Fluid Dynamics | Part IB, 2004

From the general mass-conservation equation, show that the velocity field u(x)\mathbf{u}(\mathbf{x}) of an incompressible fluid is solenoidal, i.e. that u=0\nabla \cdot \mathbf{u}=0.

Verify that the two-dimensional flow

u=(yx2+y2,xx2+y2)\mathbf{u}=\left(\frac{y}{x^{2}+y^{2}}, \frac{-x}{x^{2}+y^{2}}\right)

is solenoidal and find a streamfunction ψ(x,y)\psi(x, y) such that u=(ψ/y,ψ/x)\mathbf{u}=(\partial \psi / \partial y,-\partial \psi / \partial x).

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