Validation of boundary conditions between a porous mediumand a viscous fluidCitation for published version (APA):Lankveld, van, M. A. M. (1991). Validation of boundary conditions between a porous medium and a viscous fluid.(DCT rapporten; Vol. 1991.071). Technische Universiteit Eindhoven.

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Validation of boundary conditions between a porous medium and a viscous fluid

Ir. M.A.M. van Lankveld

i

WFW 91.071 August 1991 coaches: Dr. ir. C.W.J. Oomens

Prof. dr. ir. J.D. Janssen Eindhoven University of Technology

Contents

Summary Notation 1 IEt:t,rGdWtiÛIl

2 Literature 2.1 Early work 2.2 Generalized Darcy equation 2.3 Case study 2.4 Numerical study 2.5 Themodynamical approach 2.6 Exact solution for the slip velocity 2.7 Conclusions

3.1 The Finite Element Method 3.2 The numerical model 3.3 Calculation of the average velocities 3.4 Evaluation of the boundary conditions 3.5 Results 3.6 Discussion

3 Numerical validation

4 Conclusions References

ii iii 1 i

2 2 4 7 7 10 13 1 4 1 6

1 6

17

19 20 22 24 25 26

1

Summary

The objective of this report is to increase insight into the boundary conditions between a porous solid, filled with a fluid, on one side and a viscous fluid on the other side. In UiaTihïmkìl ûï syíîûvid joints the articular cartilage, an elastic porous soiici, is in contact with the synovial fluid, the lubrication fluid of the joint. To understand how lubrication works, it is necessary to know the boundary conditions between the porous mixture and the fluid. Literature shows several boundary conditions which are very similar. The kinematic boundary condition states the mixture velocity and the fluid velocity at the interface are equal, this is a so-called pseudo-no-slip boundary condition. The dynamic boundary condition states the shearing stress from the external fluid is divided over the mixture components according their volume fractions. The only boundary conditions that incorporate a deformable porous medium are those derived by Hou et al. (1989). FLH~~CXIIIQ~~ it has been shown that the exact value of the fluid velocity at the boundary depends on the location of that boundary. Only in a small region in the porous medium the Darcy equation has to be extended with viscous terms in order to meet the boundary conditions. Numerical calculations on microscale show it is possible to calculate the boundary Conditions. Comparison of these results with the boundary conditions formulated by Williams (1978), or Hou et al. show good agreement.

ii

Notation

Roman symbols

d

Da

F

g

h

k

K

1 n

P

P

Q

R

Re

t

U

D

S

<U>

U

U

U

X

Y

[ - I [ - I

thickness of the porous medium

Darcy number

function in Forchheimer equation

gravitational acceleration

height of channel

permeability

specific permeability

length

normal vector

pressure

pressure vector

total volume flux

drag coefficient

Reynolds number

tangent vector

velocity

Darcy velocity

slip velocity

averaged velocity

velocity vector

streamwise coordinate

transverse coordinate

k Da =

u h r Re =

iii

Greek symbols

o [ - I

Indices superscripts f

S

+

subscripts

n

nn

nt

t

X

Y

slip coefficient

coefficient

OL,,, o+rn:F. rn+n miclai n c i a i i i ~ a w

coefficient

dynamic viscosity

apparent viscosity

viscosity factor

inertia parameter

density

stress tensor

shear stress

porosity

volume fraction of the solid

basic functions

fluid solid on the fluid side, in the channel in the porous medium

h B'g

cs= [;&]i

normal to the boundary nonna1 to the boundary tangent to the boundary tangent to the boundary in the streamwise direction, tangent to the boundaq normal to the stream, normal to the boundary

1 Introduction

The present study is part of a research line on the mechanical behaviour of the human knee ioint. The reasons for this research and the final goal are described elsewhere (Schreppers: i99ij and wiii be omitted here. In the knee joint two eiemants play an important role in the lubrication: the articalar cartilage, covering the bones, and the synovial fluid which is the lubricating fluid of the knee. To understand how the joint works we will restrict ourselves first to the lubrication principles between cartilage and synovial fluid. Other elements in the knee joint that play an important role in load transmission and movement, such as the menisci, the capsule, the ligaments and the muscles, are not considered in this study.

