VISCO-ELASTIC FLUID FLOW WITH HEAT AND MASS TRASNFER IN A VERTICAL CHANNEL THROUGH A POROUS MEDIUM

Transcription

1 Volume 2, No. 1, Januar 214 Journal of Global Research in Mathematical Archives RESEARCH PAPER Available online at VISCO-ELASTIC FLUID FLOW WITH HEAT AND MASS TRASNFER IN A VERTICAL CHANNEL THROUGH A POROUS MEDIUM Rita Choudhur 1, Madhumita Mahanta 2 & Debasish De 3 1 Department of Mathematics, Gauhati Universit, Guwahati rchoudhur66@ahoo.in 2 Department of Mathematics, Girijananda Chowdhur Institute of Management & Technolog, Guwahati mmita21@gmail.com 3 Department of Mathematics, Dibrugarh Universit, Dibrugarh debasish4192@gmail.com Abstract:- A theoretical stud of two-dimensional free convective flow of a visco-elastic fluid through a porous medium bounded b a uniforml moving long vertical wav wall and a parallel flat wall in presence of heat and mass transfer has been discussed. The solutions consist of two parts: a mean part and a perturbed part. To solve the perturbed part, long wave approximation has been applied and for the mean part, the well known approximation used b Ostrach has been utilized. Solutions for the velocit, temperature, concentration, and skin friction, rate of heat transfer and rate of mass transfer have been obtained. The results are discussed for the positive values of Grashof number for heat transfer (i.e. flow on externall cooled plate). The visco-elastic effects on velocit profile, temperature field, concentration field, skin friction are analzed graphicall with the combination of other flow parameters involved in the solution. Ke-words: Visco-elastic, Grashof number, heat and mass transfer, porous medium. 2 AMS Mathematics Subject Classification 76A5, 76A1. 1. INTRODUCTION Free convection flows in vertical channels have been studied extensivel because of their importance in man engineering applications such as designing, ventilating and heating of buildings, cooling electronic components, dring several tpes of agriculture products grain and food and packed bed thermal storage. Convective flows in channels driven b temperature differences of building walls have also been studied and reported extensivel in literature. Ostrach [1] has initiated the stud of natural convection flow and heat transfer of fluids in vertical channels with constant and wall temperature. Lekoudis et al. [2] have investigated the compressible boundar laer flows over a wav wall. The Raleigh problem for a wav wall has been extended in detail b Shankar and Sinha [3]. The effects of small amplitude wall waviness upon the stabilit of the laminar boundar laer have been discussed b Lessen and Gangwani [4]. The behaviour of convection flows of a viscous incompressible fluid between a long vertical wav wall and a parallel flat wall was presented b Vajravelu and Sastri [5]. Extensive research in this discipline has been investigated b Hamada et al. [6], Ahmed et al. [7] and Singh [8] for ordinar medium. On the other hand, the studies in convection flows through porous medium bounded between channels have a wide range of scientific and engineering applications. During allo solidification, convection effects are important because the affect the solid fluid contact within a porous laer known as mush laer. Particularl, the stud of origin of flow through a porous medium is heavil based on Darc s experimental law. Based on the law, Singh and Sharma [9] have analsed three-dimensional Coutte flow through porous medium with heat transfer. The nature of natural convective heat and mass transfer along a vertical wav surface has been discussed b Jang et al. [1]. Further Jang and Yan [11] the also generalized the problem for mixed convection. The oscillator Coutte flow through porous medium has been investigated b Singh et al. [12]. Ahmed and Deka [13] to stud the effect of permeabilit of the porous medium on the free convection and heat transfer flow. Analtical studies of heat and mass transfer b free convection in a two-dimensional irregular channel are done b Fasogbon [14]. JGRMA 212, All Rights Reserved 22

