FREE CONVECTIVE HEAT AND MASS TRANSFER FLOW UNDER THE EFFECT OF SINUSOIDAL SUCTION WITH TIME DEPENDENT PERMEABILITY Aarti Tiwari 1, K.K.Tiwari 2, T.S.Chauhan 3, I.S.Chauhan 4 1 Deptt. of Basic Sciences, S.R.M.S. Women s College of Engg. & Tech., Bareilly (U.P.)India 2 Deptt. of Basic Sciences, S.R.M.S. College of Engg. & Tech., Bareilly (U.P.) India 3 Deptt. of Mathematics, Bareilly College, Bareilly (U.P.) India 4 Deptt. of Mathematics, Ganjdundwara P.G. College, Ganjdundwara (U.P.) India Abstract- In the present paper the investigation is undertaken to study the problem free convective heat and mass transfer flow of an incompressible viscous fluid over an infinite vertical porous plate embedded in a porous medium. The porous plate is subjected to transverse periodic suction velocity with the permeability fluctuating with time. The unsteadiness is due to time dependent porous medium as well as the suction velocity. With the use of necessary boundary conditions, firstly the governing equations are changed into dimensionless form by using the non-similar transformations. These dimensionless governing equations provide a set of coupled non-similar partial differential equations which after using series expansion method results to the expressions for the transient velocity, transverse velocity, temperature and concentration. During the course of discussion, it is found that the various parameters related to the problems influence the calculated resultant expressions. Lastly the profiles of transient and transverse velocity, temperature and concentration are analyzed and discussed graphically. Keywords magnetohydrodynamical flow, free convective heat transfer, mass transfer, incompressible fluid, viscous periodic flow. I. INTRODUCTION A number of scholars have devoted their extensive research on the study of free and forced convective threedimensional flow with heat and mass transfer due to its day-to-day applications in science and technology. The phenomenon of heat and mass transfer are observed in buoyancy induced motions in the atmosphere, in water bodies, quasi-solid bodies such as earth and so on. The free convective heat transfer flows play an important role in chemical engineering, turbo-machinery and aerospace technology. In industrial applications many transport exits where the transfer of heat and mass takes place 442 simultaneously as a result of combined buoyancy effects due to thermal diffusion and chemical species diffusion. The study of such flows was initiated by Lighthill [8] who studied the effects of free stream oscillations on the flow of a viscous incompressible fluid past an infinite plate. Stuart [13] further extended it to study a two-dimensional oscillatory flow past an infinite, porous plate with constant suction. Soundalgekar [14] studied the flow past an infinite vertical plate oscillating in its own plane and with the wall temperature. Also Messiha [10] investigated the twodimensional oscillatory flow when the plate is subjected to a time-dependent suction. The effects of different arrangements and configurations of the suction holes and slits have been studied extensively by various scholars and have been compiled by Lachmann [9]. The effect of transverse sinusoidal suction on the steady flow along a plane wall has been presented by Gersten and Gross[3]. The flow in the boundary layer becomes three-dimensional by considering this type of suction. Singh et al. (Singh et al. 1978) has investigated the boundary layer flow and heat transfer on a horizontal plate whose temperature differs from that of ambient fluid. Raptis [11] investigated the problem of unsteady flow through a porous medium bounded by an infinite porous plate subjected to a constant suction and variable temperature. Further Raptis and Perdikis [12] studied the unsteady two-dimensional free convective flows through highly porous medium. The problem of three-dimensional fluctuating flow and heat transfer through a porous medium with variable permeability was represented by Singh et al. [16]. Soundalgekar et al. [19] studied the effects of mass transfer on free convection flow of an incompressible dissipative fluid. The study of heat transfer in mercury and electrolytic solution past an infinite porous plate with constant suction in presence of transverse magnetic field and heat sink were presented by Sahoo et al. [17].
