Non-Newtonian fluid

A non-Newtonian fluid has a stress--strain-rate relation with nonlinear, time-dependent, history-dependent, or microstructure-dependent behavior.

A simple fluid has local stress determined by the history of deformation of a material element, rather than by spatial derivatives of that history. An isotropic incompressible simple fluid has an arbitrary isotropic pressure and an objective constitutive functional of the relative deformation gradients. Internal-variable equations can represent such a functional when the initial material state is fixed by the past history.

A viscometric flow has the local relative deformation history of steady simple shear flow, up to rigid changes of frame. For an isotropic simple fluid, symmetry leaves one shear stress and two independent normal-stress differences. The corresponding viscometric functions depend on the magnitude of the shear rate, while pressure supplies the isotropic part.

Shear banding is spatial separation into regions with different local shear rates. In simple inertialess planar shear, equal shear stress can coexist with different rates when the homogeneous constitutive curve is nonmonotonic. Gradient terms and boundary conditions can determine the interface and selected stress; a local multivalued flow curve alone does not provide that selection. Shear banding need not be confused with the ordinary smooth shear-rate variation in a pressure-driven channel.

A generalized Newtonian fluid has instantaneous scalar constitutive law $\tau =\eta (\dot{\gamma })\dot{\gamma }$, with viscosity depending on the current shear rate but carrying no memory.

Differential shear viscosity measures the incremental shear-stress response around a given positive base shear rate. It differs from the secant shear viscosity mu. A regular locally stable stress-to-rate inversion needs positive differential viscosity. Zero differential viscosity makes a first-order stress perturbation singular, and negative differential viscosity identifies a decreasing constitutive branch.

Perturb the Couette flow between walls moving at +-U by a small pressure gradient -G. The mean shear rate remains U/h, and the stress variation across the gap is -Gy. Linearizing the monotone generalized Newtonian fluid law divides this variation by differential viscosity, giving the displayed velocity and flux per unit width. Require $Gh\ll (U/h){\eta }_d(U/h)$ and a smooth constitutive law on the perturbed rates. For a power-law fluid, the denominator is $nk(U/h{)}^{n-1}$.

The Weissenberg-Rabinowitsch equation recovers the wall shear rate of a generalized Newtonian fluid from a measured pressure-drop versus flow-rate curve. For steady fully developed flow in a horizontal circular pipe with radius $R$, length $L$ and no slip, define positive wall shear stressIf $Q({\tau }_R)$ is the volumetric flow rate, the wall shear rate isThe derivative samples the local slope of the measured flow curve.

To derive it, momentum balance gives $\tau (r)={\tau }_{R}r/R$. With $\dot{\gamma }(r)=-du/dr$ and $u(R)=0$, integrating the velocity profile givesDifferentiating ${\tau }_R^{3}Q$ proves the formula. This assumes a local stress/shear-rate relation at the specified measurement conditions; elastic memory or wall slip requires additional modeling.

For a power-law fluid with $\tau =K{\dot{\gamma }}^n$, $n>0$, one has $Q\propto {\tau }_R^{1/n}$ andThe Newtonian case $n=1$ recovers $4Q/(\pi R^3)$. The correction is useful because a non-Newtonian wall shear rate cannot generally be inferred from the Newtonian expression alone.

Thixotropy is a reversible decrease of viscosity caused by sustained deformation, followed by recovery while the material rests. It requires an evolving microstructural state and therefore depends on shear history.

A yield-stress fluid can sustain a finite shear stress without steady flow, but flows when the applied loading exceeds its yield stress. The ideal Bingham plastic treats the unyielded region as rigid and gives, after yield,Here $\dot{\gamma }$ is the shear rate, ${\tau }_y$ the yield stress and ${\eta }_p$ the post-yield dynamic viscosity. Real materials can undergo small elastic deformations below the flow threshold; an ideal rigid branch does not describe every part of their response.

A pressure-driven experiment makes the spatial distinction visible. Where the local shear stress is below threshold, a plug flow of a yield-stress fluid can move without internal shearing, while adjacent yielded layers deform. If every point remains below threshold and the walls have no wall slip, the ideal material does not flow. Slowly evolving microstructure and thixotropy make the apparent threshold depend on preparation, waiting time and the loading protocol.

A Bingham plastic has $\dot{\gamma }=0$ for $|\tau |<{\tau }_c$ and $\tau ={\tau }_c\mathrm{sgn}\dot{\gamma }+\mu \dot{\gamma }$ after yield.

