Past exam of the fluids and waves course of the University of Oxford / 2014 / 3 / a

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- Solution a

Solution

The total velocity fields on the two sides are

u_{+} = (U, 0) + \nabla ϕ_{+}, u_{-} = \nabla ϕ_{-} .

Each base flow is divergence-free. Incompressibility therefore makes both disturbance velocity potentials satisfy the Laplace equation:

ϕ_{+, xx} + ϕ_{+, yy} = 0, ϕ_{-, xx} + ϕ_{-, yy} = 0.

At the fixed horizontal walls, impermeability requires zero vertical velocity:

ϕ_{+, y} (x, h, t) = 0, ϕ_{-, y} (x, - h, t) = 0.

The discontinuity in tangential base velocity at the interface is a vortex sheet. The assumption of irrotational flow is understood separately in each open fluid region; it cannot hold distributionally across that sheet when

U \neq = 0

.

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