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NUMERICAL SIMULATION OF POWER-LAW

FLUID FLOWS IN A POROUS CHANNEL

Jess Alfonso Puente Angulo ([email protected])

Maria Laura Martins-Costa ([email protected])

Heraldo da Costa Mattos ([email protected])

Laboratory of Theoretical and Applied Mechanics, Mechanical Engineering Graduate Program

Universidade Federal Fluminense

UFF

Presented by: Jess Alfonso Puente Angulo


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1. INTRODUCTION

PROBLEM: Power-law fluid flows in a porous channel limited by two

impermeable flat plates.

MODEL: mixture theory approach fluid saturated porous media

fluid and porous matrix treated as superimposed continuous constituents

of a binary mixture - each of them occupying its whole volume.

NUMERICAL APPROXIMATION: Runge-Kutta method coupled with ashooting strategy.


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1. INTRODUCTION

Time independent fluids

Shear-thinning

or

pseudoplastic

fluid

Bingham Plastic

Shear-

thickening or

dilatant fluid

Newtonian


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1. INTRODUCTION

Mixture Theory


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2. MECHANICAL MODEL

Mass and Momentum Equations

Porous channel

( ) 0F FF F F F F F F F

t t

vv v v T m g

Problem statement

Hypothesis

Rigid porous matrix

Incompressible fluid

Steady state fluid

Impermeable flat plates


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2. MECHANICAL MODEL

Constitutive model

2

2

=

I

I

-p

K

D

D

F

F

F F

F F

T I +D

m v

D

D .D

2

, , ,

I

I

1

DK

with

n K

and

D

F Fm v


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2. MECHANICAL MODEL

Constitutive model

0 0

1 0

K

K

F

F

m

m

1

a

a

b

K


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2. MECHANICAL MODEL

The generalized permeability function Kis then given by

-12 1

2

-12 1

2

4 3 12 1 3 6

or using the definition of

1 4 3 12 1 3 6

n n

n

n nan

a

nK Kn K

K

nK b n K

F

F

v

v

If n=0 then K K


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2. MECHANICAL MODEL

Permeability and Porosity behaviour

Identification of the parameter a

1 1

1 2

0.2, 0.3

1000 and 10000K D K D

Permeability versus porosity


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2. MECHANICAL MODEL

Neglecting gravity effects and considering fully developed steady

state horizontal flow

2

0

2 ( ) 0

0 on

F

nn

F F F F F

F

p

y H

v

I D D D v v

v

Defining


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2. MECHANICAL MODELThe governing equations for the flow are:

Two-point boundary-value problem

Analytical solution for maximum flow velocity

1

2 1

max

1

ndpw

dx

yat0

02

12 22

22

2

Hw

HyHwwdy

wd

dy

dwn

dx

dp nn

n


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2. MECHANICAL MODEL

Dimensionless variables

* * * *

2 1

max max

, , w = , pF nx y w p

x yL H w w

H


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2. MECHANICAL MODEL

Dimensionless expression for porosity and permeability

2 2

12 1

2 2

1

combining some equations and simplifying

63 2 1

1 4 3 1

n

nana

n a a

H

bb n

H n


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3. NUMERICAL PROCEDUREIn order to apply the Runge-Kutta method coupled with a shooting

strategy

Variables redefinition

1 2;dwz w zdy

The boundary value problem is transformed into an equivalent Initial

Value Problem

2 2 222 1 1

1

112

2

02 1

0

n n ndz dpz z zz at y Hdy n dx

withz at y Hdz

zdy


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3. NUMERICAL PROCEDUREThe two-point boundary-value problem is transformed into an equivalent

Initial Value Problem

22 22

2 1 1

1

212

2

02 1 ;

nn ndz dp

z z zz at y Hdy n dx

for H y H such thatz t at y Hdz

zdy

Find the root of a scalar function represented as ,where for a given t, the value of is the value of the variable at

point y = -H. Essentially this procedure is a shooting technique.

2( ) ( ; )t t z y H t ( )t 2z


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4. RESULTSFlow of a power-law fluid through a porous plane channel

Variation of the parameter with the permeability, for distinct power-law

indexes


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First estimative of the derivative for shear-thinning behavior


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Velocity profiles for shear-thinning behavior


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Dimensionless velocity profiles for shear-thinning behavior


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First estimative of the derivative for shear-thickening behavior


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First estimative of the derivative for shear-thickening behavior


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Velocity profiles for shear-thickening behavior


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Dimensionless velocity profiles for shear-thickening fluids (n0)


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Behavior of the maximum velocity in a porous channel.


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One-dimensional shear stress versus shear strain


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5. CONCLUSIONS

The proposed model is adequate to study power-law fluid flows for

different values of n.

The fourth-order Runge-Kutta method coupled with a shooting strategy

proved to be suitable to solve this kind of problems.

The fluid constituent velocity is strongly influenced by the shear-

thickening and by the shear stress increase.


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ACKNOWLEDGMENTS

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