The chemical concentration ate, t) {M f L in 1 space dimension) satises the following dif fusion problem With a constant ux, J = D%, of chemical...

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. The chemical concentration ate, t) {M f L in 1 space dimension) satisfies the following dif— fusion problem With a constant flux, J = —D%, of chemical entering at the boundary5: = [1:%=D%, 12:10, tbl}IC: u(1', 0) = 0BC: ofiw t) = —A u[oo t} = a33: ’ ’ 1 (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 15(39, t)- Identify the similarity variable 13. Choose asimilarity variable that vanishes at I = l}. (c) Using the similarity variable and the dimensionally reduced form of u, transform thediffusion problem into an ODE boundary value problem, as done in class. Clearly statethe boundary value problem including the boundary conditions. (d) Solve the ODE and derive the solution u(:r.', t). Hint: One solution of the ODE is n -check. A second solution can be constructed by the method of reduction of order (lookit up in any ODE book). (e) Graph your solution if D = l ,r" 2 and A = 1 for a sequence of four interesting times onthe same axes.