Total Force Acting on a Cube of Fluid
For a incompressible fluid at rest there are no shear or deforming forces and Any stresses are normal to any surface drawn in the fluid and the pressure inside the fluid is the same in all directions.
Sincethough the fluid is incompressible the densityneed not be the same everywhere. We can writ
Draw a cube in the fluid.
The pressure on the face A isand the force isThe total force in the x direction is then
The total force is
This force must be balanced by an external force. If the external forces are due to a potential fieldthe force per unit mass is
The total force is thensince the fluid is in equilibrium.
Ifis constant then