A vortex sheet is a term used in fluid mechanics for a surface across which there is a discontinuity in fluid velocity, such as in slippage of one layer of fluid over another. While the tangential components of the flow velocity are discontinuous across the vortex sheet, the normal component of the flow velocity is continuous. The discontinuity in the tangential velocity means the flow has infinite vorticity on a vortex sheet. At high Reynolds numbers, vortex sheets tend to be unstable. In particular, they may exhibit Kelvin–Helmholtz instability. The formulation of the vortex sheet equation of motion is given in terms of a complex coordinate z = x + i y {\displaystyle z=x+iy} . The sheet is described parametrically by z ( s , t ) {\displaystyle z(s,t)} where s {\displaystyle s} is the arclength between coordinate z {\displaystyle z} and a reference point, and t {\displaystyle t} is time. Let γ ( s , t ) {\displaystyle \gamma (s,t)} denote the strength of the sheet, that is, the jump in the tangential discontinuity. Then the velocity field induced by the sheet is
∂ z ∗ ∂ t = − ı 2 π ∫ − ∞ ∞ γ ( s ′ , t ) d s ′ z ( s , t ) − z ( s ′ , t ) {\displaystyle {\frac {\partial z^{*}}{\partial t}}=-{\frac {\imath }{2\pi }}\int \limits _{-\infty }^{\infty }{\frac {\gamma (s',t)\mathrm {d} s'}{z(s,t)-z(s',t)}}}
The integral in the above equation is a Cauchy principal value integral. We now define Γ {\displaystyle \Gamma } as the integrated sheet strength or circulation between a point with arc length s {\displaystyle s} and the reference material point s = 0 {\displaystyle s=0} in the sheet.
Γ ( s , t ) = ∫ 0 s γ ( s ′ , t ) d s ′ a n d d Γ d s = γ ( s , t ) {\displaystyle \Gamma (s,t)=\int \limits _{0}^{s}\gamma (s',t)\mathrm {d} s'\qquad \mathrm {and} \qquad {\frac {\mathrm {d} \Gamma }{\mathrm {d} s}}=\gamma (s,t)}
As a consequence of Kelvin's circulation theorem, in the absence of external forces on the sheet, the circulation between any two material points in the sheet remains conserved, so d Γ / d t = 0 {\displaystyle \mathrm {d} \Gamma /\mathrm {d} t=0} . The equation of motion of the sheet can be rewritten in terms of Γ {\displaystyle \Gamma } and t {\displaystyle t} by a change of variable. The parameter s {\displaystyle s} is replaced by Γ {\displaystyle \Gamma } . That is,
∂ z ∗ ∂ t = − ı 2 π ∫ − ∞ ∞ d Γ ′ z ( Γ , t ) − z ( Γ ′ , t ) {\displaystyle {\frac {\partial z^{*}}{\partial t}}=-{\frac {\imath }{2\pi }}\int \limits _{-\infty }^{\infty }{\frac {d\Gamma '}{z(\Gamma ,t)-z(\Gamma ',t)}}}
This nonlinear integro-differential equation is called the Birkoff-Rott equation. It describes the evolution of the vortex sheet given initial conditions. Greater details on vortex sheets can be found in the textbook by Saffman (1977).
Diffusion of a vortex sheet Once a vortex sheet, it will diffuse due to viscous action. Consider a planar unidirectional flow at t = 0 {\displaystyle t=0} ,
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