CONSTITUTIVE RELATION FOR NEWTONIAN FLUID

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CONSTITUTIVE RELATION FOR NEWTONIAN FLUID
The Cauchy equation for momentum balance of a
continuous, deformable medium
combined with the condition of symmetry of the
stress tensor
yields the relation
Further applying the condition of
incompressibility (? const., ?ui/?xi 0), it
is found that
(Why?)
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CONSTITUTIVE RELATION FOR NEWTONIAN FLUID
But how does the stress tensor ??ij relate to the
flow?
?21 ?12
moving with velocity U
u1
fluid
x2
x1
fixed
- ?21 - ?12
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
Empirical result for steady, parallel (u2 0)
flow that is uniform in the x1 direction
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CONSTITUTIVE RELATION FOR NEWTONIAN FLUID

• Newtons 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

where ? is the viscosity (units N s m-2).
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CONSTITUTIVE RELATION FOR NEWTONIAN FLUID
Alternative 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,
where ? denotes the molecular kinematic diffusity
of momentum, ? L2/T.
We now show that
so that the kinematic diffusivity of momentum
the kinematic viscosity.
momentum source
fluid
momentum sink
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CONSTITUTIVE RELATION FOR NEWTONIAN FLUID
?x2
x2
?x1
x1
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(?u1?x1?x21) net convective inflow rate of
momentum net surface force gravitational force
or equivalently
?/?t(?u1?x1?x21) net convective inflow rate of
momentum net diffusive rate of inflow of
momentum gravitational force
Now the net convective inflow rate of momentum is
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CONSTITUTIVE RELATION FOR NEWTONIAN FLUID
FD,mom, 12
?12
?x2
x2?x2
?x1
FD,mom, 12
?12
The gravitational force in the x1 direction is 0.
x2
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
net surface force
Equivalently,
net diffusive rate of inflow of momentum
Thus ?/?t(?u1?x1?x21)
The only way that this could be true in general
is if
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CONSTITUTIVE RELATION FOR NEWTONIAN FLUID
Generalization to 3D where denotes the
viscous stress tensor,
According to the hypothesis of plane Couette
flow, we expect a relation of the form
However, note that
Here ?ij denotes the rate of strain tensor and
rij denotes the rate of rotation tensor. (See
Chapter 8.)
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CONSTITUTIVE RELATION FOR NEWTONIAN FLUID
We 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.
Most general possible linear form
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)
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CONSTITUTIVE RELATION FOR NEWTONIAN FLUID
Set
to obtain
and thus the constitutive relation for a
Newtonian fluid
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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