Distribution of pressure inside a capsule

Question

How would pressure of an ideal gas be distributed over the inside of a capsule (a cylinder with semi-spheres on the ends)? What about the strain on the material? Is there a general formula for how much?

Answer 1

If the cylinder is stationary, it is according to the hydrostatic pressure equation:

$\vec{\nabla} p = \rho \vec{g}$

You can derive this equation by eliminating all velocities in the Navier-Stokes equations. If gravity is oriented in the negative $y$ direction, the equation becomes:

$\frac{dp}{dy} = -\rho g$

EDIT:

If you integrate this equation you get an expression for the pressure at any point in the cylinder:

$p = -\rho g y + C$

where $C$ is a constant that you pick to satisfy boundary conditions. Basically the points on the surface of the capsule that are closer to earth have higher pressure than the points that are further away. (Think about what your ears feel if you dive to the bottom of a swimming pool.)

As for the distribution over the semi-sphere ends, the pressure acts normal to the surface. Using this fact you can integrate the equation for the pressure over these surfaces to get the forces.