BME Transport HW 9 both parts
Transport in BME (BME 495 or BME 461)
IUPUI, Fall 2008
E. Morris (instructor)
Handed out Nov 18, 2008 à Due Nov. 25, 2008
Homework #9 Convection of solute through pores – theoretical development
(1). Please derive the expression for shear stress, t from a STEADY STATE momentum (force) balance on the “sheaths” of fluid flowing through a rigid tube – as we discussed in class yesterday.
Recall that you must balance pressure forces “in” with forces “out”.
The forces we covered were hydraulic (i.e., pressure) and shear. Pressure acts on the face (drawn in gray) of the ‘sleeve’ and shear acts on the longitudinal surfaces (dashed red (inner) and blue (outer)).
(2.) Given the parabolic velocity profile in the rigid tube that we derived yesterday, calculate the volumetric flow rate over the whole cross-section of the tube. That is, find the volumetric flow rate (length3/time) which takes into account the range of linear velocities (length/time) of each of the “sleeves” flowing through the tube. Volumetric flow rate, Q, can be calculated as the product of linear velocity v and cross-sectional area, A.
The velocity profile (i.e., the velocity of each sleeve at a given radius, r) is as follows:
(The arrows –pointing to the right- representing the velocities at the various radii.)
When you get to Q, the volumetric flow rate averaged over all radii for a rigid cylindrical tube, you will have arrived at Poiseuille’s (pwa-‘zayeh’s) equation.
(3.) On Thursday, we studied a problem – transport of LDL in the artery wall - that required the diffusion-convection-degradation equation in 1D. Attached, are corrected programs that originally appeared in Dunn et al, Numerical Methods for Biomedical Engineers that solve a similar problem – transport of cells in a medium in 1D – numerically.
Please run the two programs multiple times and create plots of the cell density profiles (Note: when prompted at the command line the following holds: in the Dirichlet case, ‘1’ means yes, in the Neumann case, ‘9’ means yes. ‘0’ always means ‘no’. You can probably accept the default settings (first question at command line) but then modify Diffusion and Convection by answering the second set of questions in the affirmative.
(a.) Create multiple plots of the output of each case. What is the difference in appearance between the Neumann condition problem and the Dirichlet condition problem. What is the difference between those two types of boundary conditions and how do their differences explain the respective appearances of the cell density profiles.
Now focus on the Neumann condition
(b.) Create plots of multiple solutions on top of each other for the Neumann case.
Here is an example plot of two overlayed families of solutions (two different sets of parameters - in different colors-, each at multiple times).
(c.). Modify the code to indicate the times of the solutions in a family.
(d.) Vary the ratio of Diffusion to Convection. (You will have to vary things slightly because of stability limitations of the “explicit” method implemented to solve the equations. DO NOT GIVE ME PLOTS OF WILDLY OSCILATING CURVES THAT REACH 1018. THAT IS GARBAGE PRODUCED WHEN THE ALGORITHM GOES UNSTABLE. Technically, this occurs when the off-diagonal terms in the coefficient matrix are negative. Take a numerical methods class in grad school if you are interested…)
(e.) Modify the code to calculate and DISPLAY the corresponding Peclet number (see #4 below) in the same color as the curves.
The programs have a “hold on” command at the end, so you can overplot multiple families as I have done in the example output.
(f.) What is the difference, visually, between high, moderate (near 1) and low Peclet number systems? Is this consistent with what we have learned in class?
(4.) I have modified the equation in the Dunn program to include degradation (or other loss) of cells. The state equation in the Neumann case is the same as we discussed on Thursday in relation to LDL transport in rectangular coordinates.
mD is the motility of the cells (entirely analogous to diffusivity of molecules)
v is drift velocity
k is degradation rate constant
(a.) Nondimensionalize the equation above (as we did at the end of class yesterday) using
these dimensionless variables (ignore boundary conditions for now):
(b.) What are the dimensionless parameters in the result and what do they represent?
(By the way, the Saidel text has 1/Pe because he divided by VC0/L; Pe = vL/D)
(5.)
(a.) Modify the Neumann condition program again to plot the value of the dimensionless ratio of diffusion time constant to degradative time constant. Which dimensionless parameter from your work, above, is this? Show that it is dimensionless. What is a typical name given to such a dimensionless parameter that compares diffusion to degradation processes (use books (how quaint!) or Google if you must.)
(b.) Produce plots of families of (labeled) curves to show the effect of high degradation relative to diffusion.
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