Fully Developed Flow of Two Viscous Immiscible Fluids Through a Channel with Heat Transfer

Fully Developed Flow of Two Viscous Immiscible Fluids Through a Channel with Heat Transfer

Abdul Mateen

Department of UG & PG Studies and Research in Mathematics, Government College Gulbarga. INDIA

Abstract

An analytical solution is presented for the problem of fully developed flow and heat transfer of two viscous, incompressible immiscible fluids through a channel taking into an account the viscous dissipation. The channel walls are maintained at two different and constant temperatures. The transport properties of both the fluids are assumed to be constant. Exact solutions are given for the equation governing the flow and heat transfer for both the region with suitable boundary and interface conditions. The variation of velocity and temperature profiles with respect to various parameters such as Prandtl number, Eckert number, ratio of viscosity and thermal conductivity are presented graphically and discussed.

Keywords flow and heat transfer, channel flow, immiscible fluids.

1. Introduction

   Recent years, the requirement of modern technology has stimulated interest in flow and heat transfer studies which involves interaction of several phenomena such as heat exchangers, transport of heat or cooled fluids, chemical processing equipment and micro-electronic cooling. The problem of flow and heat transfer was extensively investigate by many researcher with different hypothesis[1-7].

   All the mentioned studies pertain to a single-fluid model. Most of the problems relating to the petroleum industry, plasma physics, magneto- fluid dynamics, etc., involve multi-fluid flow situations. In modeling such problems, the presence of a second immiscible fluid phase adds a number of complexities as to the nature of interacting transport phenomena and interface conditions between the phases. In general, multi-phase flows are driven by gravitational and viscous forces. There has been some theoretical and experimental work on stratified

   fluids layers is studied analytically by Umavathi et al. [12] and also Umavathi et. al. [13-15] have presented analytical solutions for unsteady/oscillatory two-fluid flow and heat transfer in a horizontal channel. Hishyar and Abdullah [16] presented two immiscible fluids in contact with each other on a solid boundary, have been studied: The asymptotic solution near contact line is fined for different cases. Stamenkovi, M. ., et. al.[17] investigates the magnetohydrodynamic flow of two immiscible, electrically conducting fluids between isothermal and insulated moving plates in the presence of an applied electric and inclined magnetic field with the effects of induced magnetic field.

   Keeping in view the wide area of practical importance of multi-fluid flows as mentioned, the objective of this study to investigate the viscous incompressible and immiscible fluids flow and heat transfer thorough a channel.

2. Mathematical Formulation

   Consider viscous flow of two immiscible fluids in a horizontal channel. The region 0 y h (Region-I) is filled with a viscous fluid having density 1 , dynamic

   viscosity 1 , specific heat at constant pressure

   1

   1

   CP thermal conductivity k1 and the region

   2

   2

   h y 0 (Region-II) is filled with a different viscous fluid having density 2 , dynamic viscosity 2 , specific heat at constant pressure CP and thermal conductivity k 2 .

   The flow of both regions is assumed to be fully developed and fluid properties are constant and driven by a common pressure

   laminar flow of two immiscible fluids in a horizontal pipe [8-10] Loharsbi and Sahai [11] studied two-

   gradient ( p

   x

   ). The two plates are maintained

   phase MHD flow and heat transfer in a parallel plate

   channel, with one of the fluids being electrically

   at constant temperatures

   T at y h and

   w

   w

   1

   conducting. Fully developed flow and heat transfer in horizontal channel consisting of an electrically conducting fluid layer sandwiched between two

   T at y h

   w

   w

   2

   Under these assumptions and taking Region-I

   1 2 0

   and

   C C

   p

   p

   p

   p

   1 2

   Cp the governing

   d 2u

   1 P 0

   (2.8)

   equations of motion and energy are given by:

   Region-I

   dy 2

   d 2

   du 2

   1 Ec Pr 1 0

   (2.9)

   1

   1

   2u p

   1 y 2 x 0

   (2.1)

   dy 2 dy

   0

   0

   Region-II

   2T

   u 2

   d 2u2 P

   2(T2.10)

   u 2

   k 1 1 0

   (2.2)

   dy 2

   K 1 1 0

   (2)

   1 y 2

   1 y

   1 y 2

   1 y

   d 2

   Ec Pr du 2

   Region-II

   2

   2 0 (2.11)

   2

   2

   2u p

   2 y 2 x 0

   (2.3)

   dy 2

   dy

   2T

   u 2

   where

   2

   is the ratio of viscosities and

   k2 2 2 2 0

   (2.4) 1

   y 2

   y

   k2

   k1

   is the ratio of thermal conductivities.

