Governing Equations and Const i tut ive Relat ionships for Biolog i ca l ly Relevant Fluids

While the balance laws are no different from the equations derived in chapter 3, it is

sometimes useful to consider alternative forms. In fluid mechanics, the balances of mass,

linear momentum, and angular momentum are usually written,

,

or, in indicial form,

.

For the first law of thermodynamics and the Clausius-Duhem form of the second law of

thermodynamics, we have,

!ρ + ρ∇iv =

∂ρ∂t

+∇ρiv + ρ∇iv =∂ρ∂t

+∇iρv = 0

ρ !v = ρ ∂v

∂t+∂v∂x

v⎛⎝⎜

⎞⎠⎟= divT + ρb

T = TT

!ρ + ρ

∂vi

∂xi

=∂ρ∂t

+∂ρ∂xi

vi + ρ∂vi

∂xi

=∂ρ∂t

+∂ρvi

∂xi

= 0

ρ !vi = ρ

∂vi

∂t+∂vi

∂x j

v j

⎛

⎝⎜

⎞

⎠⎟ =

∂Tij

∂x j

+ ρbi

Tij = Tji

ρ !ε = ρ ∂ε

∂t+∇εiv⎛

⎝⎜⎞⎠⎟= TiD + ρr − divq

2

,

which may also be represented in indicial form,

,

ρ0 !η = ρ0

∂η∂t

+∇ηiv⎛⎝⎜

⎞⎠⎟=ρ0rθ

−Divq0θ

ρ !ε = ρ ∂ε

∂t+∂ε∂xi

vi⎛⎝⎜

⎞⎠⎟= TijDij + ρr − ∂qi

∂xi

ρ0 !η = ρ0

∂η∂t

+∂η∂xi

vi⎛⎝⎜

⎞⎠⎟=ρ0rθ

−1θ∂q0i∂Xi

3

Newtonian Flows and the Navier-Stokes Equations In fluid mechanics, many texts derive the balance laws with the inherent assumption

that the flow is Newtonian, making it difficult for students to derive the governing equations

for more complicated rheological behaviors. And, while Newtonian flows are important,

there are numerous examples in biomedical engineering where the flows are distinctly non-

Newtonian. These include, but are not limited to turbulence in the upper airways, mucus and

bile flows, and flow through the gut (Table 8.1). We will start by considering incompressible

and compressible Newtonian flows and illustrate the process for obtaining the Navier-Stokes

equations. Subsequently, we will consider nonlinear fluids and utilize the same process to

derive the governing equations.

Table 8.1 – Examples of biological fluid flows and their characteristics

Flow Flow Regime Linear vs.

Nonlinear

Compressibility

Blood in ?? mm arteries Laminar Nonlinear Incompressible

Blood in ?? mm arteries Laminar Linear Incompressibility

Plasma skimming layer Laminar Linear Incompressible

Upper airways Turbulent Nonlinear Compressible

Lower airways Laminar Linear Slightly compressible

Sweat/tears Laminar Linear Incompressible

Mucus Laminar Nonlinear Incompressible

Bile Laminar Nonlinear Incompressible

Synovial fluid Laminar Linear Incompressible

Bioreactors Laminar or turbulent Linear Incompressible

Flow through the gut Laminar Nonlinear Incompressible

Aqueous and vitreous

humors of the eye

Laminar Linear Incompressible

Urine Laminar Linear Incompressible

Canaliculi Laminar Linear Incompressible

Interstitial flows Porous media Darcy flows Incompressible at

microscale, compressible

at macroscale

4

For incompressible, Newtonian fluids, the constitutive law has the form,

T = − pI + 2µD

where p is the fluid pressure and µ is the viscosity. Clearly this equation satisfies the required

invariance under superposed rigid body motions. Inserting it into our governing differential

equations, we note that the Balance of Mass equation is unchanged. But, introducing this

equation into the Balance of Linear Momentum yields,

ρ ∂v

∂t+∂v∂x

v⎛⎝⎜

⎞⎠⎟= −∇p + 2µdiv D( ) + ρb ,

or, in component form,

ρ

∂vi

∂t+∂vi

∂x j

v j

⎛

⎝⎜

⎞

⎠⎟ = −

∂p∂xi

+ µ∂2vi

∂x j2 +

∂2v j

∂xi∂x j

⎛

⎝⎜

⎞

⎠⎟ + ρbi .

The Balance of Angular Momentum is identically satisfied. THERMO

If we then decide to consider a constitutive law that allows for compressibility of the fluid,

we might choose something like,

,

where p is the fluid pressure, λ is the second viscosity coefficient (sometimes called the

dilatational viscosity or the volume viscosity), and µ is the dynamic viscosity. To

demonstrate that this satisfies invariance requirements, we first recall the definition of a

superposed rigid body motion,

T = − pI + λtr D( )I + 2µD

5

While Newtonian fluids can be used to model a wide range of biomedically relevant

fluids including, but not limited to, plasma, tears, cell culture medium, air in the lower

airways, and lymph. But, to describe flows in the upper airways, and the flows of mucus and

bile, and flow through the gut, it is necessary to introduce more complicated constitutive

laws. First, we will consider the potential variables that can be included in the constitutive

laws and then examine the possible ways that they can be combined.

