UNSTEADY FREE CONVECTION BOUNDARY-LAYER FLOW PAST AN IMPULSIVELY STARTED VERTICAL SURFACE WITH NEWTONIAN HEATING

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1 FLUID DYNAMICS UNSTEADY FREE CONVECTION BOUNDARY-LAYER FLOW PAST AN IMPULSIVELY STARTED VERTICAL SURFACE WITH NEWTONIAN HEATING R. C. CHAUDHARY, PREETI JAIN Department of Mathematics, University of Rajasthan Jaipur (INDIA) Received May 16, 006 An exact solution of the unsteady free convection boundary-layer flow of an incompressible fluid past an infinite vertical plate with the flow generated by Newtonian heating and impulsive motion of the plate is presented here. The resulting governing equations are non-dimensional zed and their solutions were obtained in closed form with the help of Laplace-transform technique. A parametric study of all involved parameters is conducted and a representative set of numerical results for the velocity, temperature and skin-friction is illustrated graphically and physical aspects of the problem are discussed. Key words: natural convection, incompressible fluid, heat transfer, Newtonian heating. INTRODUCTION The study of unsteady boundary layer is useful in several physical problems such as flow over a helicopter in translation motion, flow over blades of turbines and compressors, flow over the aerodynamic surfaces of vehicles in manned flight, etc. The unsteadiness in the flow field is caused either by time dependent motion of the external stream (or the surface of the body) or by impulsive motion of the external stream (or the body surface). When the fluid motion over a body is created impulsively, the inviscid flow over the body is developed instantaneously but the viscous layer near the body is developed slowly and it becomes fully developed steady state viscous flow after certain instant of time. Unsteady laminar free convection past an infinite vertical plate for Prandtl number Pr = 1.0 in case of a step change in wall temperature with time was derived by Illingworth [1] and for Pr 1.0, he derived the solution in integral form. Siegel [] studied the unsteady free convection flow past a semi-infinite vertical plate under step-change in wall temperature or surface heat flux by employing the momentum integral method. He first pointed out that the initial behavior of the temperature and velocity fields for a semi-infinite vertical Rom. Journ. Phys., Vol. 51, Nos. 9 10, P , Bucharest, 006

2 91 R. C. Chaudhary, Preeti Jain flat plate is the same for the doubly infinite vertical flat plate i.e. in the case of a temperature field it is the same as a solution of an unsteady one-dimensional heat conduction problem. Soundalgekar [3] first presented an exact solution to the flow of a viscous incompressible fluid past an impulsively started infinite vertical plate by Laplace-transform technique. An excellent review of existing theoretical and experimental work can be found in books by Stuart [4], Telionis [5] and Pop [6]. Theoretical studies on the laminar natural convection heat transfer from a vertical plate continue to receive attention in the literature due to their industrial and technological applications. Martynenko et al. [7] investigated the laminar free convection from a vertical plate maintained at a constant temperature, which is also the temperature of the surrounding stationary fluid. Perdikis [8] studied free convection effects on flow past a moving plate. Camargo et al. [9] presented a numerical study of the natural convective cooling. Recently Raptis et al. [10] studied the free convection flow of water near a moving plate. The unsteady free convection flow with heat flux and accelerated boundary condition was investigated by Chandran et al. [11]. Das et al. [1] analysed the flow problem with periodic temperature variation and Muthucumaraswamy [13] considered the natural convection with variable surface heat flux. Recently, Chandran [14] presented natural convection with ramped wall temperature. In all the studies cited above the flow is driven either by a prescribed surface temperature or by a prescribed surface heat flux. Here a some-what different driving mechanism for unsteady free convection along a vertical surface is considered in that it is assumed that the flow is also set up by Newtonian heating from the surface. Heat transfer characteristic are dependent on the thermal boundary conditions. In general there are four common heating processes specifying the wall-to-ambient temperature distributions, prescribed surface heat flux distributions, conjugate conditions where heat is specified through a bounding surface of finite thickness and finite heat capacity. The interface temperature is not known a priori but depends on the intrinsic properties of the system, namely the thermal conductivity of the fluid and solid respectively. Newtonian heating, where the heat transfer rate from the bounding surface with a finite heat capacity is proportional to the local surface temperature and is usually termed conjugate convective flow. This configuration occurs in many important engineering devices, for example: (i) In heat exchangers where the conduction in solid tube wall is greatly (ii) influenced by the convection in the fluid flowing over it. For conjugate heat transfer around fins where the conduction within the fin and the convection in the fluid surrounding it must be simultaneously analyzed in order to obtain the vital design information. (iii) In convective flows set up when the bounding surfaces absorbs heat by solar radiation.