   where u is the x-component of fluid velocity and T is the fluid temperature. The subscripts 1 and 2 correspond to region-I and region-II, respectively. The boundary conditions on velocity are the no-slip boundary conditions which required that the x-

   component of velocity must vanish at the wall. The

   The hydrodynamic and thermal boundary and interface conditions for both fluids in non-dimensional form become

   u1 h 0

   u2 h 0

   boundary conditions on temperature are isothermal

   u1 0 u2 0

   (2.12)

   conditions. We also assume the continuity of velocity, shear stress, temperature and heat flux at

   the interface between the two fluid layers at y=0.

   u1

   y

   u2

   y

   at y 0

   The hydrodynamic boundary and interface conditions for the two fluids can then be written as

   1 h 1

   2 h 0

   u1 h 0

   0 0

   (2.13)

   u h 0 1 2

   2 1 2

   at y 0

   u1 0 u2 0

   (2.5)

   y y

   u1 u2

   at y 0

   1 y 2 y

   The thermal boundary and interface conditions on temperature for both fluids are given by

   T1 h Tw1

3. Solution

   Equations (2.8) to (2.11) are solved exactly for u1, u2 ,1 and 2 using the conditions

   T2 h Tw2 T1 0 T2 0

   (2.6)

   (2.12) and (2.13). The solutions of velocity and temperature for both the regions are

   T T

   u C C y P y 2

   k1 1 k 2 2

   at y 0

   1 1 2 2

   y y P 2

   To make equations dimensionless, we use the

   u2 C3 C4 y 2 y

   following quantities

   D D y l y 2 l y3 l y 4

   u U u * y hy *

   Twi Tw2

   1 1 2 1 2 3

   i 0 i

   Tw1

   Tw2

   2 D3 D4 y l4 y 2 l5 y3 l6 y 4

   i

   i

   h 2 P

   P U x

   1C p

   Pr

   k

   The constants appearing in the above solutions

   are defined in the Appendix section.

   1 0 1

   U 2

   Ec 0

   C p (Tw1 Tw2 )

   (2.7)

4. Results and Discussions

   In this section representative flow and heat transfer of two immiscible fluids through a horizontal channel are presented and discussed

   for various parametric conditions. Exact solutions are obained for the governing equations. The solutions are depicted graphically in Figs. 1 to 5 for different values of viscosity ratio on the flow and thermal conductivity ratio, Prandtl number and Eckert number on temperature field. The parameters are fixed as 1 except the varying one, Pr=0.7, Ec=0.2.

   that the flow can be controlled by considering different fluids having different properties.

   1.0

   Region-I

   1.0

   0.5

   y 0.0

   2.0 1.0

   0.5

   0.5

   =0.25

   Region-I

   0.5

   y 0.0

   Region-II

   = 2.0

   1.0

   0.5

   0.25

   -0.5

   -0.5

   -1.0

   0 1 2 3 4

   u

   Fig.1 Velocity profile for different viscosity ratio

   Region-I

   -1.0

   0.0 0.2 0.4 0.6 0.8 1.0

   Fig.3 Temperature profile for different conductivity ratio

   Figure 1 show that velocity profiles are suppressed for large values of viscosity ratios. The flow profile is large in region-I compare to region-II for values of viscosity ratios less than one. The flow profile is large in region-II compare to region-I for values of viscosity ratio greater than one. The flow profiles almost remain the same in both the regions for equal values of viscosity ratios and similar effects on the temperature field as shown in Fig 2.

   1.0

   0.5

   y 0.0

   Pr=0.0001 0.71

   5.0

   Region

   7.0

   1.0

   Region-I

   -0.5

   = 2.0

   0.5 1.0

   0.5

   0.25

   -1.0

   Region 0.0 0.4 0.8 1.2 1.6 2.0

   Fig. 4 Temperature profile for different Prandtl number Pr

   y 0.0

   Region-II

   -0.5

   1.0

   Region-I

   -1.0

   0.0 0.2 0.4 0.6 0.8 1.0

   Fig. 2 Temperature profile for different viscosity ratio

   0.5

   Ec =0.2

   0.4

   0.6

   0.8

   Fig. 3 depicts the effect of thermal conductivity ratio, as the ratio increases the magnitude of suppression is large in region-II compared to region-I. This is obvious because the upper plate is maintained at a low temperature compared to region-I.

   Figures 4 and 5 display the effect of Prandtl number and Eckert number respectively on temperature filed. It is seen that temperature is increases with increase in Prandtl number as well as Eckert number. Since the values of Prandtl number are very small for liquid and metals and it is very high for highly viscous fluid. Thus one can conclude

   y 0.0

   -0.5

   -1.0

   0.0 0.4 0.8 1.2 1.6

   Fig.5 Temperature porfile for different values of Eckert number Ec

   Acknoelegment

   The author would like to thank UGC-New Delhi for the financial support under UGC-Minor Research Project.