For fixed plates $y=\pm h$, no slip, viscosity ${\mu }_0>0$, yield stress ${\tau }_0\ge 0$ and $G=-p_{,x}>0$, force balance gives shear stress $-Gy$. The central plug flow of a yield-stress fluid has half-width $y_0={\tau }_0/G$. If $y_0<h$, the velocity isIt is even in $y$; its derivative is $(-Gy+{\tau }_0\mathrm{sign}y)/{\mu }_0$ in the yielded layers and zero in the plug. If $Gh\le {\tau }_0$, there is no flow. Odd velocity is incompatible with this pressure-driven no-slip solution at nonzero flow.

A yield surface separates yielded material, where stress exceeds the yield criterion and deformation occurs, from an unyielded plug where the deformation rate vanishes.

For a Bingham layer on a stationary cylinder of radius $a$ with a shear-free cylindrical surface at $b$, an axial pressure gradient $-G$ produces ${\tau }_{rz}=G(b^2/r-r)/2$. Flow starts when $G(b^2-a^2)/(2a)$ exceeds the yield stress.

A thin Bingham layer of thickness $h$ between a stationary wall and a moving body gives $u_b=(h/\mu )(|{\tau }_b|-{\tau }_c{)}_{+}\mathrm{sgn}{\tau }_b$.

A Herschel–Bulkley fluid is unyielded for $|\tau |\le {\tau }_y$ and obeys $\tau =\mathrm{sgn}(\dot{\gamma })({\tau }_y+K|\dot{\gamma }{|}^n)$ after yield. The parameters $K$ and $n$ are its consistency and power-law index.

Where the stress in a yield-stress fluid remains below the yield stress, the shear rate vanishes and the material moves as an undeformed plug bounded by yielded layers.

A slump test for yield stress probes a material by allowing a molded sample to deform under its own weight. Fill a known mold on a level support, remove the mold, and measure the decrease in height or the spread of the final deposit. Gravity supplies the driving stress; resistance associated with the yield stress can arrest the motion before the sample becomes a thin layer.

The measured distance is not itself a stress. Inferring yield stress requires a calibrated relation or a mechanical model using the initial geometry, final shape and density. The final slump alone generally cannot identify both yield stress and dynamic viscosity: a time-resolved measurement provides additional information. A modified concrete test, for example, records the time required for a specified fall as well as the final slump.^[ref] The procedure is useful as a simple field measurement, with interpretation dependent on sample history and geometry.

A power-law fluid has shear stress $\tau =K|\dot{\gamma }{|}^{n-1}\dot{\gamma }$. It is shear thinning for $n<1$, Newtonian for $n=1$, and shear thickening for $n>1$.

For a thin, shear-free coating on a horizontal cylinder, lubrication theory balances tangential gravity against the transverse shear stress gradient. With angle measured counterclockwise from the rightward horizontal and wall speed $U=\Omega a>0$, the flux is $Uh-\mathrm{sgn}(\cos \theta )\frac{n}{2n+1}(\rho g|\cos \theta |/k{)}^{1/n}{h}^{(2n+1)/n}$. On the uphill side this function has a finite maximum. The tightest restriction is at $\cos \theta =1$, where differentiation gives $h_{\ast }=[k{U}^n/(\rho g){]}^{1/(n+1)}$ and the displayed global steady-flux bound. This is the leading gravity-driven model without higher-order pressure-gradient or capillary corrections.

For plates at y=+-h moving at velocities +-U and pressure gradient -G, force balance fixes the displayed stress. The power-law fluid law gives $u_y=(Gh/k{)}^{1/n}\mathrm{sgn}(\alpha -y/h)|\alpha -y/h{|}^{1/n}$. Integrating from the lower no-slip wall yields $u=-U+hn(Gh/k{)}^{1/n}[|\alpha +1{|}^{1+1/n}-|\alpha -y/h{|}^{1+1/n}]/(n+1)$. The upper wall fixes alpha uniquely for k,n,G,U>0. Absolute values are essential when the shear reverses inside the gap.

Shear thinning is the decrease of effective viscosity with increasing shear rate.

Viscoelasticity combines recoverable elastic response with dissipative and time-dependent response. Two specimens may react similarly to a rapid pull but behave differently when the same load is maintained. In a material creep test, constant stress produces evolving strain; in a stress relaxation test, fixed strain produces evolving stress.

The Maxwell model makes the distinction explicit by placing an elastic spring and viscous element in series. In simple shear strain it obeysAfter a small step to fixed ${\gamma }_0$, its shear stress decays as $\tau (t)=G{\gamma }_0{e}^{-t/\lambda }$ with viscoelastic relaxation time $\lambda =\eta /G$. Under constant shear stress ${\tau }_0$, it instead continues to deform, with $\gamma (t)={\tau }_0/G+{\tau }_{0}t/\eta$. These are predictions of this particular model: a Kelvin-Voigt model has bounded creep and different short-time behavior. Small-amplitude oscillatory shear measures the same competition through storage modulus and loss modulus, rather than assuming that one constant dynamic viscosity characterizes the material.

stress is a linear causal functional of deformation history: in simple shear, $\sigma (t)={\int }_{-\infty }^{t}G(t-t^{\prime })\dot{\gamma }(t^{\prime })d{t}^{\prime }$. Superposition holds and complex viscoelastic modulus is amplitude-independent within the small-deformation regime.

