Flow Of A Couple Stess Fluid Through A Porous Layer Bounded by Parallel Plates

Flow Of A Couple Stess Fluid Through A Porous Layer Bounded by Parallel Plates

Vol. 1 Issue 10, December- 2012

1K. Nanda Gopal, 2S. Sreenadh and 1A.G. Vijaya kumar

1 Department of Mathematics, Sree Vidyanikethan Engineering College, A. Rangampet, Tirupati, A.P, INDIA

2 Department of Mathematics, S.V. University, Tirupati, A.P, INDIA

Abstract The flow of a couple stress fluid through a porous layer bounded by parallel plates is investigated. The expres- sions for the velocity and the temperature are obtained in terms of exponential functions. The mass flow rate and its frac- tional increase are determined. The effect of permeability and couple stress parameters on the velocity and temperature are discussed.

Keywords Couple stress fluid, Parallel plates, Porous layer, Mass flow rate.

1. The flow of a non-Newtonian fluid through and past porous media is of wide spread importance in various branches of science and technology. The flow of a non

   Newtonian fluid through and past porous media in of wide spread importance in various branches of science and Technology. With the growing importance of non-Newtonian fluids in modern technology, industries, the investigations on such fluids are desirable. During recent years the theory of polar fluids has received much attention and this is because the traditional Newtonian fluids cant precisely describe the characteristics of the fluid flow with suspended particles. The study of such fluids have applica- tions in a number of processes that occur in industry such as the extrusion of polymer fluids, solidification of liquid crystals, cooling of metallic plate in a bath, exotic lubrica- tions and colloidal and suspension solutions. In the category of non-Newtonian fluids couple stress fluid has distinct features , such as polar effects. The theory of polar fluids and related theories are models for fluids whose micro structure is mechanically significant. The constitutive equa- tions for couple stress fluids were given by Stokes (1966). The theory proposed by Stokes is the simplest one for micro fluids, which allows polar effects such as the presence of couple stress, body couples and non-symmetric tensor. Couple stresses are found to appear in noticeable magnitude in fluids with very large molecules. The couple stress ef- fects are considered as a result of the action of one part of a deforming body on its neighbourhood. This theory has wide results and applications in mechanics of bio fluids, colloidal fluids, liquid crystals and for pumping fluids such as syn- thetic lubricants. This theory has wide results and applica- tions in mechanics of bio-fluids, colloidal fluids liquid crystals and for pumping fluids such as synthetic lubricant. The theory of Stokes has applied for the study of some sim- ple lubrication problems see Bujurke and Jayaraman, (1982).

   Since the long chain hyaluronic acid molecules are found as additives in synovial fluids, they are modeled as couple stress fluids in human joints. The presence of small amounts of additives in a lubricant can improve the bearing perfor- mance by increasing the lubricant viscosity and thus pro- ducing an increase in the load capacity. These additives in a lubricant also-reduce the coefficient of friction and increase the temperature range in which bearing can operate.

   Based on the couple stress theory of Stokes, Valanis and Sun (1969), Chaturani and Kaloni (1976), Chaturani and Upadthya (1976) have proposed various theoretical models obtained from these three models are in good agreement with experimental results.

   Further Chaturani and Pralhad (1981) studied a three layered flow model for blood flow and they assumed that the top and bottom layers consist of plasma and the middle layer consist of red-cell suspension (couple-stress fluid). Recently Malashetty and Umavathi (1999) discussed the effects of couple stresses on the free Convection flow in a vertical channel. Free convection flow of an electrically conducting couple stress fluid and a couple stress fluid for the radiating medium in a vertical channel has been studied by Umavathi (1999,2000).

   Keeping in mind the importance and applications of non Newtonian (Couple stress) fluids, the flow of a couple stress fluid through a porous layer bounded by parallel plates is investigated. The expressions for the velocity and the tem- perature are obtained. The mass flow rate and its fractional increase are determined. The effect of permeability and couple stress parameters on the velocity and temperature are discussed.

