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The Deborah number compares a material's viscoelastic relaxation time with the time scale of an imposed process. A small value allows substantial stress relaxation during that process; a large value can reveal stored elasticity. In small-amplitude oscillatory shear, one commonly uses $\mathrm{De}=\omega \tau$. Large frequency does not by itself invalidate a linear viscoelastic fluid approximation if the deformation amplitude remains sufficiently small.

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.

The usual elastic secondary flow around a rotating sphere is inward towards the equator, then poleward beside the sphere and outward near both poles. It is superposed on the primary azimuthal rotating sphere in Stokes flow. Tensile normal stresses along curved azimuthal streamlines act as hoop stresses, pulling liquid towards the rotation axis. Incompressibility supplies the axial return flow. Unlike the primary rotation, this meridional circulation does not reverse when the rotation is reversed.

The direction assumes the usual polymeric sign $c={\psi }_1+2{\psi }_2>0$ and negligible inertia. A general second-order fluid has circulation proportional to $c$: a negative value reverses the arrows and $c=0$ eliminates the leading elastic circulation. Inertial secondary circulation is a different contribution, so slow rotation alone does not determine which contribution dominates. The sketch shows the elastic contribution for $c>0$:

For a quantitative check of the sketch, take sphere radius $a$, angular velocity $\Omega$ and $K=\Omega a^3$. The primary azimuthal velocity is $v_{\varphi }=K\sin \theta /r^2$, with $\theta$ measured from the rotation axis. Its only nonzero strain components in the spherical orthonormal basis are $E_{r\varphi }=E_{\varphi r}=-3K\sin \theta /(2r^3)$. The leading elastic secondary circulation around a rotating sphere has the stream functionwith $v_r=H(3{\cos }^2\theta -1)/r^2$ and $v_{\theta }=-H^{\prime }\sin \theta \cos \theta /r$. The bracket is $(1-a/r{)}^2(1+2a/r)>0$ outside the sphere. Thus $v_r<0$ at the equator and $v_r>0$ near the poles when $c>0$, and both components vanish on the sphere. The diagram uses these velocities, not independently chosen arrows.

Two other manifestations of the same elastic normal-stress differences are rod climbing, in which inward hoop stresses raise a free surface around a rotating rod, and die swell, in which relaxation of the axial stress and stored polymer deformation allows an extrudate to expand sideways after leaving the die. These are not explained by a scalar shear viscosity alone. A Newtonian extrudate can also swell through velocity-profile rearrangement, so the elastic contribution should not be identified with every instance of swelling.

The second-order fluid is a slow, weak-flow expansion of the objective constitutive history of an isotropic simple fluid. Material frame indifference requires objective tensor rates. For an incompressible flow, the leading isotropic stress is arbitrary pressure, the linear deviatoric term is $2\mu E$, and at the next order the independent contributions can be represented by $E^2$ and an objective rate of $E$. Isotropic scalar contributions are absorbed into the pressure. Their coefficients can be expressed through the two low-rate normal-stress differences, givingHere $\stackrel{△}{E}$ is the lower-convected derivative, not a division by $E$. This local expansion is appropriate when deformation and temporal change over the relevant viscoelastic relaxation time are small: both the Weissenberg number and the appropriate Deborah number must be small. It captures leading normal stresses in slow flows, but not arbitrary finite-strain history, rapid transients or strongly nonlinear shear thinning. In particular its restrictions are stronger than small-amplitude linear oscillatory response at arbitrary frequency.

For steady simple shear flow $\mathbf{v}=(\dot{\gamma }y,0,0)$, use component-first $L_{ij}={\partial }_j{v}_i$, so the paper's $\nabla v$ is $L^T$. The relevant tensors areSubstitution givesThus the first and second normal-stress differences are $N_1={\psi }_1{\dot{\gamma }}^2$ and $N_2={\psi }_2{\dot{\gamma }}^2$, and the shear stress is $\mu \dot{\gamma }$.