2 The objective of the paper is to stud the visco-elastic effects on free convective flow with heat and mass transfer through porous medium. The visco-elastic fluid is characterized b second order fluid whose constitutive equation [Colemann and Noll, 196] is given b where is stress tensor, is the hdrostatic pressure, s are kinematic Rivlin-Ericksen tensors, are material constants describing viscosit, elasticit and cross-viscosit respectivel. From thermodnamic consideration is assumed to be negative [Coleman and Markovitz, 1962]. The solutions of pol-isobutulene in cetane at 3 C simulate a second order fluid (Kapur et al. [15]). 2. FORMULATION OF THE PROBLEM We consider the two-dimensional stead laminar free convective flow of a second-order fluid along a vertical channel in a porous medium. Let axis be taken verticall upwards and parallel to the flat wall and axis is taken perpendicular to it. The wav wall is represented b and the flat wall is considered as. Let and be respectivel the temperature and the molar species concentration of the fluid at wav wall and and be respectivel the equilibrium temperature and equilibrium molar species concentration of the fluid at the flat wall. The geometrical consideration of the problem is given in Figure 1. Our investigation is restricted to the following assumptions: (i) All the fluid properties except densit in the buoanc force are constant. (ii) The dissipative effects are neglected in the energ equation. (iii) The volumetric heat source/sink in the energ equation is constant. (iv) The wavelength of the wav wall is large ( is small). (v) The level of species concentration in the fluid is so small that the Soret and Dufour effects are neglected. (vi) The wav wall is assumed to move with a uniform velocit c. With the above mentioned assumptions, the fluid is governed b the following equations: Equation of Continuit: u x v g JGRMA 212, All Rights Reserved 23

3 = = d Figure 1: Geometrical consideration of the problem. Equations of motion: Equation of Energ Equation of Species concentration The boundar conditions to the problem are taken as The following non-dimensional variables are introduced for non-dimensionalizing the governing equations of motions. JGRMA 212, All Rights Reserved 24

4 The dimensionless equations governing the fluid flow are obtained as follows: The relevant boundar conditions of the problem are 3. METHOD OF SOLUTION To solve the equations (2.8) to (2.11), we use the perturbation technique where the velocit profiles, temperature field, concentration field and pressure are assumed as follows where first order quantities,,,, are small compared to the zeroth order quantities,,,,. According to Ostrach (1952), the change in the pressure gradient in the direction along the flow is assumed to be zero x p ps, p s is the fluid pressure in static condition. Substituting the equations (3.1) into equations (2.8)- (2.11) and equating the like terms, we get Zeroth order equations: JGRMA 212, All Rights Reserved 25

5 First order equations: The boundar conditions pertinent to the above zeroth and first order differential equations are To solve the equations (3.6) to (3.9), we introduce a stream function defined as where, For small values of λ, we consider JGRMA 212, All Rights Reserved 26

6 Eliminating the pressure term from (3.6) and (3.7) and using equations (3.11) and (3.12) to the order of following differential equations: we get the Again using equations (3.11) and (3.12) into the equations (3.8) and (3.9) and equating the like terms, we get The related boundar conditions for solving the equations (3.14) to (3.21) are given as follows: The ordinar differential equations (3.14) to (3.21) are solved subject to the boundar conditions (3.22). The solutions and the constants are not presented here for the sake of brevit. 4. RESULTS AND DISCUSSIONS The intention of the present paper is to investigate the visco-elastic effect in stead two-dimensional free convective flow of a visco-elastic fluid through a porous medium bounded b a uniforml moving long vertical wav wall and a parallel flat wall. The corresponding results for Newtonian fluid can be deduced from the obtained results b setting α=, α / =. Its non-zero values (α= -.5, α / =.5) and (α= -.1, α / =.1) characterize the visco-elastic fluid flow mechanisms. The real parts of the JGRMA 212, All Rights Reserved 27