2. Consider the flow of a couple stress fluid through a porous medium bounded by parallel plates. The permeability of the

   porous medium is taken as k . The lower and upper plates

   Vol. 1 Issue 10, December- 2012

   are maintained at fixed temperatures T1 and T2 respectively. The x-axis is taken along the central line of the channel and y-axis perpendicular to it. The width of porous channel is 2h as shown in figure 4.1

   To derive the basic equations of the problem, we make the following assumptions.

   1. The flow in the x-direction is driven by a constant

      x x ; y y ; u u ; p

      h h U

      T T2

      T1 T2

      p ;

      u2

      pressure gradient.

   2. The flow is steady and fully developed with neg- ligible body forces so that all the physical quantities except the pressure are functions of y only.

In view of the above dimensionless quantities, the equations (1.1)-(1.8) take the following form. The asterisks (*) are neglected here after.

u

x 0

(1.9)

4u

y4

where

2 2u

b y2

• c2u Pa2

  (1.10)

  2 a2 2

  b ; c

  a ; a

  2

  2

  Da

  p

  ;

  Figure1: Physical model

  Under these assumptions the basic equations of the flow are given below.

  Basic Equations

  u

  P Re p

  x

  d2

  dy2 Ee Pr

  where

  du 2

  dy

  (1.11)

  x 0

  4 2 u

  4u 2u P

  y y K x

  2 2

  (1.1)

  (1.2)

  u = 0 at y = 1 (1.12)

  u=0 at y = – 1 (1.13)

  2u

  T u

  2 0

  (1.3)

  y2 0 at y = ±1 (1.14)

  y k y

  Boundary Conditions

  u = 0 at y = h (1.4)

  0

  1

  at y = – 1 (1.15)

  at y = 1 (1.16)

  u = 0 at y = – h (1.5)

  d2u

  dy2

  0 at y h

  (1.6)

  Solving equation (1.10) subject to the boundary conditions (1.12) (1.14) we obtain the velocity field as.

  T T2

  at y h

  (1.7)

  A y A y B y B y Pa (1.17)

  2

  u C e 1 C e 1 C e 1 C e 1

  T T1

  at y h

  (1.8)

  1 2 3 4 c2

  Where

  1.4 Non- dimensionalization of the flow quantities

  We introduce the following quantities in order to make

  C1 C2

  FeA1 eA1 e2A e2A

  the basic equations and boundary conditions dimensionless. 1 1

  C3 C4

  G e e2B

  B1 eB1

  • e2B

  ure 1.5. For fixed a = 2, = 0.2, Da V=ol0. 1.0I3ss,uew1e0,oDbesceermvbeer- 2012

  that for a given y, the velocity increases with the increasing

  P. This may be due to the increase in the Reynolds number

  Pa2b2

  F c2

  1 1

  G

  a2

  a2 b2

  Pa2

  c2

  of the porous layer.

  From equation 1.18, we have calculated temperature as a function of y fo different values of Darcy number Da, for fixed a = 1.2, = 0.01, Pr = 0.7, Ec = 0.01, Q = 5, and is shown in figure 1.6. We observe that for a given Da, the

  temperature increases with the increasing in y the tempera-

  A1

  b2

  b4 4c2 2

  2

  B1

  b2

  b4 4c2 2

  ture attains the maximum and minimum values at the lower and upper boundaries of the porous layer. For a given y, we notice that the temperature decreases with the increasing Darcy number Da.

  P Re p

  x

  a2 h

  , b2 a2 ,

  2 a2

  c

  Da

  The variation of temperature with y is calculated

  from equation 1.18 for different values of Prandtl number

  Solving equation (1.11) subject to the boundary conditions (1.15) (1.16) we obtain the temperature as.

  where

  1. Mass flow rate

     The dimensionaless mass flow rate of the flow of a couple stress fluid through the porous medium bounded by parallel plates is given by

     M 1udy

     1

     Pr and is shown in figure 1.7. We observe that for a given y ,

     the temperature increases with increasing Prandtl num-

     ber.

