Fluid Dynamics: Flame front in a channel

[Mythbusters]

A flame front accelerates exponentially in a channel of width 2R with non-slip boundary conditions at the walls. Flame acceleration creates pressure gradient -deltaP/density=exp(sigma*t) (x-direction) directed parallel to the channel walls.

Assuming a plane-parallel flow, find velocity distribution in the channel.

I started with the Navier-Stokes equation and because of the plane-parallel flow and pressure gradient I came to this equation:

du/dt=exp(sigma*t)+my*du²/dy²

But now I don't know how I will continue, someone has an idea? Maybe I'm on the wrong track?

Mythbuster - "There's your problem"

 

[NateD]

The Navier-Stokes equation is not applicable for this problem because it does not take into account the boundary conditions. Instead, we need to use the continuity equation and the momentum equation.The continuity equation states that the flux of mass through any area of a fluid must remain constant, so we can use this to calculate the velocity distribution in the channel. The momentum equation takes into account the pressure gradient and the non-slip boundary conditions to solve for the velocity profile. The velocity distribution in the channel can be found by solving the following equations: Continuity equation: ∂u/∂t + ∂(uv)/∂x = 0 Momentum equation: ρ(∂u/∂t) + ρ(u∂u/∂x) + ∂P/∂x = μ (∂2u/∂y2) where u is the velocity in the x-direction, v is the velocity in the y-direction, P is the pressure, Ρ is the density and μ is the dynamic viscosity. By solving these equations with the given boundary conditions, we can find the velocity distribution in the channel.