The characteristic time for a delayed strain response to approach equilibrium under a step in stress. For a Kelvin-Voigt model with elastic modulus $E$ and viscosity $\eta$, the equation $\eta \dot{\epsilon }+E\epsilon ={\sigma }_0$ gives $\epsilon =({\sigma }_0/E)(1-e^{-t/{\tau }_r})$, where ${\tau }_r=\eta /E$. A relaxation time instead describes decay of stress after an imposed strain; the two times need not coincide in a standard linear solid model.

A four-element linear viscoelasticity model with a spring $E_0$, a Kelvin-Voigt model branch $E_1,{\eta }_1$, and a free dashpot ${\eta }_0$ in series. Adding their strains under common stress gives the displayed creep compliance, with $\tau ={\eta }_1/E_1$. There is an instantaneous elastic response, a delayed recoverable response and steady viscous flow.^[ref] Named after Johannes Martinus Burgers, it extends a Maxwell model with a delayed elastic branch, or a standard linear solid model with a free dashpot.

For a load ${\sigma }_0$ maintained for time $t_h$, the two recoverable contributions behave as in recovery after a finite-duration stress load, while the free dashpot retains ${\sigma }_0{t}_h/{\eta }_0$. HenceThis residual response models viscous flow; it does not by itself establish a plastic yield threshold or a damage mechanism in a real specimen.

Anelasticity is a delayed but recoverable strain response caused by relaxation of internal degrees of freedom. After a small imposed stress is removed, the strain can return to its original value over a finite time rather than instantaneously. Under oscillatory loading, this delay causes a phase lag and dissipation, measured by a loss modulus or loss tangent. Stress-induced redistribution of interstitial impurities in a metal gives the Snoek effect. This solid-state response is distinct from an approximation to fluid mass conservation carrying a similar name.

The Snoek effect is anelasticity caused by stress-induced redistribution of mobile interstitial defects among differently oriented sites in a crystal. An applied stress favors orientations whose local distortion best accommodates it. At low frequency the atoms can track that bias and provide additional relaxed compliance; at high frequency their populations are nearly frozen. Dissipation is largest between these regimes. For a weak single relaxation, the loss tangent has the approximate shape $\Delta \omega \tau /[1+(\omega \tau {)}^2]$, with $\tau ={\tau }_0{e}^{E_a/(k_{B}T)}$ for thermally activated hopping. The peak is near $\omega \tau =1$. This describes a standard mechanism for interstitial impurities in body-centred cubic metals, not every damping mechanism in every alloy. ^[ref]

A standard linear solid model has a finite instantaneous stiffness, a nonzero long-time stiffness and one exponential stress-relaxation mode. A Kelvin-Voigt model with spring $G_0$ and viscosity $\eta$ in series with spring $G_1$ givesThese expressions follow by adding series strains under common stress. Its storage modulus is $G_L+(G_H-G_L)x^2/(1+x^2)$ and its loss modulus is $(G_H-G_L)x/(1+x^2)$ for $x=\omega \tau$. Its relaxation modulus is $G(t)=G_L+(G_H-G_L)e^{-t/\tau }$, while its creep compliance is $J(t)=G_1^{-1}+G_0^{-1}(1-e^{-G_{0}t/\eta })$. Thus creep retardation and stress-relaxation times differ. This solid has bounded creep and cannot represent terminal fluid flow.

A spring $E_0$ in series with a parallel spring-dashpot branch $E_1,\eta$ has creep compliance $J(t)=1/E_0+(1/E_1)(1-e^{-E_{1}t/\eta })$. It includes immediate and delayed recoverable strain, but no unbounded viscous flow. Large strain, damage or anisotropy require additional constitutive structure.

Apply constant stress ${\sigma }_0$ at $t=0$ and remove it at $t=t_h$. For series spring and Voigt-branch creep, the series spring recovers its strain ${\sigma }_0/E_0$ immediately on unloading. The Kelvin-Voigt model branch has retardation time $\tau =\eta /E_1$ and must keep continuous strain because an instantaneous dashpot displacement would require an impulse. Its post-unloading equation is $\eta \dot{\epsilon }+E_1\epsilon =0$. Solving with the branch's strain at $t_h$ gives the title formula. Thus all remaining strain eventually recovers: this standard linear solid model has no permanent set.