For the velocity-preservation results, the definitions imply $\stackrel{△}{E}=\stackrel{\circ }{E}+2E^2$. Consequently the non-Newtonian stress can be writtenwhere $\stackrel{\circ }{E}$ is the Jaumann derivative. Let ${\mathbf{v}}_N$ solve the steady incompressible Stokes flow problem with the prescribed boundary velocities. The supplied identity says that $\nabla \cdot {\stackrel{\circ }{E}}_N$ is curl-free. If ${\psi }_1+2{\psi }_2=0$, then $\nabla \cdot S_N=\nabla \Phi$ for a scalar $\Phi$ on the ordinary simply connected flow domain. The momentum equation at ${\mathbf{v}}_N$ isHence the Newtonian velocity itself is a solution of the second-order constitutive model, with altered pressure $p=p_N+\Phi$. For nontrivial topology one also needs the curl-free force to admit a single-valued pressure; this is implicit in the usual boundary-value problem.

On the regular weakly non-Newtonian branch, Uniqueness of Stokes flow fixes the velocity correction. At first order it solves homogeneous Stokes flow, with zero boundary velocity, after the gradient force has been absorbed into pressure. Multiplying its equation by the correction and integrating gives $2\mu \int E_1:E_{1}dV=0$, hence the correction vanishes. Higher perturbative corrections vanish in turn because the previous velocities remain ${\mathbf{v}}_N$. The velocity is $\mathbf{v}={\mathbf{v}}_N$ in this second-order-fluid solution; only the pressure and stresses change. This is Newtonian velocity preservation in a second-order fluid. It does not assert global uniqueness of distant nonlinear solution branches or exact equality for a real fluid whose higher-order constitutive terms have been omitted.

For a planar incompressible velocity $(u(x,y),v(x,y),0)$, the rate-of-strain tensor has the formAll quantities are independent of $z$, so $\nabla \cdot E_N^2=\nabla (a^2+b^2)$. The extra quadratic term is therefore also a pressure gradient. Together with the supplied identity for the Jaumann derivative, the entire non-Newtonian force is a gradient for arbitrary ${\psi }_1,{\psi }_2$. The same boundary-value and weak-branch argument proves $\mathbf{v}={\mathbf{v}}_N$ for every such planar second-order-fluid flow with prescribed boundary velocities, without requiring ${\psi }_1+2{\psi }_2=0$. Specifying stresses or a free surface instead of velocities would not give the same conclusion automatically, because the extra stresses affect those boundary conditions.

In the Giesekus model, $\lambda$ is the viscoelastic relaxation time: after deformation stops, polymeric stress relaxes on that timescale. The parameter $\eta$ has dimensions of dynamic viscosity and is the model's zero-rate viscosity scale. Since $\lambda {\mathbf{\tau }}^2/\eta$ already has dimensions of stress, $\alpha$ is dimensionless.

The symbol $\stackrel{▽}{\tau }$ is the upper-convected derivativeThe convective and velocity-gradient terms account for translation, rotation, and affine stretching of material elements. They make the constitutive law objective under time-dependent rigid changes of observer.

In small-amplitude oscillatory shear, expand the measured shear stress asComparison with the definitionof the storage modulus and loss modulus givesand henceThus the measured phase directly gives the loss tangent.

At low angular frequency, a standard viscoelastic fluid has time to relax and responds mainly as a viscous liquid: stress is approximately in phase with strain rate, so $\delta \to \pi /2$ as $\omega \to 0$. At high frequency it cannot relax during one cycle and responds mainly as an elastic solid, so stress is approximately in phase with strain and $\delta \to 0$ as $\omega \to \infty$.

For a material dominated by one viscoelastic relaxation time $\lambda$, the crossover between these regimes occurs when $\omega \lambda$ is of order one. Measure the frequency ${\omega }_c$ at which $G^{\prime }=G^{\prime \prime }$, equivalently $\delta =\pi /4$, and estimateA broad or multi-peaked crossover instead indicates a spectrum of relaxation times.

The Weissenberg number measures deformation accumulated over a viscoelastic relaxation time, often as the product of that time and a shear rate or an extension rate. Its precise rate convention should be stated. Unlike a Deborah number based on externally imposed temporal variation, it can remain nonzero in a steady flow. A weak-flow constitutive expansion generally requires both slow temporal change and small deformation over the memory time.