7A (Fluid Dynamics)

R = x^{2} + y^{2}

, direct differentiation gives

\frac{\partial u}{\partial x} = \frac{- 2 x ( x ^{2} - 3 y ^{2} )}{R ^{3}}, \frac{\partial v}{\partial y} = \frac{2 x ( x ^{2} - 3 y ^{2} )}{R ^{3}},

\nabla \cdot u = 0

. Likewise,

\frac{\partial v}{\partial x} = \frac{- 2 y ( 3 x ^{2} - y ^{2} )}{R ^{3}} = \frac{\partial u}{\partial y},

so the two-dimensional vorticity

\partial_{x} v - \partial_{y} u

vanishes. The flow is therefore incompressible and irrotational away from the origin.

b

With the convention

u = \frac{\partial ψ}{\partial y}, v = - \frac{\partial ψ}{\partial x},

ψ (x, y) = \frac{y}{x ^{2} + y ^{2}},

up to an additive constant. Differentiation recovers both velocity components.

ϕ

, one requires

u = \partial_{x} ϕ

and

v = \partial_{y} ϕ

. Integration gives

ϕ (x, y) = - \frac{x}{x ^{2} + y ^{2}},

There are no discussions about this article yet.

There are currently no matching articles.