exam2_soln - Fall 2005 10.34 Exam II. Wed. Nov. 9, 2005...

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Fall 2005 10.34 Exam II. Wed. Nov. 9, 2005 (SOLUTION) Read through the entire exam before beginning work, and budget your time. You are researching drug eluting stents, and wish to understand better how the drug release rate depends upon the properties of the fluid in which it is immersed. You develop a model using the simplified geometry shown in the figure below. The stent is placed on a solid impermeable surface, and to simply the flow problem, we assume that the drug stent is very thin and does not affect the local nature of the flow. Also, we assume that the stent is very wide in the out-of-plane direction so that we can neglect variation in the z direction. The local velocity of the fluid goes from zero at the solid surface (no-slip BC) to the superficial velocity V at a distance from the surface. δ b We assume a linear velocity profile, v () x y = ------ ≤≤ , b Vy 0 y δ δ b (EQ 1) Vy δ > , b Given this velocity profile, we wish to model the external mass transfer resistance to release of drug from the stent. We solve the 2-D convection/diffusion equation 2 2 c c c c t = – v + D + D x x x 2 y 2 (EQ 2) xx yB to obtain cxy ) at steady state. on the domain x L ≤≤ R and 0 ( , Fall 2005 10.34 Exam II. Wed. Nov. 9, 2005 November 10, 2005 1
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B is a distance sufficiently far away from the surface that we can assume that the drug con- centration is zero there, cx , B ) = 0 . At the left “inlet” boundary x = x L , we use the BC ( L , y ) = 0 . At the right “outlet” boundary, we use the BC c ⁄ ∂ x ( = 0 . At the solid sur- ( x R , y ) xL and the remaining regions are imperme- face y = 0 , the stent occupies the region 0 ≤≤ able to the drug. At the stent surface, we use the boundary condition c ( 0 ≤≤ , 0 ) = c eq , where c is the drug concentration in the fluid that is in equilibrium with the stent. At all eq other regions on the impermeable surface, we use the no-flux BC c ⁄ ∂ y = 0 . ( x , 0 ) ,,, , ,, , B . The parameters for this problem are V δ b Dc eq Lx L x R NOTE: This problem was inspired by an actual numerical study performed by a student here at MIT. In that case, she was looking at the spatial distribution of drug release into the underlying tissue by solving a reaction/diffusion equation in the underlying surface underneath the stent to model drug uptake. She also solved, coupled to the convection/diffusion equation in the fluid stream, the Navier-Stokes equations to compute the exact laminar velocity profile around the stent. In practice, the drug release into the underlying tissue is not spatially uniform, but is concentrated just below and downstream of the stent. Here, I have simplified the problem by making the underlying surface impermeable to the drug and by neglecting any perturba- tion in the flow field by the stent. ( You wish to compute the steady state concentration profile , y ) using finite differences.
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exam2_soln - Fall 2005 10.34 Exam II. Wed. Nov. 9, 2005...

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