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Fluid Statics - Variation of Pressure from Point to Point (Continued)

The pressure forces acting on an element determine the pressure gradient in each direction.

The free body diagram below shows the pressure forces on each face of the element of volume, δxδyδz. Note that the pressure force is the product of pressure on each face times its area. The symbol, g, represents the acceleration of gravity in the minus z-direction. Sum forces in each direction to determine the value of each pressure gradient ∂P/∂x, ∂P/∂y, and ∂P/∂z.

i.e. ∂P/∂x is the change in pressure in the x-direction over a distance δx

Σ Fx = P δy δz – (P+δP/δx) δy δz = 0 Result: ∂P/∂x = 0

Σ Fy = P δx δz – (P+δP/δy) δx δz = 0 Result: ∂P/∂y = 0

Σ Fz = P δy δz – (P+δP/δz) δx δy - γ δx δy δz = 0 Result: ∂P/∂z = - γ = - ρ g

To summarize, the fluid pressure varies linearly with depth (z-direction). If x and y represent

axes in the horizontal directions, then there is no change in pressure horizontally since both pressure

gradients, ∂P/∂x, and ∂P/∂y are zero.

Key Concepts:

Pressure at any given point in a fluid is the same in every direction.

Pressure changes from point to point if there is a pressure gradient. i.e. p = ˗ ρgz
Fluid pressure increases with depth (i.e. gravity is in the ˗ z direction).

Click here for examples.

Click here for calculation of pressure forces on submerged plates.

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