Further Singh and Takhar [18] have analyzed the effects of periodic suction velocity on three-dimensional viscous fluid with heat and mass transfer. Also, Guria and Jana [4] studied the effect of buoyancy forces and time dependent periodic suction on three-dimensional flow past a vertical porous plate. Hayat et al.[5] studied the effect of thermal radiation on the flow of a second grade fluid. Jain and Sharma[7] and Jain and Gupta[6] have studied three dimensional coutte flow with slip boundary conditions and suction velocity vary sinusoidaly. Al-Odat and Al-Azab[2] have presented the influence of Chemical Reaction on the transient MHD free convection over a moving vertical plate. Aboeldahab and Azzam[1] have studied unsteady three dimensional combined heat and mass transfer for convective flow over a stretching surface. The aim of the present paper is to bring out the effects of free convective heat and mass transfer of threedimensional flow of a viscous incompressible fluid past an infinite vertical porous plate in presence of transverse sinusoidal suction velocity oscillating with time and time dependent permeability. II. NOMENCLATURE = Dimensional concentration of the fluid (kg.m 3 ) = Dimensional concentration at the plate = Dimensional fluid concentration in the free stream (K) = Specific heat at constant pressure (J. kg -1. K -1 ) = Chemical Molecular diffusivity = Thermal Grashof number = Mass Grashof number = Mean permeability (m 2 ) = Transverse permeability (m 2 ) = Wavelength of permeability distribution (m) = The pressure (Pa) = Prandtl number K = Reynolds number S = Schmidt number = Dimensional temperature of the fluid (K) = Dimensional temperature at the plate (K) = Dimensional fluid temperature in the free stream (K) = ) Dimensional velocity components along the = Dimensionless molar concentration = Dimensionless temperature = Thermal conductivity (W/m.K) = Density (kg/m 3 ) = Coefficient of viscosity (kg/m.s) = Kinematic viscosity (m 2 /s) = Frequency parameter (S 1 ) = subscripts which represents the quantity evaluated at wall conditions = subscripts which represents the quantity evaluated at free stream conditions = Coefficient of volume expansion for heat transfer = acceleration due to gravity = zeroth order temperature = zeroth order concentration III. FORMULATION OF THE PROBLEM Consider the heat and mass transfer flow of a viscous incompressible fluid past an infinite vertical porous plate with transverse sinusoidal suction velocity fluctuating with time. The porous plate lies perpendicular to the - plane with -axis along the plate in the upward direction. The normal to the plane of plate is taken along -axis and directed into fluid flowing laminarly with a uniform free stream velocity. All the fluid properties are considered constant except the effect of the density variation with concentration and temperature in the body force term. Using Boussinesq and boundary layer approximation, the governing equations of heat and mass transfer flow for continuity momentum energy and concentration are given by Singh and Sharma, Varshney and Singh = Suction velocity (m/s) = directions (m/s) 443
where,, are the velocity components along,, directions respectively. The suction velocity is assumed oscillating sinusoidally with time as where and are the respective wavelength and amplitude of suction variation. The permeability of the porous medium fluctuates with time in the following form Using above non dimensional quantities and eq. ; eq. to reduces to The relevant boundary conditions of the problem are The non dimensional quantities are introduced as follows 444
with the transformed boundary conditions as with corresponding boundary conditions to solve these equations as Under the boundary conditions equations to are, the solutions of the IV. SOLUTION TO THE PROBLEM When the amplitude of the permeability variation is very small i.e., then the solutions of the nonlinear partial differential equations - subject to the boundary condition are of the following form When, with where where stands for,,,, and. To the zero the order, the problem reduces to the steady two dimensional convective heat and mass transfer governed by the following equations Substituting eq. into eq. to and equating the coefficients of, the following system of linear partial differential equations of first order are obtained 445
where stands for and. The prime in denotes differentiation with respect to. Equations for and have been chosen so that the continuity equation is satisfied. Substituting these equations into equations to and solving them subject to the boundary condition, we obtain the solutions of and as with the boundary conditions The equation (27) to (32) are the linear partial differential ones, which describe the free convective three dimensional flow with heat and mass transfer. The equations (27), (29) and (30) govern the cross flow and the equations like (28), (31) and (32) govern the main flow, the temperature and the species concentration, respectively. The solutions of the equations (27) to can be obtained in terms of complex numbers, the real part of which will have physical significance and therefore and will be considered as where and respectively are the real positive roots of the equations 446