Several lubrication models for synovial joints have been developed. To decide which of them is best, it is necessary to know the boundary conditions between the porous cartilage layer and the contacting synovial fluid.

This report presents a survey of the literature about the contact between a porous medium and a fluid (chapter 2) and a numerical study on microscale to validate some of the models from the literature (chapter 3).

1

2 Literature

2.1 Early work

AA e2dy study ^E honnday conditions between a porolls medium and a viscous fluid is performed by Beavers and Joseph (1967). Their analysis and experiments are focused on a rectangular channel with a porous wall in which fluid flows under the action of a pressure gradient, see figure 2.1.

1

T h

Y

X

Figure 2.1 Velocity profile for the rectilinear flow in a horizontal channel formed by a permeable lower wall and an impermeable upper wall. (Beavers and Joseph, 1967)

Beavers and Joseph assume the slip velocity is related to the tangential stress. Then the Poiseuille velocity in the channel and the Darcy velocity in the porous wall can be coupled through the following equation:

2

Here u is the slip velocity, that is the local averaged tangential velocity just outside the porous medium, u the velocity inside the porous wall, given by Darcy’s law, k the permeability of the porous medium [m2]. The slip coefficient a is a dimensionless constant depending on the geometry of the interstices. With this boundary condition an analytical solution for the slip velocity can be derived:

S

D

h where /3 is a dimensionless constant, /3 = - and q is the viscosity of the fluid.

By measuring the increased flow rate and comparing this to the flow rate in a channel with impermeable walls, the slip coefficient for two natural porous materials can be calculated. The conclusion is that the slip coefficient a is directly related to the average pore diameter. When the height of the channel is of the order of h = 4(2k), so the average size of the pores in the material is approximately equal to the height of the channel, the assumption of rectilinear flow breaks down and the slip condition is no longer valid.

&

Taylor (1971) and fiehardson (19’71) study the slip condition for a flow along a simplified model of a porous medium (see figure 2.2). Above this model, consisting of a flat disk with well-defined concentric grooves is a Couette flow. The pressure gradient is zero. The asymptotic solution for a small gap and a large gap gives the torque as a function of the geometric parameters. Also k and a can be expressed in the geometric parameters. Taylor conducts experiments with the idealized porous medium, and measures the shear stress of the fluid above the disk. Richardson calculates the flow parameters both inside and outside the porous medium. The value of a

from the calculations agree with the experimental results. The conclusion is that a is only a function of the geometric parameters. When there is no mean pressure gradient, as in this study, the values of a are meaningful even when the gap size g is smaller than the distance s between the planes of the porous model.

3

1

Figure 2.2 Model of the porous material and the gap with fluid flow above, (a) linear model; (b) circular model. (Taylor, 1971)

2.2 Generalized Darcy equation

In figure 2.1 it can be seen that the slip velocity us is not equal to the Darcy velocity uD. Darcy's law is an empirical law which also can be derived from micromechanics. The Navier-Stokes equations are solved in well defined media, neglecting the inertia terms. After averaging over a certain volume we obtain the Darcy equation:

which is only valid for statistically homogeneous, isotropic materials and low Reynolds numbers, where the Reynolds number is based QD the fluid flow inside the pores and on the pore size. For higher Reynolds numbers the inertia terms can ai^ longer be neglected. In the boundary layer the viscous term plays an important role for the transfer of shear stress. Several authors developed a higher order Darcy's law to account for these effects.

Forchheimer (1901) formulated an extension of Darcy's law with a second order term in the velocity to desribe intertia effects in the porous medium:

4

- Vp -2. - bu2=O (2.4)

where b depends both on the structure of the porous medium and on the Reynolds number.

Brinkman (1947) proposes an extension of Darcy's law in order to describe flow through a dense swarm of particles. This problem is related to the fluid flow through porous media and an adjoining fluid fíow:

- v p + q* v2u -?u = o

where q* is an apparent viscosity in the porous medium representing the contribution of the solid particles to the transport of the momentum in the fluid, considering the finite size of the inclusions. When the fluid transport is hindered by the solid, q* is smaller than q, else q is larger. In the first approximation q* is assumed equal to the viscosity q. The second term is the viscous stress tensor. The third term, the Darcy term, represents the distributed resistance of the solid inclusions. This extended Darcy's law is used by other authors to describe the flow in the boundary layer in a porous medium.

