That seems to be a homework. The theory of the BL over a flat plate is illustrated in any textbook of fluid mechanics. The pressure gradient is shown to be a known term in the Blasius theory.
An important assumption of BL theory, dp/dy=0. Pressure stays unchanged across the BL.
There is no curvature in the flow direction(x-direction), therefore the pressure gradient is zero.
In y-direction, also the pressure gradient is supposed to be zero, because of the small thickness of the boundary layer.
for more information use the bellow textbooks:
- Fox RW, McDonald AT, Mitchell JW. Fox and McDonald's introduction to fluid mechanics. John Wiley & Sons; 2020 Jun 30.
- Gerhart PM, Gerhart AL, Hochstein JI. Munson, Young and Okiishi's Fundamentals of Fluid Mechanics. John Wiley & Sons; 2016 Sep 13.
The question seems to be wrong. It could have been "how does the pressure gradient change in the boundary layer over a flat plat?"
There is no assumption on dp/dy but only a result from the dimensional analysis of the NSE in the proper lenght scales. The result is that dp/dy->0 for Re->Inf but is never exactly zero.
If you consider the Prandtl hypothesis, the pressure may be considered as being equal to zero.
The pressure across the boundary layer does not change. The pressure is impressed on the boundary layer it value can be calculated by hydrodynamic consideration.
Pressure gradient across boundary layer, dp/dy is almost zero for flat plate. However along the flow, I.e. dp/dx could vary. Its value can be found by solving boundary layer equations, which is Blasius solution and available in literature.
In almost parallel flows the transversal pressure gradient is directly related to turbulence gradient;
Academic resources on turbulent flows are provided on:
SINGLE PHASE AND MULTIPHASE TURBULENT FLOWS (SMTF) IN NATURE AND ENGINEERING APPLICATIONS | Jamel Chahed | 3 publications | Research Project (researchgate.net)
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