     The variation of temperature with y is calculated

     from equation 1.18 for different values of Eckert number Ec and is shown in figure 1.8. We observe that for a given y, the emperature increases with the increasing Eckert number.

     Fig. 1.2 Velocity profiles for different values of couple stress parameter a, with fixed values of

     0.2, Da 0.03, P 5,

     c a a

     c a

     a 2Pa2

     2 1 e e 2 3 e e

     a a c2

  2. Inferences

  From equation 1.17 we have calculated velocity as a function of y for different values of couple stress parame- ter a for fixed Da = 0.1, P = 2, and = 0.5 and is shown in figure 1.2. We observe that the velocity increases with the

  increasing in y in 0 y 0.5

  increasing y for 0.5 y 1.

  and it decreases with the The velocity attains the

  maximum value at the central line of the channel. For a given y, we notice that the velocity increases with increas- ing couple stress parameter a.

  The variation of velocity U with y is calculated from equ- ation 1.17 for different values of porosity and is shown in figure 1.3. for fixed a = 2, P = 5 and Da

  = 0.03. We observe that for a given y, the velocity increases with the increasing . This is due to increase in the poros- ity of the porous layer.

  The variation of velocity U with y is calculated from equation 1.17 for different values of Darcy number Da and is shown in figure 1.4. We observe that for a given y, the velocity increases with increasing Da. This may be due to the increase in the permeability of the porous layer.

  The variation of velocity U with y is calculated from equation 1.17 for different values of P and is shown in fig-

  Fig. 1.2 Velocity profiles for different values of couple stress parameter a, with fixed values of

  0.2, Da 0.03, P 5,

  Fig. 1.3: Velocity Profiles for different values of , with fixed values of a 2, Da 0.03, P 5,

  Fig. 1.6 : Temperature profiles for different values of Da, with fixed values of

  a 1.2, 0.001, P 0.7, E 0.1, Da 0.03, Q 5

  Fig. 1.4: Velocity profiles for different values of Da, with fixed values of a 2, 0.002, P 5,

  Fig.1.7: Temperature profiles for different values of Pr with Fixed values of

  a 1.2, 0.001, E 1, Da 0.03, Q 5

  F

  Fig. 1.5: Velocity profiles for different values of P with

  fixed values of

  a 2, 0.002, Da 0.03

  Fig. 1.8: Temperature profiles for different values of Ec with fixed values of

  a 1.2, 0.001, Da 0.03, Q 5

  1. s y 2 s

     1 cosh 2ay s1 2 cosh 2by

     4a 2 2 4b2

     Ec Pr s2

     y 2

     2

     s3

     s3 cosh(a a b

     2

     2 cosh(a b) y

• b) y

  • F1 y F2

a b

s s s

1 cosh 2a 1 2 cosh 2b

4a 2 2 4b2

2

s2

s3

a b2 s

cosh(a b)

3

2 cosh(a b)

a b

F 1 and F 1 E P H

1 2 2 2 c r

s 4a2c 2 s 4c 2b2 s 4abc c

1 1 2 3 3 1 3

1. 1. CHATURANI , P. and. PRALHAD. R.N. Biorhe- ology, 12, 283 (1981).

   2. CHATURANI, P. , and UPADTHYA. V.S. , Bi- orheology, 16, 109 (1979).

   3. CHATURANI. P, and KALONI. P.N. , Biorhe- ology, 13, 243 (1976).

1. UMAVATHI. J.C., and MALASHETTY. M.S., Int. J. Nonlinear Mech. 34, 1034 (1999).

2. VALANIS, K.C., and SUN. C.T, Biorheology, 6, 85 (1969).