Pressure gradient across flat plate with flow is zero?

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Discussion Overview

The discussion centers around the pressure gradient in the flow direction for fluid flow over a flat plate, specifically questioning why the pressure gradient (dp/dx) is considered to be zero in this scenario. Participants explore concepts related to fluid motion, boundary layers, and the implications of viscosity and acceleration in fluid dynamics.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant questions the assertion that the pressure gradient in the flow direction is zero, suggesting that a pressure gradient is necessary to initiate fluid motion.
  • Another participant proposes that fluid motion can occur due to the movement of the plate through the air, indicating that a pressure gradient is not required for flow to exist.
  • A participant inquires whether viscosity necessitates a pressure gradient to maintain constant flow velocity, implying a connection between viscosity and flow behavior.
  • Another participant counters that inviscid flow outside the boundary layer does not require a pressure gradient to maintain motion, as viscous effects are limited to the boundary layer.
  • A participant raises a question regarding the Navier-Stokes equation and the treatment of acceleration in fluid flow, specifically whether the time derivative term can be neglected under certain conditions.
  • A later reply clarifies that the time derivative term in the Navier-Stokes equation does not vanish if velocity changes at a single point in space, but does vanish if the velocity is constant at that point over time.

Areas of Agreement / Disagreement

Participants express differing views on the necessity of a pressure gradient for maintaining flow and the implications of viscosity. The discussion remains unresolved regarding the conditions under which the time derivative term in the Navier-Stokes equation can be neglected.

Contextual Notes

Participants reference concepts such as the Blasius boundary layer and potential flow, indicating that assumptions about flow conditions and definitions may vary. The discussion involves nuanced technical arguments that are not fully settled.

Urmi Roy
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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?
 
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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.
 
Last edited:
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?
 
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.
 
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!
 
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.
 

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