7 solutions implied throughout the discussions. The results are discussed with fixed values of λ (=.5), m(=3), a(=1), q(=4), K(=2), A(=1.5) and ε(=.1). Here we have considered onl positive values of Grashof number for heat transfer (Gr > ) which corresponds to an externall cooled plate. The concept of Gr > is extensivel used in the mechanism of cooling problems of various electronic components. 1.5 u (Pr=3, α=, α'=) (Pr=3, α=-.5, α'=.5) (Pr=3, α=-.1, α'=.1) (Pr=5, α=, α'=) (Pr=5, α=-.5, α'=.5) (Pr=5, α=-.1, α'=.1) Figure 2: Velocit profile against for Gr=2, Gm=1, Sc=2, m=3, a=1, q=4, K=2, A=1.5 at. Velocit profiles are discussed graphicall in figures 2 and 3. The patterns of temperature field and concentration field are noticed in figure 5 and 4 respectivel and variations of shearing stress for various cases are analsed in figures 5 to 9. Figures 2and 3 depict the velocit profile against for various values of visco-elastic parameters. The figures also demonstrate the effects of Prandtl number and Schmidt number for both Newtonian and non-newtonian fluid flow sstems. u (Sc=2, α=, α'=) (Sc=2, α=-.5, α'=.5) (Sc=2, α=-.1, α'=.1) (Sc=4,α=, α'=) (Sc=4, α=-.5, α'=.5) (Sc=4,α=-.1, α'=.1) Figure 3: Velocit profile against for Gr=2, Gm=1, Pr=3, m=3, a=1, q=4, K=2, A=1.5 at. JGRMA 212, All Rights Reserved 28

8 ϕ (α=, α'=) 3.9 (α=-.5, α'=.5) 3.85 (α=-.1, α'=.1) Figure 4: Concentration against for Gr=2, Gm=1, Pr=3, Sc=2, m=3, a=1, q=4, K=2, A=1.5 at. Prandtl number plas a significant role in simultaneous momentum and thermal diffusion problems as it is defined as the ratio of momentum diffusivit to thermal diffusivit. The Prandtl number is analogous to the Schmidt number in mass transfer concept. In our stud, we have considered Pr>1 for various fluid flows. Phsicall, it means that momentum diffuses at a greater rate than heat. When Pr increases, the viscosit of the fluid will raise to make it thick. Then a diminishing trend in the speed is noticed for both Newtonian and non-newtonian fluid flows (Figure 2). The enhancement of Schmidt number also increases the stickiness and hence both fluid flows experience a decelerating tendenc (Figure 3). The variations of Newtonian and non-newtonian fluids against the displacement factor are also noticed in figures 2 and 3. The figures suggest that during the non-zero values of α and α / i.e. for non-newtonian fluids, the velocit will be enriched in comparison with the simple Newtonian fluid flows. So it is concluded from the figures that the absolute values of visco-elastic parameter raises the speed of complex fluids. The concentration field against the distance is plotted in figure 4 for both simple and complex fluids. The concentration of both fluids will be amplified along with the increasing values of. Also, it is observed that the concentration of the Newtonian fluids is of higher order in comparison with the set of non-newtonian fluids. θ Pr=3 Pr= Figure 5: Temperature against for Gr=2, Gm=1, Sc=2, m=3, a=1, q=4, K=2, A=1.5 at. JGRMA 212, All Rights Reserved 29

9 σ x (=, α=, α'=) (=, α=-.5, α'=.5) (=, α=-.1, α'=.1) (=1, α=, α'=) (=1, α=-.5, α'=.5) (=1, α=-.1, α'=.1) Pr Figure 6: Shearing stress against Pr for Gr=2, Gm=1, Sc=2, m=3, a=1, q=4, K=2, A=1.5 at. Temperature profiles for Pr=3 and Pr=5 against are shown in figure 5. The increasing values of Prandtl number raise the temperature of the fluid in Newtonian cases but the effect of visco-elastic parameters are negligible in the temperature field. The dimensionless viscous drag on the surface of the bod due to the motion of the fluid is known as the skinfriction. To get the phsical behaviour of the viscous drag, the graphs of shearing stress against for viscous values of flow parameters have been investigated. Figure 6 deals with the characteristics of shearing stress against Pr at both the walls for phase angle λx = The graphs illustrate that the enrichment of Pr modifies the drag force to make the fluid thicker at the flat wall but a reverse nature is noticed at the wav wall. The presence of visco-elasticit enhances the drag force experienced b the non-newtonian fluid at the flat wall in comparison with the Newtonian fluid but then it shows a reverse effect at the wav wall. The variation of shearing stress against Pr for the phase angles & at the wav wall has been shown in figure 7. Increasing values of phase angle lead to subdue the shearing stress of both tpes of fluid. Figure 8 demonstrates the nature of shearing stress at both the wav walls for the phase angle λx =. The effects of Schmidt number (Sc) on the viscous drag of Newtonian as well as non-newtonian fluids at both the walls are revealed in figure 9. The results are investigated at the phase angle. Schmidt number characterizes the simultaneous effect of momentum diffusion and concentration diffusion σ x (α=, α'=) (α=-.5, α'=.5) (α=-.1, α'=.1) (α=, α'=) (α=-.5, α'=.5) (α=-.1, α'=.1) Pr λx=π/2 λx= Figure 7: Shearing stress against Pr for Gr=2, Gm=1, Sc=2, m=3, a=1, q=4, K=2, A=1.5 at. JGRMA 212, All Rights Reserved 3

