Engr Help

Pressure Variation – Rigid Body Motion

Key Concept: Fluid can translate and rotate in a container as a "chunk" of fluid. i.e. If you

accelerate at a constant rate in your car, the surface of the gasoline in the tank slopes.

In a Nut Shell: A fluid element undergoing rigid body acceleration responds to the pressure

gradient, del(P) and the body force, γ k. Rigid body acceleration may be translation or rotation.

(del is the vector operator < ∂/∂x, ∂/∂y, ∂/∂z >) or del(P) = < ∂P/∂r, (1/r)∂P/∂θ, ∂P/∂z >

The equation of motion is: (See figures below.)

-del (P) - γ k = ρ a

In component form: - ∂P/∂x = ρ a_x , - ∂P/∂y = ρ a_y , - ∂P/∂z = γ + ρ a_z

In component form: ∂P/∂r = ρrω², ∂P/∂θ = 0, ∂P/∂z = - γ

where ∂P/∂x = pressure gradient in the x-direction, ∂P/∂r = pressure gradient in the r-direction

∂P/∂y = pressure gradient in the y-direction, (1/r)∂P/∂ θ = pressure gradient in the θ -direction

∂P/∂z = pressure gradient in the z-direction,

ρ = fluid mass density

γ = fluid specific weight

a_x , a_y , a_z = components of acceleration in the x, y, and z-directions

a_r , a_θ , a_z = components of acceleration in the r, θ , and z-directions

and gravity is in the – z – direction