with,, being given by In the above equations the quantities are defined as 447
when A = 0.5, = 0.71,, S =0.30,, t = 1, ω = 5. It is noticed that an increase in the Reynolds number leads to an increase in the flow velocity within the boundary layer. This is because due to the dominance of viscous forces over the inertia forces. Moreover, the behaviour of Reynolds number gets reversed to the temperature and concentration profiles. Moreover, it is interesting to observe that the thickness of velocity, thermal as well as concentration boundary layer becomes thicker with the increasing values of permeability parameter. Figures 4 and 5 illustrates the variation of transient velocity and temperature with various Prandtl numbers for diffusing Helium in air (species diffusivity > momentum diffusivity, S = 0.30) when A = 0.5, = 0.1, = 0.5,,, t = 1, ω = 5. With increasing values of Prandtl number, there is a clear decrease in flow velocity i.e. the flow is decelerated through the boundary layer transverse to the plate when the plate is cooled by the free convection currents ( ). encapsulates the ratio of momentum diffusivity to thermal diffusivity for a given fluid. It is also the product of dynamic viscosity and specific heat capacity divided by thermal conductivity. Higher fluids will therefore posses higher viscosities (and lower thermal conductivities) implying that such fluids will flow slower than lower fluids. As a result the velocity will be decreased substantially with increasing Prandtl number. Also, it is observed that (Figure 5), the fluid temperature is reduced monotonically when the Prandtl number is increased. As the smaller values of Prandtl number are equivalent to increase in the thermal conductivity of the fluid and therefore, heat is able to diffuse away from the heated surface more rapidly for higher values of Prandtl number. Moreover, Figure 5 also shows that the effect of higher results into the thinner thermal boundary layer as the higher Prandtl number fluid has a lower thermal conductivity. V. RESULTS AND DISCUSSION The effects of Reynolds number and Permeability parameter on the transient velocity, temperature and concentration have been presented in Figures 1, 2 & 3 448 The transient velocity and concentration has been exhibited in the respective Figures 6 & 7 with different values of Schmidt numbers (S) for conducting air ( ) when A = 0.5, = 0.1,,,, t = 1, ω = 5. S quantifies the relative effectiveness of momentum and mass transport by diffusion. Higher values of Sc amount to a fall in the
chemical molecular diffusivity i.e. less diffusion therefore takes place by species transfer. In the present study we have performed calculations for Prandtl number ( ), so that. Physically this implies that the thermal and species diffusion regions are of different extents. An increase in will suppress concentration in the boundary layer regime. Higher will imply a decrease of molecular diffusivity causing a reduction in concentration boundary layer thickness. Lower will result in higher concentrations i.e. greater molecular (species) diffusivity causing an increase in concentration boundary layer thickness. For = 1.0, the momentum and concentration boundary layer thicknesses are of the same value approximately i.e. both species and momentum will diffuse at the same rate in the boundary layer. Velocity, u, as shown in Figure 6 is found to decrease strongly with an increase in Schmidt number from 0.30 (helium diffusing in air), 0.66 and 0.78 (in all these cases species diffusivity > momentum diffusivity) through to 2.62 (species diffusivity < momentum diffusivity). Similarly there is a strong reduction in species concentration values ( ) as shown in Figure 7 with a rise in. Concentration profiles follow a smooth decay from the wall (plate) to the edge of the boundary layer; velocity profiles however as in earlier graphs peak close to the plate and then descend thereafter towards the free stream. FIGURE-1 Effects of and on transient velocity FIGURE-2 The behaviour of thermal Grashof number ( ) and mass Grashof number ( ) on the transient velocity has been shown in the Figures 8 & 9, respectively. For the case of thermal buoyancy vanishes. For it is present. A strong acceleration in the flow is induced with a rise in from 0 through 1 (thermal buoyancy and viscous forces equivalent) to 5 and 10 (for which thermal buoyancy forces exceed viscous hydrodynamic forces in the boundary layer). Like thermal Grashof number, it is observed form Figure 9 that a positive increase in from 0 through 1 to 5 and 10, clearly accentuates velocities i.e. accelerates the flow. is proportional to the mass buoyancy generated by free convection currents in the regime. Increasing buoyancy will therefore aid the flow. As expected, the fluid velocity increases and the peak value becomes more distinctive due to increase in the buoyancy force represented by or. Effects of and on temperature FIGURE-3 Effects of and on concentration 449