Bingham Fluids

Need to check the tau term. Is it invariant? (Actually a discontinuous

change at tau_0). How do you characterize the objectivity of that? Should be okay

Power Law Fluids

others??

T = − pI + τ0 + 2µD

T = − pI + kDn

6

Planar Couette Flow Couette type flows are generated by one surface moving past another with a layer of

fluid in between (Fig. ??). One of the best biological applications is the lubrication layer of

fluid that separates two diarthroidal joints in the absence of actual contact between the two

joints. While the actual process of cartilage lubrication is considerably more complicated,

Couette flow provides a good place to start.

Figure  ??  (A)  When  the  knee  is  extending,  the  lower  leg  moves  relative  to  the  femur  and  the  two   cartilage   surfaces   move   past   each   other,   creating   a   thin   layer   of   fluid   lubrication  (exaggerated   in   the   exploded   view).  While   the   actual  mechanisms   of   joint   lubrication   are  considerably   more   complicated,   and   the   joint   exhibits   some   level   of   curvature,   planar  Couette   flow   provides   an   approximation   for   flows   between   two   cartilage   surfaces   (B).  Adapted  from  an  image  initially  created  by  Patrick  J.  Lynch,  medical  illustrator.    

7

For planar flows of this type, we begin by describing the velocity field using

Cartesian coordinates,

,

where forms the usual fixed orthonormal basis. The components of the acceleration

vector can then be written by employing the chain rule for differentiation,

,

and the rate of deformation tensor has the general form,

.

For planar Couette flow it is usually assumed that the flow has no velocity components in

the E2 and E3 directions, there is no variation in the E3 direction (i.e. partial derivatives with

respect to x3 are negligible) and that the flow is steady. The latter assumption implies that the

partial derivatives with respect to time are identically zero. One may certainly argue that,

over the time span that most joint loading problems occur, the assumption of steady state

flow simply cannot apply. But it is useful starting point and will at least provide us with

v = v1E1 + v2E2 + v3E3

Ei{ }

a1 =∂v1

∂t+∂v1

∂x1

dx1

dt+∂v1

∂x2

dx2

dt+∂v1

∂x3

dx3

dt

a2 =∂v2

∂t+∂v2

∂x1

dx1

dt+∂v2

∂x2

dx2

dt+∂v2

∂x3

dx3

dt

a3 =∂v3

∂t+∂v3

∂x1

dx1

dt+∂v3

∂x2

dx2

dt+∂v3

∂x3

dx3

dt

Dij =12

2∂v1

∂x1

∂v1

∂x2

+∂v2

∂x1

∂v1

∂x3

+∂v3

∂x1

∂v2

∂x1

+∂v1

∂x2

2∂v2

∂x2

∂v2

∂x3

+∂v3

∂x2

∂v3

∂x1

+∂v1

∂x3

∂v3

∂x2

+∂v2

∂x3

2∂v3

∂x3

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥⎥

8

approximations of the actual flow. Given these assumptions, the acceleration field simplifies

to,

and the rate of deformation tensor can be written as,

9

Once we have simplified the kinematics as far as we dare, the next step is to apply the balance of mass equation which, in Cartesian coordinates, takes the form,

,

and when we employ the previously discussed kinematic assumptions it becomes,

!ρ + ρdivv = !ρ + ρ

∂v1

∂x1

+∂v2

∂x2

+∂v3

∂x3

⎛

⎝⎜⎞

⎠⎟= 0

10

The next step is to introduce our constitutive law. For all the archetypal flows

discussed in this section, we will utilize the relationship for an incompressible Newtonian

fluid, , where µ is the fluid viscosity. The components of the stress tensor

can then be written as,

where p is the hydrostatic pressure. It is now possible to incorporate the results so far into

the balance of linear momentum, ,

,

T = − pI + 2µD

ρ !v = divT + ρb

0 =∂T11

∂x1

+∂T12

∂x2

+∂T13

∂x3

= −∂p∂x1

+ µ∂2v1

∂x22

0 =∂T21

∂x1

+∂T22

∂x2

+∂T23

∂x3

= µ∂2v1

∂x1∂x2

−∂p∂x2

0 =∂T31

∂x1

+∂T32

∂x2

+∂T33

∂x3

= −∂p∂x3

11

It is important to note that, at this stage, it is usually assumed that the velocity of the moving surface propagates into the body of the liquid because of its viscosity, generating the flow

field without any external pressure gradient, (i.e. ). Provided that this is a valid

assumption, the pressure gradient in every direction is zero and the total fluid pressure must remain constant. The remaining equation can then be integrated directly with respect to x2,

,

where A1 and A2 are constants of integration that can be solved by applying the appropriate

boundary conditions,

,

which ultimately leads to,

.

This result is interesting for a few reasons. First, the velocity in the E1 direction is a linear

function of the x2 coordinate. Second, even though the flow field initially propagates due to

viscous interactions, the fluid viscosity is not part of the final solution. It should be noted,

however, that viscosity will play a role in a number of different aspects of the flow as it

develops including the amount of force required to accelerate the moving surface and how

quickly the flow field reaches steady state.

