Oct 7: Friction, viscosity and shear stress
Governing
equations: the math and physics of oceanography
So far in class you
have covered the concepts of:
·
conservation of mass
·
conservation of scalar quantities (like temperature and salt)
o
the role of advection, mixing and air-sea fluxes
The equations that
govern fluid motion describe the
influences of different forces that add or remove momentum to a fluid. The
equations of motion are therefore essentially a statement of
·
conservation of momentum
o
acceleration and advection – material derivatives “following the fluid”
o
pressure force and pressure gradients
o
Coriolis (not really a force at all, just a matter of how you
look at it)
o
gravity and hydrostatic pressure
The final step to a
complete description of the governing equations is consideration of friction.
Frictional forces
transfer momentum from the wind to the ocean surface, generating currents and
waves and causing mixing.
Friction also
explains the drag on a fluid that will eventually bring it to rest if all the
driving forces, such as wind, were removed.
Friction acts by
transferring momentum from one fluid parcel to an adjacent parcel by internal
stresses. These stresses arise because a real fluid is viscous.
Friction is the
ultimate sink of energy in fluid flow. Friction dissipates the kinetic energy
and momentum of fluids.
If fluids had no
viscosity, or were inviscid, the wind blowing over a flat sea surface would
have no effect, and fluid flow could be put into perpetual motion by the
application of pressure forces and gravity.
But in real fluids,
friction and stress are important.
You’ve already been
introduced to the concept of molecular diffusion of heat and salt in the governing
equations.
Molecular
diffusivity is a property of the fluid.
Mixing and stirring
on small scales typically appear to act in the same way as molecular diffusion,
but with a larger “eddy” diffusivity that parameterizes the net effect of small
eddies and turbulence in a fluid that mix scalar quantities.
Eddy diffusivity is
a property of the flow, not of the fluid itself.
In molecular
diffusion, the fluxes of tracer across the faces of a fluid element are given
by, e.g.
and it is the divergence of the flux that can cause a
net gain or loss of heat in the fluid element:
Momentum will
diffuse though a fluid in much the same manner as heat and salt, only momentum
is a vector and this gives arise to stresses which are a little different to
the analogous fluxes of tracers of heat and salt.
The experiments of
Hagen (1839) and Poiseuille (1840) of steady flow through a long pipe showed
that the discharge (flow rate in m3/s) is proportional to the pressure
difference at the ends of the pipe and the 4th power of the tube
diameter.
This result was
consistent with hypotheses concerning two fluid properties that were suggested
by observations:
Fluid immediately
adjacent to the wall of the pipe was not moving; the so-called “no slip”
property or boundary condition.
u=0 at z=0
The shear stress,
per unit mass, within fluids is proportional to the “rate of strain” of the
fluid.
in a Newtonian
fluid
where n
is the
kinematic viscosity 1 x 10-6 m2 s-1 for water.
Examples of
different velocity (and hence stress) profiles.
Left: flow between
two flat plates with the top plate moving a
Right: flow with a
free surface where no stress is applied, hence 