Saffman (1971) developes a generalized Darcy equation for non uniform flows in non homogeneous porous media. The boundaries are considered as discontinuous changes of properties of the porous media. For example at the boundary between a viscous fluid and a porous medium the porosity and the permeability change from the values of respectively unity and infinity to the actual values of the porous medium. Saffman gives a theoretical justification for the slip condition of Beavers and Joseph:

(2.11)

The exact value of the slip velocity depends on the location of the interface, if the interface is shifted through a distance s-&, the value of c is increased by s.

Dagan (1979) also developes a generalization of Darcy% law for non uniform flows and uses the Beavers and Joseph slip condition to demonstrate this generalized law.

5

Williams (1978) investigates the flow of an incompressible viscous fluid through a porous medium incorporating terms which account for capillary forces, drag forces and viscous shear effects. The equation of motion is:

(2.12) -Vp+2qhV(@Vu)-@- 'p r i~Vcp+be=pq(~u+Vu*u) D d

Tne first term is an indeterminant pressure, the second term a viscous term, the third term is the Darcy term, the fourth term is associated with Fick's law, where D is the diffusion coefficient, be is an external force, and the right hand side represents inertial forces. In order to determine the viscosity factor A, a Poiseuille flow over a semi-infinite porous medium is considered. This viscosity factor h only depends on the solid parameters, and is only significant near a boundary. Two assumptions with regard to the boundary conditions are made. Firstly the boundary condition for the fluid velocity becomes via an averaging procedure:

u - y 4 + - (P uy=o- (2.13)

Secondly, the fluid at the boundary takes up a part of the shearing stress from the external

stream (t = cp q - and this must equal the force exerted by the interior fluid at the

boundary (t = h @ q - ), therefore:

du T ly4+ = va% Iy=o-

(2.14)

With a Poiseuille flow above the porous medium and an extended Darcy flow in the porous medium Williams is able to derive the Beavers and Joseph slip condition. The square root of the viscosity factor h turns out to be equal to the slip coefficient a. It is possible to measure h using a simple viscometer. The upper wall has a velocity U and there is no pressure gradient (see figure 2.3). The solution for the fluid velocity in the channel yields the shear force per unit area to drive the shearing:

1 with 8 = /4(u). The measurement of the shear force then yields A.

6

Figure 2.3 Measurement of the viscosity factor h. (Williams, 1978)

2.3 Casestudy

Beavers and Joseph assume the porous wall to be semi-infinite. Rudraiah (1985) studies the same problem, however with a wall of finite thickness. At one side the wall is in contact with a Newtonian fluid. At the other side the wall is either bounded by an impervious solid wall or by a static fluid. The fluid in the channel is described by the Navier-Stokes equation, the flow in the porous wall by the Brinkman equation. The boundary conditions are as formulated by Williams. This yields a solution for the fluid velocity in the channel and in the porous wall. Analogous to Beavers and Joseph, Rudraiah formulates an analytical solution for the slip velocity. One of the main conclusions is that the boundary layer is very thin, depending on the value of A and @, and can be described by the Brinkman equation. Outside this layer the velocity is almost constant and approaches the Darcy velocity.

2.4 Numerical study

Larson and Higdon (1986 and 1987) examine the microscopic flow in 2-dimensional geometries that are representative for real porous media. First they study the flow through infimite regular lattices of solid inclusions under the action of a pressure gradient. The p q c x e is io detemine the permeability as a function of several geometric parameters concerning the shape of the inclusions and their orientation relative to each other. The lattice is divided in cells, each cell in the porous medium contains one inclusion. The equation for the velocity is solved as an interior flow problem in each cell and the boundary conditions are passed from cell to cell.

7

U - -

Figure 2.4 Streamlines for pressure driven flow through infinite square lattices of circular cylinders. (a) porosity cp = 0.9; (b) cp = 0.6; (c) cp = 0.6, pressure gradient directed 4" from horizontal; (d) cp = 0.6, pressure gradient directed at 45" to horizontal. (Larson and Higdon, 1987)

The main goal of their study however is to describe the flow mear a surface of a porous medium. They consider a simple shear flow over a semi-infinite lattice to determine the slip velocity and the thickness of the boundary layer. Above the porous medium are only pure-fluid cells.

