Differential Analysis
of Fluid Flow
In a Nutshell: Imagine the control
volume being shrunk to a point. Then
the differential form for conservation of mass at any
given point becomes: (in Cartesian
coordinates) ∂ρ /∂t + ∂(ρu)
/∂x + ∂(ρv)
/∂y + ∂(ρw)
/∂z = 0
where
ρ is the fluid density, u,v, and w are the
components of fluid velocity Conservation of mass
in cylindrical polar coordinates ( r, θ, z ): ∂ρ
/∂t +
(1/r)∂(rρvr) /∂r +
(1/r)(ρvθ)
/∂θ + ∂(ρvz)
/∂z = 0
where
ρ is the fluid density, vr, vθ,
vz
are the components of fluid velocity Likewise,
the differential form for conservation
of linear momentum at any given point, assuming inviscid
flow becomes: ρ [∂u
/∂t + u ∂u /∂x + v ∂u/∂y + w ∂u
/∂z] = ρ gx
-
∂P/∂x ρ [∂v
/∂t + u ∂v /∂x + v ∂v/∂y + w
∂v /∂z] = ρ gy
- ∂P/∂y ρ [∂w
/∂t + u ∂w /∂x + v
∂w/∂y + w ∂w
/∂z] = ρ gz
- ∂P/∂z These
equations go by the name, Euler’s equations of motion for an inviscid fluid. Again
for steady flow, the acceleration terms
∂u /∂t , ∂u /∂t, and ∂u /∂t are
zero. The
acceleration terms u ∂u
/∂x , v ∂u/∂y, and w
∂u /∂z etc are “convective"
acceleration terms.
They are nonlinear and make solution
difficult except for special cases. The
terms ρ gx
, ρ gy , ρ gz result from gravitational forces on the
fluid element. The
terms ∂P/∂x , ∂P/∂x , and ∂P/∂x are pressure gradients acting on the fluid
element. |
Copyright © 2019 Richard C. Coddington
All rights reserved.