structural:scholar_edition
Strong form

Consider the incompressible Navier–Stokes equations for a Newtonian fluid of constant density ρ {\textstyle \rho } in a domain Ω ⊂ R d (d = 2, 3) {\displaystyle \Omega \subset \mathbb {R} ^{d}\quad (d=2,3)} with boundary ∂ Ω = Γ D ∪ Γ N, {\displaystyle \partial \Omega =\Gamma _{D}\cup \Gamma _{N},} being Γ D {\textstyle \Gamma _{D}} and Γ N {\textstyle \Gamma _{N}} portions of the boundary where respectively a Dirichlet and a Neumann boundary condition is applied (Γ D ∩ Γ N = ∅ {\textstyle \Gamma _{D}\cap \Gamma _{N}=\emptyset }): { ρ ∂ u ∂ t + ρ (u ⋅ ∇) u − ∇ ⋅ σ (u, p) = f in Ω × (0, T) ∇ ⋅ u = 0 in Ω × (0, T) u = g on Γ D × (0, T) σ (u, p) n ^ = h on Γ N × (0, T) u (0) = u 0 in Ω × { 0 } {\displaystyle {\begin{cases}\rho {\dfrac {\partial \mathbf {u} }{\partial t}}+\rho (\mathbf {u} \cdot \nabla)\mathbf {u} -\nabla \cdot {\boldsymbol {\sigma }}(\mathbf {u},p)=\mathbf {f} &{\text{ in }}\Omega \times (0,T)\\\nabla \cdot \mathbf {u} =0&{\text{ in }}\Omega \times (0,T)\\\mathbf {u} =\mathbf {g} &{\text{ on }}\Gamma _{D}\times (0,T)\\{\boldsymbol {\sigma }}(\mathbf {u},p){\hat {\mathbf {n} }}=\mathbf {h} &{\text{ on }}\Gamma _{N}\times (0,T)\\\mathbf {u} (0)=\mathbf {u} _{0}&{\text{ in }}\Omega \times \{0\}\end{cases}}} u {\textstyle \mathbf {u} } is the fluid velocity, p {\textstyle p} the fluid pressure, f {\textstyle \mathbf {f} } a given forcing term, n ^ {\displaystyle {\hat {\mathbf {n} }}} the outward directed unit normal vector to Γ N {\textstyle \Gamma _{N}}, and σ (u, p) {\textstyle {\boldsymbol {\sigma }}(\mathbf {u},p)} the viscous stress tensor defined as: σ (u, p) = − p I + 2 μ ε (u). {\displaystyle {\boldsymbol {\sigma }}(\mathbf {u},p)=-p\mathbf {I} +2\mu {\boldsymbol {\varepsilon }}(\mathbf {u}).} Let μ {\textstyle \mu } be the dynamic viscosity of the fluid, I {\textstyle \mathbf {I} } the second-order identity tensor and ε (u) {\textstyle {\boldsymbol {\varepsilon }}(\mathbf {u})} the strain-rate tensor defined as: ε (u) = 1 2 ((∇ u) + (∇ u) T). {\displaystyle {\boldsymbol {\varepsilon }}(\mathbf {u})={\frac {1}{2}}\left(\left(\nabla \mathbf {u} \right)+\left(\nabla \mathbf {u} \right)^{\mathrm {T} }\right).} The functions g {\textstyle \mathbf {g} } and h {\textstyle \mathbf {h} } are given Dirichlet and Neumann boundary data, while u 0 {\textstyle \mathbf {u} _{0}} is the initial condition. The first equation is the momentum balance equation, while the second represents the mass conservation, namely the continuity equation. Assuming constant dynamic viscosity, using the vectorial identity ∇ ⋅ (∇ f) T = ∇ (∇ ⋅ f) {\displaystyle \nabla \cdot \left(\nabla \mathbf {f} \right)^{\mathrm {T} }=\nabla (\nabla \cdot \mathbf {f})} and exploiting mass conservation, the divergence of the total stress tensor in the momentum equation can also be expressed as: ∇ ⋅ σ (u, p) = ∇ ⋅ (− p I + 2 μ ε (u)) = − ∇ p + 2 μ ∇ ⋅ ε (u) = − ∇ p + 2 μ ∇ ⋅ = − ∇ p + μ (Δ u + ∇ ⋅ (∇ u) T) = − ∇ p + μ (Δ u + ∇ (∇ ⋅ u) ⏟ = 0) = − ∇ p + μ Δ u. {\displaystyle {\begin{aligned}\nabla \cdot {\boldsymbol {\sigma }}(\mathbf {u},p)&=\nabla \cdot \left(-p\mathbf {I} +2\mu {\boldsymbol {\varepsilon }}(\mathbf {u})\right)\\&=-\nabla p+2\mu \nabla \cdot {\boldsymbol {\varepsilon }}(\mathbf {u})\\&=-\nabla p+2\mu \nabla \cdot \left\\&=-\nabla p+\mu \left(\Delta \mathbf {u} +\nabla \cdot \left(\nabla \mathbf {u} \right)^{\mathrm {T} }\right)\\&=-\nabla p+\mu {\bigl (}\Delta \mathbf {u} +\nabla \underbrace {(\nabla \cdot \mathbf {u})} _{=0}{\bigr)}=-\nabla p+\mu \,\Delta \mathbf {u}.\end{aligned}}} Moreover, note that the Neumann boundary conditions can be rearranged as: σ (u, p) n ^ = (− p I + 2 μ ε (u)) n ^ = − p n ^ + μ ∂ u ∂ n ^. {\displaystyle {\boldsymbol {\sigma }}(\mathbf {u},p){\hat {\mathbf {n} }}=\left(-p\mathbf {I} +2\mu {\boldsymbol {\varepsilon }}(\mathbf {u})\right){\hat {\mathbf {n} }}=-p{\hat {\mathbf {n} }}+\mu {\frac {\partial {\boldsymbol {u}}}{\partial {\hat {\mathbf {n} }}}}.}