ME 523
Numerical Solutions Applied To Heat Transfer And Fluid Mechanics Problems


Classification of partial differential equations

The contents of this page are simply a review of a classification method that you should have seen in an undergraduate PDE class.  The lecture will focus more on material in your first reading assignment (see the ANGEL Lessons tab) from an Oak Ridge National Laboratory web site, providing more general techniques for classifying PDEs. Read sections 4 and 5 of this material carefully.

 

Boundary Value Problems (Equilibrium Problems)

Mathematical Model (valid in region R) : Conservation Equation
Boundary conditions (Neumann and/or Dirichlet
Classification: Elliptic Equations

Applications:

Steady State Heat Conduction
Invicid Flow (Potential)
Viscous Flow
 
Initial Value Problems (Propagation Problems)  Mathematical Model (valid in region R, with possible moving boundary) : Conservation Equation
Boundary conditions
Initial Conditions
Classification: Parabolic, Hyperbolic

Applications:

Transient Pressure Waves
Propagation of Momentum
Propagation of Stresses
How can we classify a mathematical model?
 

where a, b, c, and f can be constants or functions of

If

 
b2 – 4 ac < 0 Equation is Elliptic
b2 – 4 ac = 0 Equation is Parabolic
b2 – 4 ac > 0 Equation is Hyperbolic
 

Example: Laplace or Poisson Equation  (x, y)

Laplace Poisson Classification a = 1, b = 0, c = 1   b2 – 4 ac = 0 – 4 (1) (1) = -4   Equations are Elliptic   Example: Fourier Equation  Heat Conduction   We have 2 space variables and one time variable.   Neglect one space variable since both space variables are of the same form. (Second Derivatives)
  a = 0, b = 0, c = k   b2 – 4 ac = 0 – 4 k 0 = 0   Equation is Parabolic  
Example: Navier-Stokes Equation - Steady State and Incompressible Flow Variables: u(x, y), v(x, y), p(x, y)

Continuity Equation:

Momentum Equations:  

 
b2 – 4 ac = 0 – 4    Equations are Elliptic  
 

Transformation of the Mathematical Model

Use coordinate transformation:

Where C is a constant

 

Variable Transformation

Example: Navier-Stokes Equations   Introduce Stream Function – Vorticity Continuity equation is automatically satisfied!

Take derivative of "x" momentum equation with respect to y
Take derivative of "y" momentum equation with respect to x

Eliminate the pressure term

Resultant expressions are:
 

Stream Function equation Vorticity equation Note: Boundary conditions must also be transformed.

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Created by  Frank Schmidt
Maintained by John Mahaffy : jhm@psu.edu