Question

Relate flow to pressure and volume.  How do you change pressure in order to cause flow?

Answer 1

As the question is very generic and no fluid and flow conditions are mentioned, would first attempt to explain the fundamentals and deduce a much simpler case of incompressible steady state flow.

Fundamental laws relevant here are conservation of mass and conservation of linear momentum. One can follow Reynolds's transport theorem to get complete derivations for the same.

Conservation of mass states that rate of change of mass of a system is equal to rate of deformation of control volume and net change of mass flow rate into the control volume(out -in). Simplifying for a fixed control volume and steady state flow in one dimension it simplifies to

outflow=inflow i.e.

density_input x Area_input x Velocity_input = density_output x Area_output x Velocity_output

if an incompressible flow is there the density component is removed from the above equation.

Now moving to the linear momentum equation

it's a vector equation and V is a vector quantity

what this means is the sum of all forces action can be divided into bulk and surface components. This equation can be written in 1 D where the equation's second component becomes net flow rate difference.

same we can write in a differential form

which says

Gravity force per unit volume + pressure force per unit volume + viscous force per unit volume =

density x acceleration

Let's simplify this and remove the viscous components

integrating this along a streamline gives us the Bernoulli's equation

now as can be seen in all these equations, pressure gradient ( change in pressure) directly causes a change in kinetic quantity (see velocity ).

a very common application is flow through pipes due to the pressure gradient. Hagen-Poiseuilli equation is a famous equation which relates the same for flow through pipes