Question

Does the Bernoulli’s equation change when the fluid is at rest? How?

Answer 1

Bernoulli’s principle applies to fluids in motion. Applying it to a fluid at rest is futile. When a fluid is at rest, the Bernoulli equation calculates the pressure difference due to a liquid column.

For a static fluid, v₁ = v₂ = 0. Bernoulli’s equation in that case is:

p₁ + ρgh₁ = P₂ + ρgh₂

Further, taking h₂ as the reference height of zero, i.e., by setting h₂ = 0, we get p₂ = p₁ + ρgh₁.

This equation tells us that in static fluids, pressure increases with depth. As we go from point 1 to point 2 in the fluid, the depth increases by h₁ and, consequently, p₂ is greater than p₁ by an amount ρgh₁.

If p₁ = p₀, the atmospheric pressure at the top of the fluid, we get the familiar gauge pressure at depth h₁ = ρgh₁. Thus, Bernoulli’s equation confirms the fact that the pressure change due to the weight of a fluid column of length h is ρgh.