FIGURE-4 FIGURE-7 Effects of on transient velocity FIGURE-5 Effects of S c on concentration FIGURE-8 Effects of on transient velocity FIGURE-9 Effects of on temperature FIGURE-6 Effects of on transient velocity Effects of S on transient velocity VI. CONCLUSIONS A detailed analytical study has been carried out for the unsteady three- dimensional combined heat and mass transfer flow of an incompressible viscous fluid over a vertical porous plate embedded in a porous medium with transverse periodic suction velocity. It was found that when thermal and mass Grashof numbers were increased, the thermal and concentration buoyancy effects were enhanced 450
and thus, the fluid velocity increased. It was observed that the effect of higher Prandtl number ( = 11.4) results into the thinner thermal boundary layer as the higher Prandtl number fluid has a lower thermal conductivity. Also, it was seen that the effect of higher Schmidt number ( = 2.62) results into the thinner concentration boundary layer as the higher Schmidt number fluid has lower concentration diffusivity. In this study, it was interesting to observed that the buoyancy forces either in thermal or mass Grashof number shows the presence of overshoot in the velocity profile near the wall for lower Prandtl number fluid ( = 0.71) but for higher Prandtl number fluid ( = 11.4) the velocity overshoot is not observed, because the buoyancy forces in Gr or Gm affects more in lower Prandtl number fluid due to the low viscosity of the fluid, which increases the velocity within the boundary layer. The study noted that velocity, temperature, and concentration profiles decrease with increases in the porosity effect. The skin-friction is strongly affected by the time dependent porosity as well as suction velocity, which confirm the importance of present investigation of unsteady combined heat and mass transfer flow. It is hoped that the present work will serve as a tool for understanding more complex problems involving various physical effects investigated in this study. REFERENCES [1] Aboeldahab, E. M. and Azzam, G.E.D.A, Unsteady three dimensional combined heat and mass transfer for convective flow over a stretching surface with time dependent chemical reaction. Acta Mech., 184, 2006, p.p 121-136. [2] Al-Odat, M.Q. and Al-Azab, T.A. (2007). Influence of Chemical Reaction on the transient MHD free convection over a moving vertical plate, Emirates J. Engng. Res. 12, 3, 15-21. [3] Gersten K and Gross JF (1974). The effect of transverse sinusoidal suction velocity on flow and heat transfer over a porous plane wall. Zeitschrift fur Angewandte Mathematik und Mechanik (ZAMM), 25, pp. 399. [4] Guria M and Jana RN (2005). Hydrodynamic effect on the three-dimensional flow past a vertical porous plate. International journal of Mathematics and Mathematical Sciences, 20, pp. 3359-3372. [5] Hayat, T., Nawaz, M., Sazid, M. and Asghar, S., The effect of thermal radiation on the flow of second grade fluid. Computers and Math. with Applications, 58, 2009,p.p 369-379. 451 [6] Jain, N.C. and Gupta, P., Three dimensional free convection coutte flow with transpiration cooling. J. Zhejiang Univ. SCIENCE A, 7(3), 2006,p.p 1-8. [7] Jain, N.C. and Sharma, B., On three dimensional free con- vection coutte flow with transpiration cooling and tempera- ture jump boundary condition. Int. J. App. Mech. Eng., 14(3), 2009,p.p 715-732. [8] Lighthill MJ (1954). The response of laminar skinfriction and heat transfer to fluctuations in the stream velocity. Proceedings of the Royal Society Ser. A, [9] Lachmann GV (1961). Boundary Layer and flow control, its Principles and Applications. Vols I & II, Pergamon Press. [10] Messiha SAS (1966). Laminar boundary layers in oscillatory flow along an infinite flat plate with variable suction. Proc. Cam. Phil. Soc., 62, pp. 329-337. [11] Raptis A (1983). Unsteady free convective flow through a porous medium. International Journal of Engineering Sciences, 21, pp. 345-348. [12] Raptis A and Perdikis CP (1985). Oscillatory flow through a porous medium by the presence of free convective flow. International Journal of Engineering Sciences, 23, pp. 51-55. [13] Stuart JT (1955). A solution of the Navier-Stokes and energy equations illustrating the response of skinfriction and temperature of an infinite plate thermometer to fluctuates in the stream velocity. Proceedings of the Royal Society Ser. A, 231, [14] Soundalgekar VM (1979). Free convection effects on the flow past a vertical oscillatory plate. Astrophysical Space Science, 64, pp. 165-172. [15] Singh P, Sharma VP, and Mishra UN (1978). Fluctuating boundary layer on a heated horizontal plate. Acta Mechanica, 30, pp. 111-128. [16] Singh KD, Sharma R, and Chand K (2000). Threedimensional fluctuating flow and heat transfer through a porous medium with variable permeability. Zeitschrift fur Angewandte Mathematik und Mechanik (ZAMM), 80, pp. 473-480. [17] Sahoo PK, Data N, and Biswal S (2003). Magnetohydrodynamic unsteady free convective flow past an infinite vertical plate with constant suction and heat sink. Indian Journal of Pure and Applied Mathematics, [18] Singh AK and Takhar HS (2007). Three-dimensional heat and mass transfer flow of a viscous fluid with periodic suction velocity. International Journal of Fluid Mechanics, 34(3), pp. 267-286. [19] Soundalgekar VM, Monohar D, Nagarjan AS, and Ramakrisna S (2001). Mass transfer effects on free convection flow of an incompressible dissipative fluid. Journal of Energy Heat and Mass Transfer, 23, pp. 445-454.