (2+J! + ^r)+ 4 +(2 ` )erfc /fj i (26) = 860 t-0.30 ti.i5 / q t =m Co 1 2 E Figure 3. Velocity profiles. t = 0 sl E Figure 4. Temperature profiles. Table 3. Calculated values of the skin-friction F. /( q K,^lE M t 0 1 oo A7 May 77

9 288 V. V. RAMANA RAO AND V. BALA PRASAD In figure 3 velocity profiles are drawn for different values of I and for M = 0 and In figure 4 temperature profiles are drawn for different I. In this case also the effect of the magnetic field is to decrease the velocity at any time except t = oo and at any point of the fluid. Table 3 shows the calculated values of the skin-friction for different t and for M = 0 and It is found for any time except I = oo, as M increases, the skin-friction decreases. As in case 1, for t = oo, both the velocity profile and the skin-friction are independent of the Hartmann number. For a fixed M as I increases th. skin-friction decreases. REFERENCES 1. Ostrach, S., NACA TR 1111 (1953). 2. Ostrach, S., NACA TN 2863 (19j2). 3. Ostrach, S., NACA TN 3141 (1954). 4. Sparrow, E. M. and Gregg, J. L., Trans. Am. Soc. Mech. Eng (1956). 5. Illingworth, C. R., Proc. Camb. Phil. Soc (1950). 6. Kameswara Rao, A., Appl. Sci. Res. A (1961). 7, Gupta, A. S., Appl. Sci. Res. A9 319 (1961). 8. Jagadeesan, S., Natarajan, S. and Srinivasan, J., ISTAM Proc. (XI Congress) p. 314 (1966). il. Raman Rao, V. V., Indian J. Phys, (1971). 10. Mishra, S. P. and Mohapatra, P., ZAMM (1975). 11. Rossow, V. J., NACA TN 3971 (1957). 12. Resler, E. L. Jr. and Sears, W. R., J. Aero. Sci (1958).

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