Deriving Pressure on a Fluid Surface at Rest with Constant Density

[coverband]

Homework Statement

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

Homework Equations

Euler's equation: $$\frac{Du}{Dt}=-\frac{1}{\rho}\nabla P + F$$

The Attempt at a Solution

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

 

[hunt_mat]

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

 

[coverband]

Thanks