MATH 450 Projects
Homework #4
Consider the convection-diffusion equation given by ut +
c u x = D u xx. Here u is the temperature of, for
example, a flowing stream and the temperature is affected by both
convection (where c is the convective velocity) and diffusion (where D
is the diffusivity). (a) Use dimensional analysis to determine the
units of both c and D if u is measured in degrees Celsius, t is measured
in seconds and x is measured in meters. (b) Formulate a complete
mathematical model, including initial and boundary conditions, to
describe the temperature of a fixed segment of a flowing stream with a
constant initial temperature and constant boundary temperatures. How many
independent dimensionless quantities are needed to describe this
problem? Don't forget the boundary and initial conditions and the
fundamental length scale in the problem. (c) Write an equivalent
dimensionless law by appropriately scaling this problem.
Homework #6
Consider the convection-diffusion equation given by ut +
c u x = D u xx. Here u is the temperature of, for
example, a flowing stream and the temperature is affected by both
convection (where c is the convective velocity) and diffusion (where D
is the diffusivity). In Homework #4 you arrived at a scaled version of
this problem. One version comes from scaling t by l/c, where l is the
length scale of the problem. Another version comes from scaling t by
l2/D. Both involve the parameter P = lc/D, called the
Peclet number. Pose the problem both ways (whichever way you didn't
originally scale, do it that way now). Write both dimensionless forms
of the problem out identifying the dimensionless variables and
parameters. One form is appropriate for a convection dominated process
and one form is appropriate for a diffusion dominated process. Use
perturbation methods to find the leading order and second order problems
for each in the the appropriate small parameters.
Homework #11
Consider the biofilm growth model developed in class. (a) Use
dimensional analysis to determine the units of all the parameters if C
is measured in gm/(cm)3, f is dimensionless, t is measured
in seconds, z is measured in centimeters, and Y has no units (though it
represents mass of bacteria/mass of substrate). (b) How many
independent dimensionless quantities are needed to describe this
problem? Don't forget the boundary and initial conditions. (c) Write
an equivalent dimensionless law by appropriately scaling this problem,
using the scalings S = C/Kc, L =
delta/sqrt{D/Kmax}, x = z/sqrt{D/Kmax}, and tau =
Kmaxt.