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Hydrostatic pressure law

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  • #1
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Homework Statement


Consider a stationary fluid (u=0) with constant density and take F= (0,0,-g). Find P(z) which satisfies [tex] P=P_a [/tex] on [tex] z=h_0 [/tex], where z is measured positive upwards. What is the pressure on z=0?


Homework Equations



Euler's equation: [tex] \frac{Du}{Dt}=-\frac{1}{\rho}\nabla P + F [/tex]

The Attempt at a Solution


[tex]\frac{1}{\rho}\nabla P = (0,0,-g) [/tex] Gives the answer in the back of the book as:
then [tex] P = P_a + \rho g(h_0-z); P(0) = P_a + \rho g h_0 [/tex]. How did they get this? Thanks
 

Answers and Replies

  • #2
hunt_mat
Homework Helper
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As they say u=0, then as you correctly wrote down:
[tex]
\frac{1}{\rho}\nabla P=(0,0,-g)
[/tex]
Which means that:
[tex]
\frac{\partial P}{\partial x}=0,\quad\frac{\partial P}{\partial y}=0,\frac{\partial P}{\partial z}=-\rho g
[/tex]
Which shows that the pressure in independent of both x & y. so you are left to solve:
[tex]
\frac{\partial P}{\partial z}=-\rho g
[/tex]
Can you solve this? What are the boundary conditions that you need to use?
 
  • #3
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Thanks
 

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