To determine the slip velocity first the nominal interface of the medium has eo be defined, for example the plane intersecting the inclusion centres in the top row of the Izttice. A layer of height h is defined and the total volume flux in this layer is used to define the slip velocity.

8

flow over Q, = 0.97;

The macroscopic prediction of the volume flux above the interface is:

(2.16)

9

Brinkman equation is:

Q- = us & (1 - e -wk)

By means of a parameter study

(2.18)

Larson and Higdon conclude there is a noticeable disparity 2, +.-.na- L L a C---- d-c-:L:--- A_--- L I A -_-A- -..I-- I-2 uetw G G i l L I K 1WU UClIIIIllUliU, GALGpl dl CAllelIlCl)’ Ill&l pûrosiiies. If the interface is chosen as the plane tangent to the top row of the inclusions, the slip velocity, as calculated from the flux above the interface, will be much higher. It can be concluded the slip velocity is very sensitive to the position of the interface, as was also pointed out by Saffman. Therefore it is not possible to define a consistent value for the slip coefficient a.

The strength of the flow field decays rapidly below the interface, and below the second row of inclusions the velocity is negligible. This would mean the boundary layer in the porous medium is very thin. Although Larson and Higdon based the calculation of the slip velocity on macroscopic quantities, they state that macroscopic models are inadequate to capture detailed flow fields.

2.5 Thermodynamical approach

In all previous studies the solid was considered as rigid. Hou et al. (1989) derive the boundary conditions considering the balance laws for mass, momentum and energy for a deformable mixture containing a surface of discontinuity. The mixture velocity is defined by Hou as the sum of the constituent velocities, weighted by their volume fractions. The normal component of this mixture velocity is according to the balance of mass continuous across the boundary: [@ uf + QF us] * n = O, where @ and QF

are the volume fractions of respectively the fluid and the solid, n is the vector normal to the boundary and [ a 1 is the difference in the value of parameter a at both sides of the boundary, 1 a ]I = a+ - a-. In order to account for the viscous interactions at the interface, the following kinematic boundary condition is proposed: [@ uf + p us] * z = O, where z is the tangent vector to the boundary. Hou et al. call this a ”pseudo-no-slip” boundary coriditisn, because it permits different velocities for the Buisi and the did at the interface, with a continuous mixture velocity at both sides of the boundary. These two equations lead to the kinematic boundary condition:

[@ uf + QF us] = o (2.19)

10

The linear momentum jump condition and the energy jump condition are:

[os + of - pf uf(uf - us)]

[OW + duf - pf (ef + 4 uf.uf)(uf - US) - hs - hfll * n = O

n = 0

with e acd h tespective!y the internu! energy density 2nd helt flzx. In case of a flow tangential to the boundary this reduces to:

(2.20)

(2.21)

Using the kinematic boundary condition (equation 2.19) these jump conditions (equations 2.21.a, b and c) result in the boundary conditions for the tangential stress between two mixtures or between a mixture and a pure fluid (@ = 1):

[ - ] = o 4 t @

(2.22)

du du r q . where = q* In a pure fluid is 4, =

Equation (2.22) implies that the shear stress on the fluid on either side of the boundary is proportional to the porosity of the corresponding side. The boundary condition for the normal stress is derived in a similar manner; this results in:

[&]=o Qi

(2.23)

Equations (2.22) and (2.23) yield the equations for the dynamic boundary conditions betweer, a mixtiare and a fliiid channel (ir, the channel is sf equal to 1):

(2.24) n c F = p n &

where o+ is the stress tensor of the fluid in the channel.

11

This is consistent with the boundary conditions Williams proposed, provided that

Hou et al. use the Beavers and Joseph problem to demonstrate the boundary conditions. They derive two expressions for the fluid velocity, one in the channel, based on the Navier-Stokes equation and the boundary conditions, the other in the porous wall, based on the Brinkman equation and the boundary conditions. These velocities depend on the following - parameters:

q* = (p2 h r].

-0.a7 Y

-o.w. f;?

