Differential Analysis in Fluid Flow

Example The velocity components of an incompressible, two-dimensional velocity flow field

are give by the equations:

u = 2 xy

v = x² - y²

Show that the flow is irrotational and satisfies conservation of mass.

For irrotational flow: curl V = 0 So for 2-D ∂v / ∂x - ∂u / ∂y = 0

which yields 2 x – 2 x = 0. So the velocity flow field is irrotational. (result)

For conservation of mass for an incompressible, 2-D velocity flow field:

∂u / ∂x + ∂v / ∂y = 0

which yields 2 y - 2y = 0 So conservation of mass is preserved. (result)

Example In a steady, two-dimensional flow field the fluid density varies linearly with respect

to the coordinate, x. i.e. ρ = Ax where A is a constant. If the x component of velocity, u, is

given by u = y, find an expression for the y component, v.

For conservation of mass: ∂(ρu) / ∂x + ∂(ρv) / ∂y = 0

Now ρu = Axy so ∂(ρu) / ∂x = Ay and Ay + ∂(ρv) / ∂y = 0

So ∂(ρv) / ∂y = - Ay integration with respect to y gives

ρv = - ½ Ay² + g(x)

or (Ax) v = - ½ Ay² + g(x) Solve for v.

v = - y² / 2x + g(x)/Ax = - y² / 2x + f(x) (result)

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