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Differential Analysis of Fluid Flow

Key Concept: An alternative to examining fluid motion in a finite control volume is to"shrink"

the volume to a point and examine the conditions of fluid motion at any given point. This

approach leads to differential equations in the "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)∂(rρv_r) /∂r + (1/r)(ρv_θ) /∂θ + ∂(ρv_z) /∂z = 0

where ρ is the fluid density, v_r, v_θ, v_z 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 g_x - ∂P/∂x

ρ [∂v /∂t + u ∂v /∂x + v ∂v/∂y + w ∂v /∂z] = ρ g_y- ∂P/∂y

ρ [∂w /∂t + u ∂w /∂x + v ∂w/∂y + w ∂w /∂z] = ρ g_z- ∂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 g_x , ρ g_y , ρ g_z 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.