-0.75.

f2 where d is the thickness of the porous wall and R is the drag coefficient, R = v. Even when the fluid in the channel and the interstitial fluid are the same, q and q* may be different. For small values of 6 (6 e 1) the thickness of the boundary layer is very small, and the slip velocity approaches the Darcy velocity. For larger values of 6 (a z l), i.e. for larger permeabilities, the fluid flow in the porous wall is no longer according to Darcy's law, but to the higher order Brinkman equation.

' - - 6 2 4 . 0 -.

(€nhanc.d Ffvid ,' Transport) . .' '&o. I

e - -__.--- -0.6L-- ! ! ! ---;:*e , -1 .o

_ - - - - _.__---- Porous lïa I I _ - _ _ - - - - - - 0.0 0.1 0.1 O 3 0.4 O S 0.6

f2

11 Figure 2.6 Fluid flux profile in the channel and in the porous walk, with al. 1989)

= 2. (HOU et

Studying a Couette flow over a porous medium Hou et al. derive the Beavers and Joseph slip condition from their own boundary conditions.

Hou states that in biological soft tissues there may be an effect of the external fluid on the internal fluid, due to the fact of a larger viscosity of the fluid outside the medium, despite the low permeability.

2.6 Exact solution for the slip velocity

Vafai and Kim (1990) present an exact solution for the interface region between a porous medium and a fluid. The fluid is described by the Navier-Stokes equation, the porous medium by a combined Brinkman-Forchheimer equation:

(2.25)

The second term in this equation is the Brinkman term, which can be compared to the viscous force term in the Navier-Stokes equation. The last term is an inertia term, as formulated by Forchheimer. F is a function which depends on the Reynolds number and the microstructure of the porous medium. The porous medium is divided in two layers: the

inner part, where the Brinkman term (q' n) can be neglected and a boundary layer where the full Brinkman-Forchheimer equation is valid. The momentum equations for these regions and for the fluid are matched through the boundary conditions between the fluid and the porous medium:

d2u Y

u = u y&+ y'o-

(2.26)

The velocities in equation (2.26) are already averaged over the fluid. Notice the difference with for example Williams or Hou concerning these equations. Vafai and Kim combine the equations for the fluid layer and the inner part of the porous medium. After integrating u from y to infinity (note: not from O to y) they find the slip velocity :

__z_)_ Re A + Da-1 - us = (us - 1) + 2) + (Da)-l

with the following parameters:

13

Re = Reynolds number based on the height of the Buid channel above the porous medium and the velocity in the inner part of the porous medium;

F Q, h.

& A = the inertia parameter, A =

Da = the Darcy number, Da = k 1 = 'isz'

2.7 Conclusions

The main conclusion is that the boundary layer in the porous medium is very thin, dependent on variables such as k, a and the height of the channel. In the boundary layer the velocity can be described by the Brinkman equation. Outside this layer the velocity can be approached by the Darcy velocity. If the boundary layer is very thin, the thickness of the porous medium is not very important. As shown by Taylor and Richardson, and îater by Williams and Rudraiah, the slip coefficient a or viscosity factor h can be measured using simple viscometers. Saffman as well as Larson and Higdon pointed out that the slip velocity depends on the location of the interface, and it is difficult to define one specific value of the slip velocity.

14

Table 2.1 Summary of the boundary conditions

Beavers and Joseph

Saffman

Williams

Larson and Higdon

Hou et al.

Vafai and Kim

du u S =c%? Iy.o u - Y'O+ - ~ u y , o -

1 h

u s h + 4 f h 2 = $ $ u d y d x

u s & ( 1 - e -'"k, = $$ u dy dx

o 0 1 h

O 0

[@uf+ p u s ] = o

du du n of= @n + q*

u = u y=o+ y=o-

du du J? Iy.o+ = ay ly=o-

,e' Re A + Da-1 - u = (us - 1) + 2) + (Da)-1 S

15

3 Numerical validation

To determine the boundary conditions between a porous medium and a viscous fluid it is possible to study the fluid flow on microscale. Through an averaging method the velocity in a certain point or on a plane can be determined.