   [MRP(S)-1075/11-12/KAGUG009/UGC-SWRO]

   Nomenclature:

   CP specific heat at constant pressure

   k thermal conductivity

   P pressure

   Pr Prandtl number

   T temperature

   Tw wall temperature

   t time

   U 0 average velocity

   Greek letters

   fluid density

   viscosity of fluid

   non-dimensional temperature

   Subscripts

   1,2 quantities for region-I and region-II respectively.

5. Reference

1. Alpher, R.A., Heat transfer in magnetohydrodynamics flow between parallel plates, Int. J. Heat Mass Transfer, vol. 3, (1961), pp. 108-112.

2. Attia, H.A. and Kotb, N.A., MHD flow between two parallel plates with heat transfer, Acta Mechanica, vol. 117, (1996), pp. 215-220.

3. Chamkha, A. J., Unsteady laminar hydromagnetic fluid-particle flow and heat transfer in channels and circular pipes, Int. J. Heat and Fluid Flow, vol. 21, (2000), pp. 740- 746.

4. Haajizadeh, M. and Tien, C.L., Combined natural and forced convection in a horizontal porous channel, Int. J. Heat Mass Transfer, vol. 27, (1984), pp. 799-813.

5. Mohanty, H.K., Unsteady natural convection in horizontal channels with arbitrary wall temperatures, Int. J. Heat Mass Transfer, vol. 22, (1979) pp. 383-388.

6. Muhuri, P.K. and Maiti, M.K., Free convection oscillatory flow from a horizontal plate,Int. Heat Mass Transfer, vol. 10, (1967), pp. 717-732.

7. Thakar, H.S. and Soundalgekar, V.M., Flow and heat transfer of a micropolar fluid past a porous plate, Indian J. Pure Appl. Math., vol. 16, (1985), pp. 552-558.

8. Packham, B. A. and Shail. R., Stratified laminar flow of two immiscible fluids, Proc. Camb. Phil. Soc vol. 69, (1971), pp. 443-448.

9. Alireza Setayesh and Sahai, V., Heat transfer in developing MHD poiseuille flow and variable transport properties, Int. J. Heat Mass Transfer, vol.33, No. 8, (1990), pp. 1711-1720.

10. Malashetty, M. S. and Leela, V., Magnetohydrodynamic heat transfer in two phase flow, Int. J. Engg. Sci., vol. 30, (1992), pp. 371-377.

11. Lohrasbi, J. and Sahai, V., Magnetohydrodynamic heat transfer in two phase flow between parallel plates, Appl. Sci, Res, vol. 45, (1988), pp. 53-66.

12. Umavathi, J. C., Malashetty. M.S. and Abdul Mateen, Fully developed flow and heat transfer in a horizontal channel containing electrically conducting fluid sandwiched between two fluid layer, Int.

    J. Applied Mechanical Engineering, vol. 9, No. 4, (2004), pp. 781-794.

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14. Umavathi, J.C., Abdul Mateen, Chamkha,

    A.J. and Mudhaf, A.A., Oscillatory hartmann twofluid flow and heat transfer in a horizontal channel, International Journal of Applied Mechanics and Engineering, Vol. 11, No. 1, (2006), pp. 155178.

15. Umavathi, J.C.,., Chamkha, A.J., Mateen, A and Mudhaf, A.A., Unsteady oscillatory flow and heat transfer in a horizontal composite porous medium channel, Nonlinear Analysis: Modelling and Control, Vol. 14, No. 3, (2009), pp. 397415.

16. Hishyar Kh. Abdullah., On the Two Immiscible Fluids in Contact with Each Other on a Solid Boundary,Applied

    Mathematical Sciences, Vol. 6, No.123, pp- 61376146, (2012).

17. Stamenkovi, M. ., et. al., MHD flow and heat transfer of two immiscible fluids between moving plates. Transactions of the Canadian Society for Mechanical Engineering, Vol. 34, No. 34, (2010), pp. 351-372.

Appendix

C1 C3 ;C2

ph

2

C1 ;C hC

h 3 4

pp

2

C ph 1 ; D

hD

• k ; D

  l8 l7

  4 2 1 1

  2 7 2

  p

  D3 D1 ; D4

  D3 l8 h

  Ec Pr C 2

  ; l 2 ;

  ; l 2 ;

  1

  1

  2

  l2

  l5

  Ec Pr PC 2

  3

  Ec Pr PC4

  3

  ; l3

  ; l6

• Ec Pr P 2

  12

• Ec Pr P 2

12

Ec PrC 2

; l4 4

; l4 4

2

l7 l1p l2 p l3h4 1

l8 l4 p l5 p l6 h4