Download How we arrive at a steady state/linear velocity profile, i.e., = constant

How we arrive at a steady state/linear velocity profile, i.e., ∂u ⁄ ∂z = constant...
Ultimately, ∂u ⁄ ∂z > 0 is the “response” to the forcing which is the movement (constant force/
stress) applied to the upper plate (τzx) which sets the fluid in motion. Hence, given τ zx > 0 this
implies that ∂u ⁄ ∂z > 0 from the formula below (µ is defined to be > 0).
But what is really (physically) happening? First of all, note that there is no time derivative associated with the “linear” shear stress given by Holton on page 9, i.e.
∂u
τ zx = µ -----∂z
Because of this, the above equation can not tell us how the fluid evolves from a quiescent (calm/
no flow) state to the steady-state/equilibrium linear velocity profile. A good analogy to this would
be geostrophic balance which, like the expression above, is a diagnostic relationship (i.e., no time
dependency) -- and thus it does not predict how the flow became geostrophic! Anyway, clearly in
the example I give in class (and in Holton Fig. 1.3) the forcing due to the upper plate is in the + x
direction. If we consider a fluid parcel (chunk o’ fluid or fluid element), due to Newton’s 3rd Law,
eventually there will be a “steady” force exerted by the fluid, just beneath the fluid parcel, in the
opposite direction (i.e., -x direction) of that exerted at the top of the fluid parcel. The problem is,
initially the fluid is not in steady state! However, when the two are equal and opposite throughout
the entire fluid, we have reached an equilibrium state whereby there is a force acting (upper plate
moving) but the net force is zero. As long as the plate at the top moves at a constant velocity, the
momentum transfer (via molecular diffusion) does NOT continue, ad infinitum, to transport
momentum downward -- rather this goes on for a finite period of time until equilibrium is reached
(even though there continues to be a potential source of higher momentum molecules from above
- read on below why the downward momentum transport doesn’t continue forever).
Anyway, if we turned off the upper plate motion -- eventually the fluid would cease to move (due
to viscous dissipation). Thus, the equilibrium state is maintained only in the presence of a moving
upper boundary (constant force here).
Here’s a picture of what I believe is going on:
top plate Uo
At “equilibrium” each sublayer (there’s an
infinite # of layers in the limit) must be doing
3 things:
1. losing molecules (with the mean momentum
of the given layer).
2. gaining higher momentum molecules from
above, and
3. gaining lower momentum molecules from
the layer below.
equilibrium state
This must be going on at a rate that maintains the momentum for that layer (otherwise the linear
velocity profile would continue to evolve and we wouldn’t be at steady state). Prior to steady
state, (e.g., initially) there is zero momentum below and hence there is a net downward transport
of momentum due to #1 and #2 above. Also note that the layer adjacent to the top plate continually loses momentum via #1 and #3 above but that as long as the plate is moving this momentum
is continuously replaced at the same rate it is removed once in equilibrium.
Hope this helps clarify things some...