Consider the fully developed laminar flow of a fluid in a tube with a wall temperature ; the fluid enters at a uniform temperature . Assuming constant physical properties and axial symmetry, the dimensionless energy equation is:

,

with boundary conditions:

, ,

, , in which the dimensionless variables are given by:

,

,

,

,

,

,

,

where

and are the radial and axial coordinates, respectively,

is the tube radius,

is the tube length,

is the fluid specific gravity,

is the fluid heat capacity,

is the average laminar velocity, and

is the fluid thermal conductivity.

The dimensionless equation is solved using the built-in Mathematica function NDSolve; the effect of the Péclet number and the Brinkman number on the temperature distribution is shown.