3 3 Unsteady free convection boundary-layer flow 913 Therefore we conclude that the conventional assumption of no interaction of conduction-convection coupled effects is not always realistic and it must be considered when evaluating the conjugate heat transfer processes in many practical engineering applications. The Newtonian heating condition has been only recently used in convective heat transfer. Merkin [15] was the first to consider the free-convection boundarylayer over a vertical flat plate immersed in a viscous fluid whilst [16, 17, 18] considered the cases of vertical and horizontal surfaces embedded in a porous medium. The studies mentioned in [15 18] deal with steady free convection. Therefore, in the present paper it is proposed to study the unsteady free convection boundary-layer flow from a flat vertical plate with Newtonian heating and the solution is obtained in closed form using Laplace transform technique [19]. Literature concerning this subject can be found in books by Slattery [1] and Carslaw et al. [1]. MATHEMATICAL ANALYSIS The unsteady free convective flow of a viscous incompressible fluid past an impulsively started infinite vertical plate with Newtonian heating is considered. The x -axis is taken along the plate in the vertically upward direction and y -axis is chosen normal to the plate. Initially, for time t 0, the plate and fluid are at the same temperature T in a stationary condition. At time t > 0, the plate is given an impulsive motion in the vertically upward direction against gravitational field with a characteristic velocity U c. It is assumed that rate of heat transfer from the surface is proportional to the local surface temperature T. Since the plate is considered infinite in the x direction, hence all physical variables will be independent of x. Therefore, the physical variables are functions of y and t only. Applying the Boussinesq approximation, the flow is governed by the following equation: u t u =ν + gβ( T T ) y k T ρ C T p = t y with the following initial and boundary conditions: t 0: u = 0 T = T for ally 0: T h at t > u = U c = T y = 0 y k : u = 0 T = T as y (1) () (3)

4 914 R. C. Chaudhary, Preeti Jain 4 On introducing the following non-dimensional quantities: u t Uc yu μc c p u=, t =, y =, Pr= Uc ν ν k g T T T ν β Gr =, θ = U3 c T According to the above non-dimensionalisation process, the characteristic velocity U c can be defined as U h c = ν k Substituting the transformation (4) in equations (1) (3) leads to the following non-dimensional equations: (4) u t = u + Gr θ y (6) θ = 1 θ t Pr y The initial and boundary conditions in non-dimensional form are: t 0 : u= 0, θ = 0 for all y t > 0 : u = 1, θ = (1 +θ ) at y = 0 y : u 0 θ 0 as y (7) (8) All the physical variables are defined in Nomenclature. The energy equation (7) is uncoupled from the momentum equation (6). One can therefore solve for the temperature variable θ(y, t) whereupon the solution of u(y, t) can also be obtained, using Laplace transform technique. The solutions are derived as: y + t y Pr Pr (, ) e Pr erfc t y θ yt = erfc t Pr t (9) y t y y + Pr Pr y uyt (, ) erfc apr erfc e erfc t = t t t Pr y y a t erfc y t e t π y /4t +

5 5 Unsteady free convection boundary-layer flow 915 where Pr t e y /4t y a y erfc t + π y Pr y + t y Pr apr erfc e Pr erfc t + t t Pr y Pr y Pr + a t erfc y Pr t + e t π + a t y Pr/ 4t y π Pr e Pr erfc y Pr/ 4t y Pr t a = Gr, erfc(x) being the complementary error function defined by Pr 1 erfc( x) = 1 erf( x), erf( x) = exp ( η )dη π 0 erfc(0) = 1, erfc( ) = 0. x (10) Solution when Pr = 1 We observe that the solution for the velocity variable given by equation (10) is not valid for fluids for Prandtl number unity. As the Prandtl number is a measure of the relative importance of the viscosity and thermal conductivity of the fluid, the case Pr = 1 corresponds to those fluids whose momentum and thermal boundary layer thickness are of the same order of magnitude. The exact solution of the free convection problem when Pr = 1 is given below. It may be noted that the solution for the temperature θ(y, t) follows from equation (9) by putting Pr = 1 in that equation. However, in the case of u(y, t), the solution has to be rederived starting from equations (6) and (7). The solution obtained is given below: y Gr y (, ) erfc e y /4t y u y t t = + ( y 1)erfc t + + π t e y+ t y + erfc t t (11) Knowing the velocity field, we now study the changes in the skin-friction. It is given by: τ = μ u y y = 0 (1)

