Advanced Math, Dimensional Analysis, Diffusion

I am looking for step-by-step solutions for 3 problems (attached), on the topics of Dimensional Analysis, Differential Equations, and Diffusion.

©C. Tier, Dept. Applied Mathematics, IIT September 14, 2020 1

Math 486/522 – Homework 3 – Diffusion
Fall 2020 C. Tier

1. The chemical concentration u(x,t) (M/L in 1 space dimension) satisfies the following dif-

fusion problem with a constant flux, J = −D

∂u

∂x
, of chemical entering at the boundary

x = 0:

∂u

∂t
= D

∂2u

∂x2
, x > 0, t > 0

IC: u(x,0) = 0

BC: D
∂u

∂x
(0, t) = −A, u(∞, t) = 0.

(a) Use dimensional analysis to find the dimensions of D and A.

(b) Based on the parameters and variables in the diffusion problem, use dimensional reduc-

tion to derive a simplified form of u(x,t). Identify the similarity variable η. Choose a

similarity variable that vanishes at x = 0.

(c) Using the similarity variable and the dimensionally reduced form of u, transform the

diffusion problem into an ODE boundary value problem, as done in class. Clearly state

the boundary value problem including the boundary conditions.

(d) Solve the ODE and derive the solution u(x,t). Hint: One solution of the ODE is η –

check. A second solution can be constructed by the method of reduction of order (look

it up in Dawkin’s ODE book).

(e) Graph your solution if D = 1/2 and A = 1 for a sequence of four interesting times on

the same axes.

2. Let u(x,t) be a density (one dimensional) that satisfies a diffusion equation

2xut = Duxx, 0 < x < ∞, t > 0. (1)

with initial condition u(x,0) = 0 and boundary conditions u(0, t) = u0 and u(∞, t) = 0.

(a) Find the dimensions of D.

(b) Perform a dimensional reduction of (1) as done in class to obtain dimensionless quan-

tities.

(c) Introduce a non-dimensional density and a similarity variable into (1) to obtain, if pos-

sible, and ODE with consistent boundary conditions.

(d) Bonus: Construct the solution to the BVP in part (c). Good luck.

©C. Tier, Dept. Applied Mathematics, IIT September 14, 2020 2

3. (Reaction-Diffusion) Consider the model of a drug patch discussed in class. The drug dif-

fuses with diffusion coefficient D but also degrades according to a first order chemical reac-

tion with rate constant k. The chemical concentration of the drug is u(x,t) (M/L) and the

patch at x = 0 provides a concentration u(0, t) = Ae−kt, also decaying in time. The new

PDE problem for the concentration of the drug is:

∂u
∂t
= D
∂2u

∂x2
−ku, x > 0, t > 0 (2)

IC: u(x,0) = 0 (3)

BC: u(0, t) = Ae−kt, u(∞, t) = 0. (4)

(a) Identify the meaning of each term on the right side of equation (2).

(b) Find the dimensions of A, D and k.

(c) Use dimensional analysis to identify three dimensionless quantities that involve the

variables and parameters in the problem.

(d) Unfortunately these dimensionless quantities cannot be used to solve the problem di-

rectly but can help lead to the solution. First introduce a nondimensional concentration

into (2), (3), and (4) using your result (d) and state the new problem. Next eliminate

the reaction term by moving it to left side of (2) and introducing an integrating factor

similar to the one used to solve first order linear ODEs. You should see one of your di-

mensionless quantities in the integration factor. You need to define a new concentration

containing the integrating factor. Finally, introduce the your remaining dimensionless

quantity as a similarity variable (see class notes) and state the resulting ODE boundary

value problem.

(e) Solve the boundary value problem and express your solution back in terms of the orig-

inal variables.

(f) Make a sequence of graphs of your solution in (e) if D = 1/2, k = .4, and A = 2. You

can pick a sequence of 4 times and include labels for the curves.