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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 an isolated bounded conductor of uniform magnetic diffusivity $\eta >0$, contained in a sphere of radius $R$ and matched to a decaying potential field in an insulating exterior, dynamo action requires maximum stretching rate $S\ge \eta {\pi }^2/R^2$. Here $S$ bounds the largest eigenvalue of the rate-of-strain tensor throughout the flow and time. Use fluid boundary conditions eliminating the stretching boundary term, for example a no-slip boundary condition, and no imposed energy input. The proof combines the magnetic free-decay spectral bound with the magnetic energy equation. This is a necessary condition, not a sufficiency criterion; changing magnetic boundary conditions changes the spectral constant.

A constitutive equation relates continuum fields such as stress, strain, and the rate-of-strain tensor to characterize a material beyond universal balance laws.

A dilute suspension of identical rigid spheres raises the dynamic viscosity by a fraction $5\varphi /2$, where $\varphi$ is the particle volume fraction. At imposed trace-free rate-of-strain tensor $E$, a sphere of radius $a$ adds viscous dissipation $20\pi \mu a^3(E:E)/3$. Multiplication by the number density $3\varphi /(4\pi a^3)$ gives $5\mu \varphi E:E$, which added to $2\mu E:E$ proves the formula. This is the leading noninteracting-sphere term; hydrodynamic interactions enter at higher concentration.

At fixed outer boundary velocity, extending the inclusion velocity rigidly through its interior makes the particle flow an admissible competitor for the particle-free Minimum-dissipation theorem for Stokes flow. Its interior rate-of-strain tensor vanishes. For a linear background, the increase in viscous dissipation is force times relative translation, torque times relative rotation, plus the contraction of the particle stresslet tensor with background strain.

A linear Maxwell fluid has constitutive equation $\mathbf{\sigma }+\lambda {\partial }_t\mathbf{\sigma }=\mu \dot{\mathbf{\gamma }}$, with $\dot{\mathbf{\gamma }}$ twice the rate-of-strain tensor. Here $\mu$ is the zero-frequency shear viscosity and $\lambda >0$ is the viscoelastic relaxation time. Stress at fixed strain decays as $e^{-t/\lambda }$; the corresponding series spring and dashpot have shear modulus $G=\mu /\lambda$. In small-amplitude simple shear from an isotropic relaxed state, the first-order forcing is off-diagonal, so both normal-stress differences vanish at linear order. This is not a statement about the nonlinear objective model at second order in shear rate.

Under the energy-closed boundary assumptions of Backus' necessary condition for dynamo action, the magnetic energy equation gives ${\dot{E}}_B\le 2s(t)E_B$ after discarding nonnegative resistive dissipation. Integrating gives an upper bound by the time average of the largest spatial eigenvalue of the rate-of-strain tensor. Thus the field-amplitude exponent is at most its space-time supremum. The exponent of squared energy is twice the field-amplitude exponent.

A Newtonian fluid has viscous stress proportional to the local rate-of-strain tensor, with coefficients independent of that rate. For an isotropic fluid these coefficients are viscosities. Its Newtonian fluid stress tensor specifies the corresponding constitutive relation; the material and its stress tensor are distinct concepts.

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.