3.1 The Finite Element Method

The fluid velocities can be calculated with the help of a numerical method. For the calculations in this section the finite element package SEPRAN is used. The Navier-Stokes equations in dimensionless forms are defined in domain i2 with boundaries r:

- V P + = v 2 u - u ~ v u = o

div u = O

These differential equations are discretized using the Galerkin method. This leads to the set of equations:

s u + N(U) u + L ~ P = O

L u = O

where SU is the viscous term, N(u)u the convective term, L$ the pressure gradient. Lu is the velocity divergence term. The vector u contains the velocity and p the pressure unknowns. To reduce the number of unk~owos a penalty fineiion method is applied and the discretized continuity equation (3.2) is replaced by:

1 (3.3) p = - B-1 L M

&

where D is the pressure mass matrix. If E is small, the penalty function system results in the

original system. If E is too small however, the matrix might become singular.

The modified Crouzeix-Raviart element is used. This has six nodes for the velocity and one for the pressure. The accuracy for the velocity is O(h3), for the pressure O(h2), with h being a characteristic dimension of the discretization.

3.2 The numerical model

The form of the numerical model is shown in figure 3.1.

Fluid flows under the action of a pressure gradient through a rectangular channel, with height hl = 20 [mm], with a Poiseuille velocity. A part of one wall of the channel consists of a porous medium with height h2 = 10 [mm]. The geometry can be considered as a 2-dimensional geometry.

In order to keep the computing time low we will only calculate the velocity in a small part of the channel, namely a section in the fluid with a height of 4.375 [mm]. Since the boundary layer is very small, only one row of inclusions is modelled. The total length of the section is 7 [mm], including an outstream of 1.875 [mm] (figures 3.2 and 3.3).

I \ I

8 . 0 fluid channel

porous medium

Figure 3.1 Macroscopic geometry.

17

4 1 / 12

i Figure 3.2 Microscopic model.

On each boundary (figure 3.2) two boundary conditions are required, which are as follows:

1: no-slip condition: u, = uy = o 2: fully developed outstream: a, = P, uy = o

on the inclusions: no-slip condition:

3: velocity according to parabolic profile: 4:

u, = constant, uy = O fully developed parabolic profile (Poiseuille velocity): u, = %(Y), uy = 0

u, = uy = o

The boundary conditions for boundary 3 have to be checked. The influence of the slip velocity is assumed to be restricted to a small region in the channel. By moving the boundary it can be checked if this assumption is correct.

Between and just above the inclusions the mesh is locally refined (figure 3.3).

4 375

OOOO

-0 625 o O00

Figure 3.3 The mesh.

5 125 7000

3.3 Calculation of the average velocities

To calculate an average velocity there are a few possibilities:

1. Averaging on a plane, which reduces in a 2-dimensional geometry to averaging on a line: 2. Averaging in a representative volume element (RVE), which reduces in a 2-dimensional geometry to averaging in an area element. The size of the element must be large enough to keep the averaged properties of the material in the element constant, in case the RVE is moved slightly or the size is changed by a small value. If the RVE is too large, macroscopic changes in the medium can not be determined. The properties of an RVE are projected to the centroid. In a continuum the centroids for the different phases are located in one point. Near a boundary however the centre of gravity of the fluid will not coincide with the centre of gravity of the solid, because the solid is only present in a small part of the RVE. Therefore the surface of the porous material, which is defined by the solid, will not coincide with the fluid boundary.

To avoid this problem, and the problem of defining a size of the RVE, the first method is chosen. Since the velocity near the beginning and the end of the geometry is influenced by the instream and the outstream, we will use only the part of the mesh where the velocity is stable instead of averaging over the entire mesh. The following relation to calculate the averaged velocity is used:

.f u(x) dx = 12 n u(x) 'u>=T1 n

(3.4)

where 1 is chosen large enough so the averaged value of the velocity <u> is no longer a function of x, and n is the number of points used to calculate the averaged velocity.

If the plane intersects the inclusions, two velocities can be calculated:

i. The velocity is only averaged in the fluid section:

19

where li is the ith fluid section between the inclusions. This results in the pure fluid velocity.

2. The velocity is averaged over an entire plane. On the inclusions the fluid velocity is zero, therefore the averaged velocity will be smaller. This velocity is analogous to the Darcy velocity and will be used in the next sections.

