Pressure in case of Accelerating Fluid

1) Consider a vessel containing a fluid. Let the acceleration of vessel in vertical direction be, a_y. Consider a small cylindrical element dy at a height y from bottom

From figure,

P₁ A - P₂A - mg = may

P₁ A - P₂A = m (g + ay)

But (P₁ - P₂) A2 = A dy (g + ay)

- dp/ dy = - (g + ay)

2) Consider a vessel containing a fluid. Let the acceleration of vessel in horizontal direction be

From diagram

(P₁ - P₂) A = Adx (ax)

Δp/Δx = - ax

If θ is the angle made by the free liquid surface with horizontal, then

tanθ = dh/dx

But we have

(P₁ - P₂) A = Adx (ax)

(h₁ - h₂) gA = Adx (ax)

dh/ dx = aₓ/g

tanθ = aₓ/g