1

CONSTITUTIVE RELATION FOR NEWTONIAN FLUIDThe Cauchy equation for momentum balance of a continuous, deformable medium

jiij

ij

jiji

ji g

x)uu(

x)u(

t

combined with the condition of symmetry of the stress tensor

ij

ijji

ji g

x)uu(

x)u(

t

yields the relation

Further applying the condition of incompressibility ( = const., ui/xi = 0), it is found that

ij

ij

j

ij

i gx

1xuu

tu

(Why?)

2

CONSTITUTIVE RELATION FOR NEWTONIAN FLUIDBut how does the stress tensor ij relate to the flow?

21 = 12

- 21 = - 12

x2

x1

u1

moving with velocity U

fixed

fluid

Plane Couette Flow: shear stress 21 = 12 is applied to top plate, causing it to move with velocity U; bottom plate is fixed. Because the fluid is viscous, it satisfies the “no-slip” condition (vanishing flow velocity tangential to the boundary) at the boundaries:

Uu,0uHx10x1

22

Empirical result for steady, parallel (u2 = 0) flow that is uniform in the x1 direction: H

xUu 21

H

3

CONSTITUTIVE RELATION FOR NEWTONIAN FLUIDNewton’s hypotheses:• for steady, parallel flow that is uniform in the x1 direction, the relation u1/U = x2/H always holds;• an increase in U is associated with an increase in 12;• an increase in H is associated with a decrease in 12.The simplest relation consistent with these observations is:

21 = 12

21 = 12

x2

x1

u1

moving with velocity U

fixed

fluid H

2

112 x

uHU

where is the viscosity (units N s m-2).

4

CONSTITUTIVE RELATION FOR NEWTONIAN FLUIDAlternative formulation let FD,mom,12 denote the diffusive flux in the x2 direction of momentum in the x1 direction. The momentum per unit volume in the x1 direction is u1, and in order for this momentum to be fluxed down the gradient in the x2 direction,

2

1

2

112,mom,D x

uxuF

where denotes the molecular kinematic diffusity of momentum, []= L2/T].

We now show that

,F 12,mom,D12

so that the kinematic diffusivity of momentum = the kinematic viscosity.

u1fluid

momentum source

momentum sink

5

CONSTITUTIVE RELATION FOR NEWTONIAN FLUIDu1

x2

x1

x1

x2

Again, the flow is parallel (u2 = 0) and uniform in the x1 direction, and also uniform out of the page (u3 = 0) . Consider momentum balance in the x1 direction. The control volume has length 1 in the x3 direction, ,which is upward vertical. Momentum balance in the x1 direction ~

/t(u1x1x21) = net convective inflow rate of momentum +net diffusive rate of inflow of momentum + gravitational force

/t(u1x1x21) = net convective inflow rate of momentum +net surface force + gravitational forceor equivalently

Now the net convective inflow rate of momentum is

01xu1xu 2xx

212x

21

111

6

CONSTITUTIVE RELATION FOR NEWTONIAN FLUIDu1

x2

x1x1

x2

The gravitational force in the x1 direction is 0.

net diffusive rate of inflow of momentum =

net surface force =

Equivalently,

The only way that this could be true in general is if

,F 12,mom,D12

Since p at x1 is equal to p at x1 +x1 (flow is uniform in the x1 direction), the only contribution to the surface forces is 21 = 12, so that

1x1x12xx12222

1xFF 1x12,mom,Dxx12,mom,D222

Thus /t(u1x1x21) =

x2

x2+x2

12

12FD,mom, 12

FD,mom, 12

1x1x12xx1222

2

1

2

1

xu

xu

1xFF 1xx12,mom,Dx12,mom,D222

7

CONSTITUTIVE RELATION FOR NEWTONIAN FLUIDGeneralization to 3D: where denotes the viscous stress tensor,

vijijij p

j

ivij x

u~

ijijj

i r21

xu

Here ij denotes the rate of strain tensor and rij denotes the rate of rotation tensor. (See Chapter 8.)

i

j

j

iij

i

j

j

iij x

uxur,

xu

xu

21

vij

According to the hypothesis of plane Couette flow, we expect a relation of the form

However, note that

8

CONSTITUTIVE RELATION FOR NEWTONIAN FLUIDWe relate the viscous stress tensor only to the rate of strain tensor, not rate of rotation tensor, in accordance with the hypothesis for plane Couette flow.

i

j

j

iij

vij x

uxu

21~

i

j

j

iij

vij x

uxu

2AA

Most general possible linear form:

klijklvij A

Consequence of isotropy, i.e. the material properties of the fluid are the same in all directions: where A is a simple scalar,

(See Chapter 8)

9

CONSTITUTIVE RELATION FOR NEWTONIAN FLUID

i

j

j

iij

vij x

uxu

2AA

Set

2A

to obtain

i

j

j

iij

vij x

uxu2

and thus the constitutive relation for a Newtonian fluid:

i

j

j

iijij x

uxup

10

CONSTITUTIVE RELATION FOR NEWTONIAN FLUID

The Navier-Stokes equation for momentum balance of an incompressible Newtonian fluid is obtained by substituting the Newtonian constitutive relation

into the Cauchy equation of momentum balance for an incompressible fluid

and reducing with the incompressible continuity relation (fluid mass balance)

to obtain

i

j

j

iijij x

uxup

ij

ij

j

ij

i gx

1xuu

tu

0xu

i

i

ijj

i2

ij

ij

i gxx

uxp1

xuu

tu