Viscous diffusion of multiple vortex system

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The following analytical solution satisfies the viscous, incompressible continuity and momentum equations in dimension-less form in the domain $0 \le x, y \le 2\pi$ . The solution is periodic in both $x$ and $y$ coordinates.

$u(x,y,t) = -(\cos x \sin y) e^{-2t/Re}$

$v(x,y,t) = (\sin x \cos y) e^{-2t/Re}$

$p(x,y,t) = -0.25( \cos 2x \cos 2y) e^{-4t/Re}$

where $u,v$ are the Cartesian velocity components, $p$ is the pressure and $Re$ is the Reynolds number.

Viscous diffusion of multiple vortex system

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