Differential Analysis of Fluid Flow (continued)

In a Nut Shell: Fluid Kinematics (review)

General motion of a fluid element generally includes translation, rotation, linear deformation, and angular deformation. The rotation vector, ω, in general, has three components.

ω = ω_x i + ω_y j + ω_z k where

ω_z = ½ ( ∂v/∂x - ∂u/∂y) , ω_x = ½ ( ∂w/∂y - ∂v/∂z) , ω_y = ½ ( ∂u/∂z - ∂w/∂x)

The rotation vector is important in that it leads to the definition of the vorticity vector, ζ .

ζ = 2 ω = Del x V where Del is the “del operator”

Note: A special situation arises for two-dimensional fluid flow in the x-y plane when

∂v/∂x = ∂u/∂y. In that case rotation around the z-axis is zero giving rise to the condition

of irrotational flow which simplifies the analysis of complex flow fields.

Special Case of Fluid motion in the xy-plane

For steady, incompressible, two-dimensional fluid motion in the xy-plane conservation of mass

reduces to

∂u/∂x + ∂v/∂y = 0 (1)

This equation is also called the continuity equation.

It proves useful to define a new function, ψ(x,y) , called the stream function such that

u = ∂ ψ /∂y and v = - ∂ ψ /∂x , then

conservation of mass (the continuity equation) is automatically satisfied. To see this

result substitute u and v into eq. (1). Equation (1) is in Cartesian coordinates. In

polar coordinates eq. (1) becomes: ( Note here ψ = ψ(r,θ) )

v_r = (1/r)∂ψ /∂θ and v_θ = - ∂ ψ /∂r

Click here to continue this discussion.

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