3.4 Evduatiûns of the boundary conditions

Two flows are considered, Re = 300 and Re = 1000. For these two problems the velocity and pressure fields are calculated with SEPRAN. The velocity along a few lines parallel to the surface, that is on y = O, 0.1, 0.5 and 1.0, is plotted in figure 3.4. The surface is the plane tangent to the inclusions. It is clear the averaged velocity reaches a constant value after the first inclusion on the surface, which is defined on y = O. Higher in the channel the velocity hardly changes. First a small dip occurs, due to the transition from a smooth wall to the porous wall. For higher y-values the dip is located further away from the transition. After the dip a small increase of the velocity occurs.

. . . . ! . . . . I . . . . ! . . . ! . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Q.1 4.350

2.900

1450

o O00 O 0 1 4 2 8 4 1 5 6 7 0 O 0 1 4 2 8 4 2 5 6 70

Figure 3.4 Velocity along lines parallel to the surface: y = O, 8.1, Q.5, 1.0; 'Re = PûûO. (a) tangent velocity ux; (b) normal velocity uy The horizontal lines in (a) represent the Poiseuille velocity.

For higher y-values the velocity falls back to lower values at the end of the section. This phenomena occurs because of the form of the outstream. The total height of the channnel is larger and the overall velocity will be lower. The boundary however is moved from y = O on the instream to y = - 0.625 on the outstream. This means that u y a increases from zero at the instream to a positive value at the outstream.

Tn 2VQid the in,creae nf uy-û 2nd the decre2se Gf the t^td VeIGCity pf ik ut ?he oiits?re2m,

a different geometry is used. (figure 3.5)

Calculations with a shifted upper boundary show no differences. The boundary is far enough from the region we are interested in. The influence of the boundary condition, i.e. a constant velocity, is therefore small.

Figure 3.5 Geometry with a narrow outstream.

From the first calculations the geometry with only one row of inclusions seems a good choice, since the velocity below the inclusions is very small compared to the velocity above the inclusions.

It is clear from figure 3.6 that the flow above the last inclusions is influenced by the outstream. The region where the averaged flow is caicuiated is therefore smaiier.

21

0.2451. . . . ! . . . . ! . . . ! . . . . !. . . ! . . . . : . . . ! . . . . ! , , . . ! , . . . f

X Imml -+ x Imml -4

Figure 3.6 Velocity along lines parallel to the surface: y = O, 0.1, 0.5, 1.0; Re = 1000. (a) tangent velocity urn* (b) normal velocity uy. The horizontal lines in (a) represent the Poiseuille velocity.

3.5 Results

With the adapted geometry the flow for three different Reynolds numbers is studied: Re = 500, 1000, 1500. The velocities along lines with constant y-values are of the same form for all Reynolds numbers, the result for Re = 1500 are shown in figure 3.7.

The tangent velocities are averaged over the interval 2.25 s x sa 3.5. This leads io an averaged value for the velocities at the several heights. If these averaged velocities are plotted and compared to the Poiseuille velocity in the channel, it can be seen that the influence of the porous wall on the flow pattern extends for only a small height into the fluid flow. (figure 3.8) The velocity below the surface is also calulated. Since averaging has taken place over the entire area, this results into a Darcy velocity.

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11.50 cm' . . . : . . . . : . . ! . . . . : . . . .

10.35 0.5

4.600

0.2 f 3.450

0.1 1.150

0.W 0.0 1.4 2 8 4.2 5.6 7.0

Figure 3.7 Tangent velocity u, along lines parallel to the suvace: y = O, 0.2, 0.2, 0.3, 0.4, 0.5 for We = 1580.

With figure 3.8 it is possible to determine the viscosity factor h as defined by Williams (equation 2.14) and the apparent viscosity as defined by Hou et al. (equation 2.25).

The ratio between - is the same for all Reynolds numbers.

Since the value of the permeability is not known, it is not possible to compare these results with the boundary conditions of other authors.

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Re = 500 - _ _ - -

Re = 1000 Re = 1500

12

10

8

6

4

2

O -0.1 o 0.00 0.10 0.20

height in channel [mml

0.30 0.40 0.50

Figure 3.8 Comparison of the slip velocity (dashed kines) and the Foiseuille velocis, (drawn lines) above the porous medium for Re = 500, Re = 1000 and Re = 1500.

