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A planar strained shear layer is a Burgers vortex sheet. For strain $u_x=-\alpha x$, $u_z=\alpha z$ and transverse velocity $u_y=U\mathrm{erf}(x/{\delta }_s)$, viscous balance gives ${\delta }_s^2=2\nu /\alpha$. At fixed velocity jump its excess viscous dissipation per sheet area scales as $U^2\sqrt{\nu \alpha }$, which vanishes as kinematic viscosity tends to zero.

Work with an incompressible Newtonian fluid of constant kinematic viscosity. The vorticity equation is $D\mathbf{\omega }/Dt=(\mathbf{\omega }\cdot \nabla )\mathbf{u}+\nu {\nabla }^2\mathbf{\omega }$. In the axisymmetric Burgers vortex the swirl does not advect the axial vorticity, and the stretching rate is ${\partial }_z{u}_z=\alpha$. Put $s(t)={\delta }^2(t)$ and $W={\omega }_z$. The Gaussian profile hasSubstitution into ${\partial }_{t}W+u_r{\partial }_{r}W=\alpha W+\nu {\nabla }^{2}W$ matches both the constant and $r^2$ terms precisely whenFor every finite positive initial $s_0$,The strain compresses the cross-section and stretches its vorticity, while kinematic viscosity spreads the core. The attracting radius is the balance of these effects; its relaxation time is ${\alpha }^{-1}$. Total circulation stays fixed because stretching increases the vorticity as the cross-sectional area decreases.

At the steady core radius, the swirl speed is of order ${\Gamma }_0/\delta$, and its velocity gradient tensor is of order ${\Gamma }_0/{\delta }^2$. Its viscous dissipation integrated over a cross-section therefore scales asThis is per unit length and per unit density. More precisely, for the swirl contribution, integration by parts relates the integrated strain viscous dissipation to $\nu \int W^{2}dA$; the boundary term vanishes for the $1/r$ swirl velocity. ThusMultiply by density for power per physical length. Here the azimuthal velocity's circulation $\Gamma$ is ${\Gamma }_0$, as follows by integrating $W$ over the cross-section. The result is finite as $\nu \to 0$ with strain and circulation held fixed. Local core viscous dissipation grows like ${\nu }^{-1}$ while its area shrinks like $\nu$. For ${\Gamma }_0/\nu \gg 1$, the swirl gradients dominate the background strain within the core.

This finite quantity is the excess viscous dissipation due to the vortex. The uniform straining flow alone dissipates $3\nu {\alpha }^2$ per unit volume, and integrating that over an infinite cross-section diverges. The total viscous dissipation of the complete idealized flow is therefore not a finite tube power unless the background is subtracted or spatially bounded.

A Burgers vortex sheet instead smooths a fixed velocity jump $U$ across a layer of thickness ${\delta }_s\sim \sqrt{\nu /\alpha }$. Its viscous dissipation per sheet area is of orderA tube with fixed circulation supports finite viscous dissipation in a vanishingly small core; a sheet with fixed velocity jump does not. This makes strained vortex tubes plausible intense dissipative structures at high Reynolds number. It does not assert that sheets are absent from turbulence: sheets can carry inertial range organization and become unstable, producing finer vortical structures.