∂p∂x1

= 0

∂2v1

∂x22 = 0

∂v1

∂x2

= A1

v1 = A1x2 + A2

v1 x2 = 0( ) = 0

v1 x2 = h( ) =U0

v1 =

x2

hU0

12

Planar Poiseuille Flow Pressure-driven flow between two plates has two important biomedical applications.

The first is flow through the alveolar capillaries where the blood vessels appear to be

stretched around the alveolus and form two nearly parallel plates through which advected

transport of oxygen and carbon dioxide occurs almost continuously. The second application

is the parallel plate flow chamber, introduced in chapter ??. These devices are commonly

used to measure cell adhesion and to evaluate the effects of fluid shear on the short-term

response of adherent and suspended cells.

A

Figure ?? Parallel

plate flow chambers

were introduced in

chapter 1 (A) and

they are used to

measure cell adhesion

and characterize the

cellular response to

fluid shear stress.

Planar Poiseuille flow

(B) approximates flow

through parallel plate

flow chambers and

the capillaries

surrounding alveoli.

B

,

v = v1E1 + v2E2 + v3E3

13

where forms the usual fixed orthonormal basis. The general components of the

acceleration vector are,

,

and the rate of deformation tensor has the general form,

.

Ei{ }

a1 =∂v1

∂t+∂v1

∂x1

dx1

dt+∂v1

∂x2

dx2

dt+∂v1

∂x3

dx3

dt

a2 =∂v2

∂t+∂v2

∂x1

dx1

dt+∂v2

∂x2

dx2

dt+∂v2

∂x3

dx3

dt

a3 =∂v3

∂t+∂v3

∂x1

dx1

dt+∂v3

∂x2

dx2

dt+∂v3

∂x3

dx3

dt

Dij =12

2∂v1

∂x1

∂v1

∂x2

+∂v2

∂x1

∂v1

∂x3

+∂v3

∂x1

∂v2

∂x1

+∂v1

∂x2

2∂v2

∂x2

∂v2

∂x3

+∂v3

∂x2

∂v3

∂x1

+∂v1

∂x3

∂v3

∂x2

+∂v2

∂x3

2∂v3

∂x3

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥⎥⎥

14

For planar Poiseuille flow, we typically assume that there is no flow in the E2 or E3

directions and that the flow is steady. This implies that the acceleration field can be

simplified considerably,

and the rate of deformation tensor takes the form,

In addition, we can make the obvious assumption that the velocity gradients in the E3

direction should also be zero, which yields,

15

Once again, we have simplified the kinematics as far as possible, the next step is to apply the balance of mass equation which, in Cartesian coordinates, takes the form,

,

and when we employ the previously discussed kinematic assumptions it becomes,

.

.

For an incompressible, Newtonian fluid, the constitutive law has the form, ,

where µ is the fluid viscosity. Inserting the rate of deformation tensor into this expression

yields,

!ρ + ρdivv = !ρ + ρ

∂v1

∂x1

+∂v2

∂x2

+∂v3

∂x3

⎛

⎝⎜⎞

⎠⎟= 0

!ρ + ρ

∂v1

∂x1

⎛

⎝⎜⎞

⎠⎟= 0

Dij =

012∂v1

∂x2

0

12∂v1

∂x2

0 0

0 0 0

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥

T = − pI + 2µD

16

This is precisely the same form that we obtained for planar Couette flow. This implies that

any difference we observe in the final solution is due to the differences in the

boundary conditions. When we incorporate the results so far into the balance of linear

momentum, , we obtain,

,

provided we assume that the viscosity is constant. Recall that the balance of mass provided

us with the restriction, , which requires that the term must also be

identically zero,

ρ !v = divT + ρb

0 =∂T11

∂x1

+∂T12

∂x2

+∂T13

∂x3

= −∂p∂x1

+ µ∂2v1

∂x22

0 =∂T21

∂x1

+∂T22

∂x2

+∂T23

∂x3

= µ∂2v1

∂x1∂x2

−∂p∂x2

0 =∂T31

∂x1

+∂T32

∂x2

+∂T33

∂x3

= −∂p∂x3

∂v1

∂x1

= 0 µ

∂2v1

∂x1∂x2

17

.

Adding them together provides B2,

,

and subtracting them gives us B1 = 0 so that the final solution has the form,

.

Unlike the solution for planar Couette flow, this one is parabolic and depends expressly on

the viscosity of the fluid.

v1 x2 = −h2

⎛⎝⎜

⎞⎠⎟=

18µ

∂p∂x1

h2 −h2

B1 + B2 = 0

v1 x2 =h2

⎛⎝⎜

⎞⎠⎟=

18µ

∂p∂x1

h2 +h2

B1 + B2 = 0

14µ

∂p∂x1

h2 + 2B2 = 0

B2 = −1

8µ∂p∂x1

h2

v1 =

12µ

∂p∂x1

x22 −

18µ

∂p∂x1

h2 =1

8µ∂p∂x1

4x22 − h2( )