3.6 Discussion

Similar calculations were done by Reijnen (1991). Instead of studying the flow above a porous wall she considered the flow above a solid wall with cavities. A parameter study shows that the ratio between length and depth of the cavities is important, as well as the ratio between the height of the channel and the depth of the cavities. The main flow is influenced not very much by the presence of the cavities, there is a small difference between a channel with and without cavities. Inside a square cavity a recirculation area is present. The vortex is not influenced by the depth of the cavity, as long as the length of the cavity is smaller than the depth.

The coarseness of the mesh has a large influence on the accuracy of the calculations of the velocity in the cavities. Our main goal of interest however is the flow above the wall, therefore we will not look at the flow between the inclusions in much detail. This means it is not possible to calculate the Darcy velocity very accurately. Because the permeability is not known, a comparison with the analytical studies described in chapter 2 is not possible.

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4 Conclusions

Beavers and Joseph were the first to derive the boundary condition between a porous medium and a viscous fluid. This boundary condition has been validated by many other iuthûrs. Their stiUy shûw-eû a diffeïence between the slip veiocity and the Darcy velocity. If the flow in the porous medium near the boundary is described by an extended Darcy equation, for example the Brinkman equation, it is possible to match the fluid velocity in the porous medium and in the fluid channel.

Saffman concluded that the exact value of the slip velocity depends on the location of the interface. The surface of the porous medium is usually attached to the solid phase, but this never is a perfectly smooth surface. Therefore it is necessary to define a surface through or tangent to the top row of inclusions or grains of the porous medium. Rudraiah found that the boundary layer in the porous medium is very thin, and that the fluid velocity there can be described by the Brinkman equation. Larson and Higdon found two different values of the slip velocity, depending on the side from which the boundary is approached.

The numerical study with a model on microscale gives the slip velocity as a linear function of the Reynolds numbers in the considered region. The ratio of shear stress above and below the surface is constant for the several Reynolds numbers. Comparing the results with the boundary conditions of Williams or Hou we can see h and q* are constant.

So far the only boundary conditions available for a deformable mixture are those derived by Hou et al. The conditions from Hou agree well with the conditions of Williams for a rigid solid. That is why in further studies on knee joint lubrication the Hou boundary conditions will be used.

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References

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Brinkman H.C. A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles. Applied Scientific Research, vol. Al, pp. 27-34, 1949

Cuvelier C., &gal A., van Steenhoven AA. Finite Element Methods and Navier-Stokes Equations. D. Reide1 Publishing Company, Dordrecht, the Netherlands, 1986

Dagan G. The generalization of Darcy's law for nonuniform flows. Water Resources Research, vol. 15, no. 1, pp. 1-7, February 1979

Forchheimer P. Wasserbewegung durch Boden. Zeitschr. des Ver. deutsch. Ing., 45, pp. 1782-1788, 1901

Hou J.S., Holmes M.H., Lai W.M., Mow V.C. Boundary conditions at the cartilage-synovial fluid interface for joint lubrication and theoretical verifications. Journal of Biomechanical Engineering, vol. 111, pp. 78-87, February 1989

Larson R.E., Higdon J.J.L. Microscopic flow near the surface of two-dimensional porous media. Part 1. Axial flow. Journal of Fluid Mechanics, vol. 166, pp. 449-472, 1986

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Reijnen 9. Numeriek onderzoek naar de invloed van wandinhomogeniteiten op het massatransport naar en door een doorlaatbare wand. Internal report WFW 91.019, Eindhoven University of Technology, 1991

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Richardson S. A model for the boundary condition of a porous material. Part 2. Journal of Fluid Mechanics, vol. 49, part 2, pp. 327-336, 1971

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Scheppers GJ.MA. Force tïaïìsmissicsn in tiie iibio-femoral contact complex. Ph.D.-thesis Eindhoven University of Technology, the Netherlands, 1991

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Vafai KI, Kim S.J. Fluid mechanics of the interface region between a porous medium and a fluid layer - an exact solution. International Journal of Heat and Fluid Flow, vol. 11, no. 3, pp. 254-256, September 1990

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porous medium. Quarterly of Applied Mathematics, pp. 255-267, October 1978

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