Prob. 1: Deriving governing equations for Zebra mussels
In class, we have derived the mass conservation equation for a uid using a con-
trol volume approach (see *.pdf les uploaded into courseweb). Herein, you will
have the opportunity to re-work that part. Zebra mussels (http://en.wikipedia.
zm.html) were introduced into the Great Lakes in the mid-1980’s, and they have
since invaded and harmed the ecology of many North American waterways. In this
problem you will derive and analyze an equation that can (and hopefully will) be
used to predict the transport of zebra mussel larvae in rivers aected by them.
1. Along with advective ux, larvae in a river are subject to dispersion and mor-
tality (i.e. death). The dispersive ux is analogous to the heat ux given by
Fourier’s law: 某公式 , where L is the concentration of larvae (mass of
larvae per mass of uid). The sink due to larval mortality can be expressed
as a fraction of the larval mass that disappears per unit time. Assume kL
is constant, therefore show that conservation of larval mass (in one dimension,
x-direction) leads to:
Where K = kL= is the dispersion coecient
2. Argue in at least two ways that the equation you derived is correct
3. Which of the processes advection, dispersion or mortality is the most important
over a 100-Km stretch of any river if u = 0:3m=s, K 1000m2=s, and
105s1. Please discuss your answer.
4. If we want to extend this study to the Michigan Lake, what are the modication
to the equation above? are you going to consider a one-dimensional equation?
Yes or No? please elaborate your answer.
5. Can you please describe brie y what are scientists doing to mitigate the eects
of Zebra mussels? Please cite any study that you have reviewed.
Prob. 2: Proving if solutions are correct or not
In class, we have derived the general equations for conservations laws (mass and
momentum). Here there are two solutions of a certain compressible ow, please
explaing if both solutions are right or wrong. Please discuss your answer.
Where 0 is a constant reference density and U0, k, and H are constants with di-
mensions of velocity, reciprocal length, and length, respectively. Please explain your
Prob. 3: Open channel ow
Figure 1: Open channel ow
The following gure shows an open channel ow down an slope. The ow char-
acteristics are: steady, two-dimensional, incompressible, laminar ow down a solid,
impermeable plane inclined at an angle . The velocity in the x-direction (parallel
to the bottom) is given by:
Where H is the uniform thickness of the lm and = = is the kinematic viscosity.
You are asked to do the following:
Compute the velocity w in the z-direction (normal to the bottom)
Show in two ways that the shear stress on the bottom is o = gHSin()
Any idea how the above equation (Eq. 4) was found? Please discuss your
Prob. 4: Finding a missing velocity component
Two velocities components of a steady, incompressible, three-dimensional ow eld
are known, namely u(x; y; z) = ax2+by2+cz2 and w(x; y; z) = axz+byz2, where a, b,
and c are constants. The y velocity component is missing, please nd the expression
for the v in function of x, y and z.