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The axial strain rate is the axial component of the rate-of-strain tensor. For axial velocity $w(z)$ it is $E=\partial w/\partial z$. In a uniform uniaxial extensional flow, $w=Ez$ and incompressible flow requires $u_r=-Er/2$; positive $E$ stretches the axial direction and contracts the transverse directions.

For a two-dimensional incompressible flow, a Cartesian streamfunction $\psi$ defines $u={\partial }_y\psi$ and $v=-{\partial }_x\psi$. The continuity equation $u_x+v_y=0$ then holds identically.

For an incompressible flow with constant mass density, the Elsässer variables are ${\mathbf{z}}^{\pm }=\mathbf{u}\pm {\mathbf{v}}_a$, where ${\mathbf{v}}_a$ is the Alfvén velocity. Adding and subtracting the Euler momentum equation and ideal magnetohydrodynamic induction equation yields ${\partial }_t{\mathbf{z}}^{\pm }+({\mathbf{z}}^{\mp }\cdot \nabla ){\mathbf{z}}^{\pm }=-\nabla \psi$, with $\psi =p/\rho +\Phi +|{\mathbf{v}}_a{|}^2/2$ and $\nabla \cdot {\mathbf{z}}^{\pm }=0$. Thus each field is transported by the other, a useful way to expose interactions of oppositely directed Alfvén waves.

For constant density ${\rho }_0$ and body force ${\Omega }^2(3x-2qz){\mathbf{e}}_x$, dotting the incompressible flow momentum equation with ${\rho }_0\mathbf{u}$ gives the kinetic energy flux $(K+P)\mathbf{u}$ and source ${\rho }_0{\Omega }^2(3x-2qz)u_x$, where $K={\rho }_0|\mathbf{u}{|}^2/2$. Coriolis acceleration does no work. Absorbing ${\Phi }_t=-3{\Omega }^2{x}^2/2$ gives the displayed balance. For nonzero $q$, the remaining force has curl $-2{\Omega }^{2}q{\mathbf{e}}_y$, so it cannot be incorporated into a scalar mechanical potential. The model can exchange energy with its maintained vertical-shear background; at $q=0$ ordinary Jacobi energy in a shearing sheet conservation is recovered under zero boundary flux.

For constant positive order-parameter mobility $M$, dynamic viscosity $\eta$ and mass density $\rho$, deterministic Model H dynamics dissipates the sum of composition free energy and kinetic energy:Assume smooth fields and periodic or closed, no-work boundaries. Using the chemical potential of a composition field, integration by parts and incompressible flow gives $\dot{F}=\int \varphi \mathbf{v}\cdot \nabla \mu -M\int |\nabla \mu {|}^2$. Dotting the momentum equation with velocity gives ${\dot{E}}_{\mathrm{kin}}=-\int \varphi \mathbf{v}\cdot \nabla \mu -\eta \int |\nabla \mathbf{v}{|}^2$. Pressure and convective work vanish under the boundary assumptions; the two capillary transfer terms cancel. This proves both the sign of the Korteweg force and its reversible exchange of energy with the composition field.

Two deep layers of incompressible flow with upper mass density ${\rho }_1$, lower mass density ${\rho }_2$, and tangential velocities $U_1,U_2$ have this dispersion relation when surface tension is absent. Decaying velocity potentials and the linearized kinematic boundary condition determine their amplitudes; the dynamic boundary condition for an inviscid interface gives the displayed relation. For $k>0$ the squared departure of the phase velocity $c$ from the density-weighted mean velocity is $g({\rho }_2-{\rho }_1)/[k({\rho }_1+{\rho }_2)]-{\rho }_1{\rho }_2(U_1-U_2{)}^2/({\rho }_1+{\rho }_2{)}^2$. A negative value gives Kelvin-Helmholtz instability despite stable density stratification.

The Kelvin stellar modes are the potential incompressible stellar surface modes of a nonrotating self-gravitating uniform-density sphere with incompressible flow. For a regular displacement potential $U=A{r}^l{Y}_l^m$, the free-surface Lagrangian pressure perturbation condition and the surface gravity perturbation of a uniform-density sphere giveIn detail $\delta p/\rho ={\omega }^{2}U-\delta \Phi$ and $\delta p/\rho =(GM/R^2){\xi }_r$ at $r=R$, so ${\omega }_l^2=lGM/R^3-4\pi G\rho l/(2l+1)$. The $l=1$ branch is rigid translation with zero restoring force. This potential family does not exhaust vortical zero-frequency displacements: $\mathbf{\xi }=\mathbf{\Omega }\times \mathbf{r}$ is tangent to the sphere and has zero restoring force but nonzero curl.

For a constant reference density, fluid perturbation kinetic energy is proportional to $\Vert u{\Vert }^2/2$ under the displayed $L^2$ inner product. An adjoint operator for perturbation evolution must be defined relative to this inner product, including boundary conditions and the projection onto incompressible flow.

The kinetic helicity conservation law impliesTherefore $q$ is materially constant exactly when the right side vanishes. Incompressible flow together with constancy of $w-u^2/2$ along vortex lines is sufficient. In compressible flow, the density-normalized quantity satisfies $D(q/\rho )/Dt=-(\mathbf{\omega }/\rho )\cdot \nabla (w-u^2/2)$.

For isotropic incompressible flow of a Newtonian fluid, the stress tensor is the displayed expression, with pressure $p$, dynamic viscosity $\mu$, velocity $\mathbf{u}$, and identity matrix $I$. In Couette flow, its off-diagonal component is the shear stress.

The symmetric first moment of the traction exerted by a rigid inclusion, with the surface normal pointing into the inclusion. Its contraction with the background rate-of-strain tensor is the strain-related contribution to extra dissipation due to a rigid inclusion. A trace-free version is equivalent for an incompressible flow. This tensor is distinct from the axisymmetric force-dipole flow used as a far-field singularity.