6 916 R. C. Chaudhary, Preeti Jain 6 and in virtue of equation (4) the above equation (1) reduces to τ= τ = du ρu dy c y = 0 (13) Case I: Pr 1 1 Gr Pr e t /Pr (1 erf t τ= Gr t Gr Pr + t Pr 1 Pr + + π + Pr + 1 π Pr + 1 (14) Case II: Pr = 1 1 Gr t 1 e t τ= + + (1 + erf( t ). π t π (15) Another phenomenon in this study is to understand the effects of t and Pr on Nusselt number. In the non-dimensional form, the rate of heat transfer is given by: Nu = = θ(0) et /Pr 1 erf t + 1 Pr DISCUSSION AND CONCLUSION In order to discuss the effect of various physical parameters on the velocity field, thermal boundary layer, shearing stress and coefficient of rate of heat on the wall, the numerical computation of the solutions, obtained in the preceding section, have been carried out and they are represented in Figs. (1) (6). The values of Prandtl number (Pr) are taken as 0.71, 1.0 and 7.0, which physically corresponds to air, electrolyte solution and water, respectively. The effect of Prandtl number on heat temperature may be analysed from the Fig. 1. It is inferred that the thickness of thermal boundary layer is greater for air (Pr = 0.71) and there is more uniform temperature distribution across the thermal boundary layer as compared to water (Pr = 7.0) and electrolyte solution (Pr = 1.0). It is observed that the increase of Prandtl number results in the decrease of temperature distribution. The reason is that smaller values of Prandtl number are equivalent to increasing thermal conductivity and therefore heat is able to diffuse away from the heated surface more rapidly than for higher values of Prandtl number. Thus temperature falls more rapidly for water than air and electrolyte solution. The maximum of the temperature occur in the vicinity of the plate and asymptotically approaches to zero in the free stream region. Further-

7 7 Unsteady free convection boundary-layer flow 917 more it is found that the thermal boundary layer thickness with increasing values of the time. Fig. represents the dimensionless velocity profiles u(y, t) for electrolyte solution (Pr = 1) inside the boundary layer vs. distance y for different times and Grashof number Gr. The effect of increasing Grashof number is to raise the values of the velocity profiles. It is also noticed that as time passes, the velocity Fig. 1. Temperature profile.

8 918 R. C. Chaudhary, Preeti Jain 8 Fig.. Velocity profile for electrolyte solution (Pr = 1.0). increases for positive values of Gr while for negative values of Gr a reverse phenomena is observed. Figs. 3 and 4 depict the velocity profiles for Pr = 7.0 (Prandtl number for water at 0 C and 1 atmosphere pressure) and Pr = 0.71 (Prandtl number for air at 0 C and 1 atmosphere pressure), taking arbitrary values of Grashof number and time. It is revealed that for both the fluids (air and water) an increase in Gr leads to a fall in the values of velocity, which is contrary to what happened for the electrolyte solution. The velocity remains positive for

9 9 Unsteady free convection boundary-layer flow 919 Fig. 3. Velocity profile for water (Pr = 7.0). Gr < 0 and negative for Gr > 0. Further, it is investigated that in case of water, the velocity first decreases and thereafter increases with the rise in time parameter and the point of maxima on the curves gets shifted towards right for Gr < 0 while a reverse behavior is noticed for Gr > 0 i.e., the velocity first increases than decreases with the rise in t and the point of minima on the curves

10 90 R. C. Chaudhary, Preeti Jain 10 Fig. 4. Velocity profile for air (Pr = 0.71). gets shifted to the right. In case of air, a similar phenomenon is observed as in case of electrolyte solution for rising time parameter. Further, it is noticed that there are kinks in the flow field for Pr = 7.0. In Figs. 5 and 6 skin friction is plotted against time parameter for Pr = 0.71, 1.0 and Pr = 7.0, respectively. A comparative study of both the graphs shows

11 11 Unsteady free convection boundary-layer flow 91 Fig. 5. Skin friction profile for air (Pr = 0.71) & electroyte solution (Pr = 1.0). that for Gr < 0 skin-friction is more for smaller Prandtl number, showing the inverse variation of shearing stress τ with Pr. Physically, this is true because the increase in the Prandtl number is due to increase in the viscosity of the fluid, which makes the fluid thick and hence a decrease in the velocity of the fluid. The effect of increasing Grashof number is to decrease the shearing stress on the surface. It is also noticed that as the time passes the skin friction rises for Gr < 0 while it falls for Gr > 0. The value of skin friction shows that there always remains a phase lead.

