Knowlpedia

Definition

Viscous dissipation in the incompressible energy balance

Viscosity times the spatial square-integral of the velocity gradient.

For kinematic viscosity ν>0\nu>0, the viscous dissipation rate in the unit-density whole-space or periodic energy balance is

D(t)=ν∫Ω∣∇u(t,x)∣2 dx,∣∇u∣2=∑i,j∣∂jui∣2.D(t)=\nu\int_\Omega|\nabla u(t,x)|^2\,dx, \qquad |\nabla u|^2=\sum_{i,j}|\partial_j u_i|^2.

The matrix norm here is the . Total dissipation over II is ∫ID(t) dt\int_I D(t)\,dt, when finite.

Strain and gradient conventions

For divergence-free fields with vanishing boundary terms, integration by parts gives 2∫∣sym⁡∇u∣2=∫∣∇u∣22\int|\operatorname{sym}\nabla u|^2=\int|\nabla u|^2. The physical local strain dissipation is 2ν∣sym⁡∇u∣22\nu|\operatorname{sym}\nabla u|^2; it need not equal ν∣∇u∣2\nu|\nabla u|^2 pointwise. Their integrated equality uses incompressibility and the boundary conditions.