MATHEMATICA TUTORIAL, Part 2.6: Applications

Preface

In this section, we show some problems of practical significance.

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Introduction to Linear Algebra with Mathematica

Glossary

Applications

 

Rayleigh problem

In fluid dynamics, Rayleigh problem also known as Stokes first problem is a problem of determining the flow created by a sudden movement of a plane from rest, named after Lord Rayleigh and Sir George Stokes. This is considered as one of the simplest unsteady problem that have exact solution for the Navier-Stokes equations.

Consider an infinitely long plate which suddenly made to move with constant velocity U in the x direction, which is located at y = 0 in an infinite domain of fluid, which is at rest initially everywhere. The incompressible Navier-Stokes equations reduce to

\[ \frac{\partial u}{\partial t} = \nu\,\frac{\partial^2 u}{\partial^2 y} , \]
where ν is the kinematic viscosity. The initial and the no-slip condition on the wall are
\[ u(y,0) = 0, \quad u(0, t>0) = U, \quad u(\infty , t>0) = 0. \]
The last condition is due to the fact that the motion at y = 0 is not felt at infinity. The flow is only due to the motion of the plate, there is no imposed pressure gradient.

The problem on the whole is similar to the one dimensional heat conduction problem. Hence a self-similar variable can be introduced

\[ \eta = \frac{y}{\sqrt{\nu\,t}} , \qquad f(\eta ) = \frac{u}{U} . \]
Substituting this the partial differential equation, reduces it to ordinary differential equation with dependent variable missing:
\[ f'' + \frac{1}{2}\,\eta\,f' =0 , \]
with boundary conditions
\[ f(0) =1, \qquad f(\infty ) = 0 . \]
The solution to the above problem can be written in terms of complementary error function
\[ u(y,t) = U\,\mbox{erfc}\left( \frac{y}{\sqrt{4\nu t}} \right) \]

 

In Mathematica, we plot a donat

RegionPlot[1 < Abs[x + I y] < 2, {x, -2, 2}, {y, -2, 2}, ImagePadding -> 1, PlotStyle -> Blue]

 

Fisher equation

Consider the famous Fisher equation

\[ uu_t = u_{xx} + u\left( 1 - u \right) . \]

 

  1.  Rayleigh problemWikipedia
  2. Batchelor, George Keith. An introduction to fluid dynamics. Cambridge university press, 2000.
  3. Kudryashov, N.A., On one of methods for finding exact solutionsof nonlinear differential equations

 

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