- #1

skrat

- 748

- 8

## Homework Statement

Imagine two parallel plates at distance ##L##. One of them at constant temperature ##T_H## and the other one at ##T_C##. Media between the plates is air.

Find ##T## as function of distance from the plate with ##T_H## in steady state.

## Homework Equations

Navier - Stokes

Heat transfer

Mass continuity

## The Attempt at a Solution

So I started with Navier Stokes

## \rho \frac{\partial T(x)}{\partial t} + \rho \nabla \cdot (\vec v\vec v) = - \nabla P + \nabla (\mu \nabla \vec v) + \vec b##

where I assumed steady state and no external body force. In addition to that I assumed pressure ##P## is constant, therefore

## \rho \nabla \cdot (\vec v\vec v) = \nabla (\mu \nabla \vec v).##

Rewritten in one dimension, that is

##\frac{\partial ^2 v}{\partial x^2}-\frac{\rho}{\mu}\frac{\partial v}{ \partial x} = 0##

The one dimensional heat equation in steady state should be

##\rho c_p\frac{\partial}{\partial x} (T(x)v(x)) = \lambda \frac{\partial ^2}{\partial x^2}T(x)##

ASSUMING everything is ok until here, I'm not sure how to solve these two DE. Any hints?