# Steady-State Two-Dimensional Convection-Diffusion Equation

Steady-State Two-Dimensional Convection-Diffusion Equation

Consider the following two-dimensional convection-diffusion problem [1]:

1

PrRe

2

∂

∂

2

x

2

∂

∂

2

y

∂T

∂x

∂T

∂y

where , , , and . Here, and are the Reynolds and Prandtl numbers, is the unknown temperature distribution, and is the given velocity field. Take and ; then the temperature is symmetric with respect to the plane and the following Neumann conditions apply: ==0.

T(0,y)=0

T(1,y)=1

0≤x≤L=1

0≤y≤W=2

Re

Pr

T(x,y)

(u,v)

u=-sin(πx)cos(πy)

v=cos(πx)sin(πy)

T(x,y)

y=1

∂T

∂y

y=0

∂T

∂y

y=2

This Demonstration uses the finite-element capabilities of Mathematica in order to readily compute the temperature distribution for user-set values of the dimensionless numbers, and .

Re

Pr

If you increase , then you can observe that the temperature diffusion is much stronger. That is, the high temperatures generated at the right boundary penetrate much farther toward the left, which is the cold boundary.

Pr