12 9 R. C. Chaudhary, Preeti Jain 1 Fig. 6. Skin friction profile for water (Pr = 7.0). Fig. 7 shows dimensionless rate of heat transfer Nu (Nusselt number). It is observed that increasing Prandtl number enhances the heat transfer coefficient. The positive values of Nusselt number show that the heat is transferred from the surface to the medium, which results in positive heat transfer coefficient. Further, is noticed that the increase in time t leads to a decrease in Nu.

13 13 Unsteady free convection boundary-layer flow 93 Fig. 7. Nusselt number. REFERENCES 1. C. R. Illingworth, Unsteady laminar flow of gas near an infinite plate, Proc. Camb. Phil. Soc., vol. 46, pp (1950).. R. Siegel, Transient free convection from a vertical flat plate, Transactions of the American Societies of Mechanical Engineers, vol. 80, pp (1958).

14 94 R. C. Chaudhary, Preeti Jain V. M. Soundalgekar, Free convection effects on Stokes problem for a vertical plate, J. Heat Transfer, ASME, vol. 99c, pp (1977). 4. J. T. Stuart, Recent Research on Unsteady Boundary Layers, Lavel University Press, Quibec (197). 5. D. P. Telionis, Unsteady Viscous Flows, Springer-Verlag, New York (1981). 6. I. Pop, Theory of Unsteady Boundary Layers, Romanian Academy of Sciences, Bucharest (1983). 7. O. G. Martynenko, A. A. Berezovsky and Yu. A. Sakovishin, Laminar free convection from a vertical plate. Int. J. Heat Mass Transfer, vol. 7, pp (1984). 8. C. P. Perdikis and H. S. Takhar, Free convection effects on flow past a moving vertical infinite porous plate, Astrophys. Space Science, vol. 15(1), pp (1986). 9. R. Camargo, E. Luna, and C. Trevino, Numerical study of the natural convective cooling of a vertical plate, Heat and Mass Transfer, vol. 3, pp (1996). 10. A. Raptis and C. Perdikis, Free convection flow of water near 4 C past a moving plate, Forschung in Ingeneiurwesen, vol. 67 (5), pp (00). 11. P. Chandran, N. C. Sacheti and A. K. Singh, Unsteady hydromagnetic free convection flow with heat flux and accelerated boundary motion, J. Phys. Soc. Jpn. vol. 67, pp (1998). 1. U. N. Das, R. K. Deka and V. M. Soundalgekar, Transient free convection flow past an infinite vertical plate with periodic temperature variation, Journal of Heat Transfer, vol. 11, pp (1999). 13. R. Muthukumaraswamy, Natural convection on flow past an impulsively started vertical plate with variable surface heat flux, Far East Journal of Applied Mathematics, vol. 14, pp (004). 14. Pallath Chandran, N. C. Sacheti and A. K. Singh, Natural convection near a vertical plate with ramped wall temperature, Heat and Mass Transfer, vol. 41, pp (005). 15. J. H. Merkin, Natural convection boundary-layer flow on a vertical surface with Newtonian heating, Int. J. Heat Fluid Flow, vol. 15, pp (1994). 16. D. Lesnic, D. B. Ingham and I. Pop, Free convection boundary layer flow along a vertical surface in a porous medium with Newtonian heating, Int. J. Heat Mass Transfer, vol. 4, pp (1999). 17. Ibid, Free convection from a horizontal surface in porous medium with Newtonian heating, J. Porous Media, vol. 3, pp (000). 18. D. Lesnic, D. B. Ingham, I. Pop and C. Storr, Free convection boundary layer flow above a nearly horizontal surface in porous medium with Newtonian heating, Heat and Mass Transfer, vol. 40 (4), pp (003). 19. B. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover Publication, Inc., New York (1970). 0. J. C. Slattery, Advanced Transport Phenomena, Cambridge University Press (1999). 1. H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids ( nd eds.), Oxford, University Press, London (1959). NOMENCLATURE C p g Gr h k Specific heat at constant pressure Magnitude of the acceleration due to gravity Grashof number Heat transfer coefficient Thermal conductivity

15 15 Unsteady free convection boundary-layer flow 95 Pr T T t u x y Prandtl number Temperature Ambient temperature Dimensional time Dimensional velocity x -direction Cartesian coordinate along the plate Cartesian coordinate normal to the plate GREEK SYMBOLS β ρ θ μ ν τ Coefficient of volumetric expansion Density of the fluid Dimensionless temperature Coefficient of viscosity Kinematic viscosity Dimensionless skin friction

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