Download Understanding the Steady State Linear Velocity Profile in Fluid Dynamics - Prof. Steven La and more Study notes Meteorology in PDF only on Docsity! How we arrive at a steady state/linear velocity profile, i.e., = constant... Ultimately, 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 this implies that 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 associ- ated with the “linear” shear stress given by Holton on page 9, i.e. 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: 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. ∂u ∂z⁄ ∂u ∂z⁄ 0> τzx 0> ∂u ∂z⁄ 0> τzx µ ∂u ∂z -----= top plate Uo equilibrium state
