Pressure Variation – Rigid Body
Motion
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.)
In
component form: -
∂P/∂x = ρ ax ,
- ∂P/∂y = ρ ay , - ∂P/∂z = γ + ρ
az 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 ax
, ay , az =
components of acceleration in the x, y, and z-directions ar
, a θ , az =
components of acceleration in the r, θ , and z-directions and gravity
is in the – z – direction |
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