Knowlpedia

Theorem

Difference equation for two incompressible flows

The common force cancels and the nonlinear difference splits into transport and reference-gradient terms.

Statement

Let (u,p)(u,p) and (v,P)(v,P) solve the with the same force and viscosity. Set w=v−uw=v-u and π=P−p\pi=P-p. Their difference equation is

∂tw+(v⋅∇)w+(w⋅∇)u=νΔw−∇π,∇⋅w=0.\partial_t w+(v\cdot\nabla)w+(w\cdot\nabla)u =\nu\Delta w-\nabla\pi,\qquad \nabla\cdot w=0.

Equivalently, ∂tw+div⁡g=νΔw−∇π\partial_t w+\operatorname{div}g=\nu\Delta w-\nabla\pi, with g=v⊗v−u⊗ug=v\otimes v-u\otimes u and row divergence (div⁡g)i=∑j∂jgij(\operatorname{div}g)_i=\sum_j\partial_jg_{ij}.

Exact expansion

The tensor difference is g=w⊗w+w⊗u+u⊗wg=w\otimes w+w\otimes u+u\otimes w. Since both velocities are divergence-free, its divergence equals the difference of their advective terms. Expanding v=u+wv=u+w proves the nonconservative formula as well. The shared force disappears under subtraction; if forces differ, their difference remains on the right.