10 σ x (=, α=, α'=) (=, α=-.5, α'=.5) (=, α=-.1, α'=.1) (=1, α=, α'=) (=1, α=-.5, α'=.5) (=1, α=-.1, α'=.1) Pr Figure 8: Shearing stress against Pr for Gr=2, Gm=1, Sc=2, m=3, a=1, q=4, K=2, A=1.5 at. σ x Sc (=, α=, α'=) (=, α=-.5, α'=.5) (=, α=-.1, α'=.1) (=1, α=, α'=) (=1, α=-.5, α'=.5) (=1, α=-.1, α'=.1) Figure 9: Shearing stress against Sc for Gr=2, Gm=1, Pr=3, m=3, a=1, q=4, K=2, A=1.5 at. The factor Sc enhances the drag force experienced b the Newtonian fluid at the flat wall but it shows a descending pattern in the drag force formed b Non-Newtonian fluids. But in case of wav wall, the Newtonian fluid flow experiences a reverse trend in comparison with the non-newtonian fluid flows. σ x (α=, α'=) (α=-.5, α'=.5) (α=-.1, α'=.1) Gr Figure 1: Shearing stress against Gr for Sc=1, Gm=1, Pr=3, m=3, a=1, q=4, K=2, A=1.5 at. JGRMA 212, All Rights Reserved 31

11 Grashoff number is also known as free convection parameter as it is a ver important tool to explain the phenomenon of free convection. Figure 1 deals with the behaviour of shearing stress against the Grashoff number for heat transfer (Gr) at the wav wall and at phase angle.grashoff number is defined as the ratio of buoanc force to viscous force. The increasing values of free convection parameter for heat transfer subdue the viscous drag formed during the motion of both simple and complex fluids. The rate of heat transfer (Nu) and the rate of mass transfer (Sh) are not significantl affected during the variations visco-elastic parameters. The frictional forces generated b the fluid flow cause the Pressure drop and it is defined as difference between the pressure at an point on the flow field and the pressure at the flat wall and it is given bthe effect of visco-elastic parameter on the zeroth-order pressure drop ( ) is analsed with the various values of other flow parameters involved in the solution Δp 1 (Gr=4, α=-.5, α'=.5) (Gr=4, α=-.1, α'=.1) (Gr=2, α=-.5, α'=.5) (Gr=2, α=-.1, α'=.1) Figure 11: Zeroth order Pressure drop against for Sc=2, Gm=1, Pr=3, m=3, a=1, q=4, K=2, A=1.5 at. The nature of zeroth order pressure drop is discussed in figure 11. The graphs analze that the Newtonian fluid does not experience pressure drop. But the visco-elasticit creates the pressure drop of governing fluid flow. The visco-elasticit factor present in the fluid flow mechanism raises the pressure drop. Also, the pressure drop shows a parabolic deca along with the enhancement of displacement variable. The figures exhibit that the rising trend of free convection parameter for heat transfer show a growth in pressure drop of various non-newtonian fluid flows. 5. CONCLUSION The visco-elastic effects in stead two-dimensional free convective flow of a non-newtonian fluid through a porous medium bounded b a uniforml moving long vertical wav wall and a parallel flat wall along with the combination of heat and mass transfer have been investigated. Some of the vital points of the present stud are listed as below: (i) (ii) (iii) (iv) (v) (vi) The absolute values of visco-elasticit parameter accelerate the fluid flow. Enhancement of Prandtl number and Schmidt number increases the thickness of both Newtonian and non- Newtonian fluid flow sstems and hence both fluids experience a diminishing trend in the speed. The concentration of the Newtonian fluid is superior in compared to the non-newtonian fluid. Enrichment of Prandtl number modifies the drag force at the flat wall but a reverse behaviour is noticed at the wav wall. The visco-elasticit enhances the drag force at the flat wall but a reverse behaviour is noticed at the wav wall. The non-newtonian factor of complex fluid flow sstem raises the pressure drop. 6. NOMENCLATURE - stress tensor. - Rivlin- Ericksen tensor. - displacement variables. components of velocities. JGRMA 212, All Rights Reserved 32

