# Numerical Solution of Stoke's First Problem

$f''+2 \eta f-4mf=0$ where $f=1$ at $\eta=0$, $f \to 0$ as $\eta \to \infty$. The case of $m=0$ arises in the analysis of motion of fluid above a suddenly accelerated flat plate.

• Are you interested in HOW to solve this equation numerically? or you just wanna look at what the numerically solution looks like. Mar 25, 2013 at 18:45
• yeah, I want to know the details procedure of numerical solution Mar 25, 2013 at 18:54
• Non-dimensionalize around $\eta$ and let $\infty$ be approximately 5. Mar 25, 2013 at 18:57
• How can I non-dimensionalize the equation around η ? I want to solve it by shooting method ? Is it at all possible preferably using MATLAB? Mar 25, 2013 at 19:03
• Sure, you can do shooting, and you can implement shooting in MATLAB. I assume you know how to do this. The boundary layer flow profile above a flat plate starts at zero at the boundary, and as you go upwards, it pretty rapidly approaches the steady-state flow velocity. Mathematically, we treat this as the limit to infinity. However, if you non-dimensionalize your units (make $\eta$ be on the scale of unity), then you don't need to go out to infinity. It's sufficient to set $f(\eta=5) = f(\eta = \infty)$ as your boundary condition. Mar 25, 2013 at 19:16

The general solution of your differential equation is $$f(\eta) = a \text{Ai}(2^{1/3}(2m-\eta)) + b \text{Bi}(2^{1/3}(2m-\eta))$$ where Ai and Bi are Airy functions, and $a$ and $b$ are arbitrary constants. Now both $\text{Ai}(t)$ and $\text{Bi}(t)$ go to $0$ as $t \to -\infty$: according to Wolfram Alpha, \eqalign{\text{Ai}(-t) &= \frac{t^{-1/4}}{\sqrt{\pi}} \sin(2 t^{3/2}/3+\pi/4) + O(t^{-7/4})\cr \text{Bi}(-t) &= \frac{t^{-1/4}}{\sqrt{\pi}} \cos(2 t^{3/2}/3+\pi/4) + O(t^{-7/4})} So the boundary condition at $\infty$ does not restrict the solution. You need another boundary condition to determine a solution.
• Are you sure you have the sign of the $2\eta f$ term right? It would work for the equation $f'' - 2 \eta f - 4 m f = 0$ (or equivalently if you wanted $f \to 0$ as $\eta \to -\infty$). Mar 25, 2013 at 19:26