# Viscous diffusion of multiple vortex system

(Difference between revisions)
 Revision as of 08:14, 16 September 2005 (view source)Praveen (Talk | contribs)← Older edit Latest revision as of 07:40, 12 April 2007 (view source)Jola (Talk | contribs) m (Reverted edits by DmzRly (Talk); changed back to last version by Praveen) (2 intermediate revisions not shown) Line 1: Line 1: - The following analytical solution satisfies the incompressible continuity and momentum + The following analytical solution satisfies the viscous, incompressible - equations in dimension-less form in the domain $0 \le x, y \le 2\pi$. The + continuity and momentum equations in dimension-less form in the domain $0 - solution is periodic in both [itex]x$ and $y$ coordinates. + \le x, y \le 2\pi[/itex]. The solution is periodic in both $x$ and + $y$ coordinates. :$:[itex] - u(x,y,t) = -\cos x \sin y e^{-2t/Re} + u(x,y,t) = -(\cos x \sin y) e^{-2t/Re}$ [/itex] :$:[itex] - v(x,y,t) = \sin x \cos y e^{-2t/Re} + v(x,y,t) = (\sin x \cos y) e^{-2t/Re}$ [/itex] Line 14: Line 15: p(x,y,t) = -0.25( \cos 2x + \cos 2y) e^{-4t/Re} p(x,y,t) = -0.25( \cos 2x + \cos 2y) e^{-4t/Re} [/itex] [/itex] + + where $u,v$ are the Cartesian velocity components, $p$ + is the pressure and $Re$ is the [[Reynolds number]].

## Latest revision as of 07:40, 12 April 2007

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.