Pressure gradient across flat plate with flow is zero?

[Urmi Roy]

So I learned recently that pressure gradient in the flow direction for flow over a flat plate is zero. However I don't understand this, because there has to be something that sets the flow in motion in the first place, and for fluids this has to be a pressure gradient.
Could someone explain why in flow over a flat plate dp/dx=0?

 

[boneh3ad]

The fluid motion might also be caused by, for example, the plate moving through the air. In that case there would be no pressure gradient but still a flow of air over the plate relative to the plate. That's just a counter-example that proves it can happen. The reason why is that you need a pressure gradient to accelerate a flow, not have it moving in the first place. Assuming the flow has already been set into motion, it can remain in motion at a constant velocity unless it encounters a pressure gradient, hence the Blasius boundary layer.

 

[Urmi Roy]

Thanks Boneh3ad. So I can say that if there is viscosity (and hence a momentum boundary layer) we would need a dp/dx to keep the flow moving at constant velocity?

 

[boneh3ad]

No. The inviscid flow outside the boundary layer doesn't see the effects of the viscosity anyway so you don't need a pressure gradient to keep it going. Viscous effects are confined to the boundary layer where they cause the velocity to fall to zero at the wall but they don't affect the outer flow.

 

[Urmi Roy]

Great, thanks Boneh3ad, I think I got this. One remaining question I have, and occurred to me while doing problems on boundary layers is for flow over a flat plate, in case there is acceleration in the fluid (that is in potential flow), will the du/dt term in the x direction Navier-Stokes equation still go away?
Thanks!

 

[boneh3ad]

It depends on whether you mean acceleration in that question. If, at a single point in space, the velocity changes, then the $\partial /\partial t$ term does not vanish. If it varies in space but is constant at a given point for all time, then it does vanish.