12 x, dimensionless displacement variables u, v dimensionless velocities in x-, - directions ρ densit of the fluid kinematic viscosit, elasticit and cross viscosit g acceleration due to gravit. β, β* - co-efficient of volume expansions due to heat and mass transfer respectivel. - temperature and concentration variables. θ, ϕ dimensionless temperature and concentration respectivel. θ s, C s fluid temperature and concentration in static condition. θ w, C w fluid temperature and concentration at the wav wall. θ 1, C 1 fluid temperature and concentration at the flat wall. α, α / - dimensionless visco-elastic parameters. d - distance between the two walls. p dimensionless pressure. Q constant heat addition and absorption. S dimensionless heat source/ sink parameter. Pr Prandtl number. Sc Schmidt number Gr, Gm Grashof number for heat and mas transfer respectivel. amplitude of the wav wall. - frequenc of the wav wall. ε dimensionless amplitude parameter. λ dimensionless frequenc parameter. c - velocit of the wav wall. A dimensionless velocit of the wav wall. C p specific heat. k thermal conductivit of the fluid. permeabilit of the porous medium K dimensionless permeabilit of the porous medium. m wall temperature ratio. q wall concentration ratio 7. REFERENCE [1] S. Ostrach, Laminar natural convection flow and heat transfer of fluids with and without heat sources in channels with constant wall temperature, N. A. C. A. Tech. Note No. 2863, [2] S. G. Lekoudis, A. H. Nafeh & W. S. Saric, Compressible boundar laers over wav walls, Phs. Fluids, 19, 1976, [3] P. N. Shankar and Sinha, U. N., The Raleigh problem for a wav wall, J. Fluid Mech., 77, 1976, [4] M. Lessen and S. T. Gangwani, Effects of small amplitude wall waviness upon the stabilit of laminar boundar laer, Phs. Fluids, 19, 1976, [5] K. Vajravelu, K. S. Sastri, The free convection heat transfer in a viscous incompressible fluid between a long vertical wav wall and a parallel flat wall, J. Fluid Mech., 86, 1978, [6] T. T. Hamadah and R. A. Wirtz, Free convection flow in vertical channel, ASME, J. Heat Transfer, 113, 1991, 57. [7] S. Ahmed and N. Ahmed, Free convection heat transfer in a viscous incompressible fluid between a uniforml moving long vertical wav wall and a parallel flat wall, Far East J. Appl. Math., 8(2), 22, [8] K. D. Singh, Channel flow with transpiration cooling for ordinar medium, ZAMP, 5, 199, [9] K. D. Singh and R. Sharma, Three dimensional Couette flow through a porous medium with heat transfer, Indian J. Pure and Appl. Math., 32, 2, 21, [1] J. H. Jang and W. M. Yan and H. C. Liu, Natural convection heat and mass transfer along a vertical wav surface, Int. J. Heat Mass Transfer, 46, 23, [11] J. H. Jang and W. M. Yan, Mixed convection heat and mass transfer along a vertical wav surface, Int. J. Heat Mass Transfer, 47, 24, [12] K. D. Singh, M. G. Gorla and Hans Raj, A periodic solution of oscillator Couette flow through a porous medium in rotating sstem, Indian J. Pure and Appl. Math., 36, 3, 25, [13] S. Ahmed and H. Deka, Free convective flow in a vertical channel through a porous medium with heat transfer, Int. J. Appl. Math., 21, 4, 28, [14] P. F. Fasogbon, Analtical studies of heat and mass transfer b free convection in a two-dimensional irregular channel, Int. J. Appl. Math and Mech., 6, 4, 21, [15] Kapur, J. N., Bhatt, B. S. and Sachbti, N. C. S. A., Non-Newtonain fluid flow, Pragati Prakashan, Meerut, 1 st Ed., JGRMA 212, All Rights Reserved 33