GEF 2500 Problem set 3
1) (Vallis ex. 1.9)
(a)
(b)
Suppose that a sealed, insulated container consists of two compartments, and that
one of them is filled with an ideal gas and the other is vacuum. The partition
separating the compartments is removed. How does the temperature of the gas
change? Explain! Obtain the expression for the final potential temperature, in
terms of the initial temperature of the gas and the volumes of the two
compartments.
A dry parcel that is ascending adiabatically through the atmosphere will generally
cool as it moves to lower pressure and expands and its potential temperature stays
the same. How can this be consistent with your answer in (a)?
2) Couette flow:
U
z=h
z
h
ρ, ν
x
z=0
Couette flow implies that there is a moving plate at one boundary and a fixed plate at the
other. It is a very common experimental configuration, and often used to determine fluid’s
viscosity.
Consider the flow of a viscous Newtonian fluid between two parallel plates located at
and
. The flow has constant density ρ and kinematic viscosity ν. We will
assume that rotational and gravitational effects are negligible. The upper plate has been
moving with constant velocity U for a very long time. The lower plate is not moving. The
2-dimensional axis system, (x,z), has unit vectors (i, k) and corresponding velocity u and w.
There is no pressure gradient along the x-axis.
(a) Write down the continuity equation for the fluid. Integrate the equation to find an
expression for the vertical velocity. What can you say about the variation along the xaxis from this?
(b) The Navier-Stokes equation, neglecting rotational and gravitational effects, is
⃗
(⃗
)⃗
⃗
where ⃗
Simplify the equation using the information from the text and
your solution to (a). Show that the equation controlling the motion is
(c) What can you say about the pressure field in the fluid?
(d) What are the boundary conditions for the flow field?
(e) Calculate the flow field.
3)
Couette flow with pressure gradient:
L
U
z=h
z
p=p0
h
ρ, ν
p=p1
x
z=0
In the previous exercise we looked at Couette flow without any pressure gradients. Next we
consider Couette flow with a pressure gradient acting in the x direction.
Consider the flow of a viscous Newtonian fluid between two parallel plates located at
and
. The flow has constant density ρ and kinematic viscosity ν. We will assume that
rotational and gravitational effects are negligible. The upper plate has been moving with
constant velocity U for a very long time. The lower plate is not moving. The 2-dimensional
axis system, (x,z), has unit vectors (i, k) and corresponding velocity u and w. The length of the
plate in the x direction is L. We take L h. The pressure is p0 at x=0 and p1 at x=L and
independent of z.
(a) Write down the continuity equation for the fluid. Integrate the equation to find an
expression for the vertical velocity. What can you say about the variation along the x-axis
from this?
(b) From the Navier-Stokes equation, derive the equation controlling the motion for this flow
field.
(c) What can you say about the pressure in the fluid?
(d) What are the boundary conditions for this flow field?
(e) Calculate the flow field.
(f) Assume that the upper plate is at rest as well (i.e. U=0). Calculate the flow field (you need
to change your boundary conditions). What is the name of this flow field?
(g) Sketch velocity profiles for various pressure gradients listed below and various U
velocities. You can do this using MatLab or any other programming language you prefer,
or you can sketch by hand.
(i)
p0 = p1
(ii)
p1–p0 = ∞
(iii)
p1=2p0
(iv)
p1= p0
(note: the profiles need not to be exact, but try to give some qualitative characteristics )