## PASCAL'S LAW - BASIC DEFINITION AND TUTORIALS

In fluids under static conditions pressure is found to be independent of the orientation of the area. This concept is explained by Pascal’s law which states that the pressure at a point in a fluid at rest is equal in magnitude in all directions.

Tangential stress cannot exist if a fluid is to be at rest. This is possible only if the pressure at a point in a fluid at rest is the same in all directions so that the resultant force at that point will be zero.

The proof for the statement is given below.

Consider a wedge shaped element in a volume of fluid as shown in Figure above. Let the thickness perpendicular to the paper be dy. Let the pressure on the surface inclined at an angle θ to vertical be Pθ and its length be dl. Let the pressure in the x, y and z directions be Px, Py, Pz.

First considering the x direction. For the element to be in equilibrium,

Pθ × dl × dy × cos θ = Px × dy × dz

But, dl × cos θ = dz So, Pθ = Px

When considering the vertical components, the force due to specific weight should be considered.
Pz × dx × dy = Pθ × dl × dy × sin θ + 0.5 × γ × dx × dy × dz

The second term on RHS of the above equation is negligible, its magnitude is one order less compared to the other terms.

Also, dl × sin θ = dx, So, Pz = Pθ
Hence, Px = Pz = Pθ

Note that the angle has been chosen arbitrarily and so this relationship should hold for all angles. By using an element in the other direction, it can be shown that Py = Pθ and so Px = Py = Pz

Hence, the pressure at any point in a fluid at rest is the same in all directions. The pressure at a point has only one value regardless of the orientation of the area on which it is measured.

This can be extended to conditions where fluid as a whole (like a rotating container) is accelerated like in forced vortex or a tank of water getting accelerated without relative motion between layers of fluid. Surfaces generally experience compressive forces due to the action of fluid pressure.