For a standard linear solid model with $0<G_R<G_U$, put $x=\omega \tau$. Its storage modulus is $G^{\prime }=G_R+(G_U-G_R)x^2/(1+x^2)$ and its loss modulus is $G^{\prime \prime }=(G_U-G_R)x/(1+x^2)$. The latter peaks at $x=1$, whereas its loss tangent isDifferentiation puts the loss-tangent peak at $x=\sqrt{G_R/G_U}$, with height $(G_U-G_R)/(2\sqrt{G_R{G}_U})$. A loss-modulus peak and a loss-tangent peak therefore need not occur at the same frequency. For weak relaxation, $G_U/G_R$ is close to one and their frequencies are approximately equal.

A passive viscoelasticity law cannot deliver net work in a periodic steady deformation without an external source of energy. With strain $ϵ={ϵ}_0\cos \omega t$ and stress $\sigma ={ϵ}_0(G^{\prime }\cos \omega t-G^{\prime \prime }\sin \omega t)$, the cycle-averaged supplied power is $⟨\sigma \dot{ϵ}⟩=\omega {ϵ}_0^2{G}^{\prime \prime }/2$. Thus a necessary passivity condition, using the convention $e^{i\omega t}$, is a nonnegative loss modulus at every positive frequency. Equivalently the work per cycle is $\pi {ϵ}_0^2{G}^{\prime \prime }$. A stable differential equation for stress does not by itself ensure this energetic condition.

The affine linear PTT fluid considered here uses a symmetric dimensionless tensor $A$ with $\stackrel{\nabla }{A}+(1+\alpha \mathrm{tr}A)A/\tau =2E$ and polymer stress $G_{0}A$. Here $\stackrel{\nabla }{A}$ is the upper-convected derivative, $G_0,\tau >0$ and the nonlinear relaxation parameter $\alpha$ is usually positive. Linearizing about rest gives the Maxwell fluid relaxation modulus $G(s)=G_0{e}^{-s/\tau }$. The PTT family also admits other relaxation functions and nonaffine derivatives; their nonlinear predictions need not equal those of this specified version.

For uniaxial extensional flow put $q=\tau \dot{\gamma }>0$, $f=1+\alpha \mathrm{tr}A$ and $T=f-2q$. The diagonal components are $a_{11}=2q/T$, $a_{22}=a_{33}=-q/(T+3q)$. The identity $f-1=\alpha \mathrm{tr}A$ gives the cubic equation in the title. For $\alpha >0$ its physical root has $T>\max (0,1-2q)$ and connects to $T=1$ at rest. Its extensional viscosity is ${\eta }_E=G_0\tau [2/T+1/(T+3q)]$, tending to $3G_0\tau$ at small $q$ and $2G_0\tau /\alpha$ at large $q$. At $\alpha =0$ the stable steady Maxwell extension instead ends at $q=1/2$; the finite high-rate limit must not be extended to that degenerate case.

In steady simple shear flow, the tensor equation gives $a_{22}=a_{33}=0$, $a_{11}=2a_{12}^2$ and the equation in the title. A circular pipe with axial pressure gradient $\Delta p$ has $G_0{a}_{rz}=\Delta pr/2$. Thus $w^{\prime }=\Delta pr/(2G_0\tau )+\alpha \Delta p^3{r}^3/(4G_0^3\tau )$, and no-slip boundary conditions givePositive $\alpha$ produces shear thinning; the sign of the pressure gradient determines the direction of the flow.

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.

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.

This differential model separates normalized molecular orientation $B=A/\mathrm{tr}A$ from scalar stretch $\lambda$. With component-first velocity gradient $K$, it evolves by $D_{t}A=KA+A{K}^T-(A-I)/{\tau }_1$ and $D_t\lambda =\lambda B:K-(\lambda -1)/{\tau }_2$. Its relaxed state is $A=I$, $B=I/3$, $\lambda =1$. The displayed constant-relaxation stretch law is an idealization; variants can add saturation or nonlinear relaxation.

For extension rate $s>0$ and $x=s{\tau }_1<1/2$, the steady conformation tensor is $\mathrm{diag}((1-2x{)}^{-1},(1+x{)}^{-1},(1+x{)}^{-1})$. Its normalized orientation has axial component $(1+x)/[3(1-x)]$ and transverse components $(1-2x)/[3(1-x)]$. Stretch is finite only if $s^2{\tau }_1{\tau }_2<1-x$, when ${\lambda }_{\infty }=(1-x)/(1-x-s^2{\tau }_1{\tau }_2)$ and the displayed extensional viscosity follows. Above $x=1/2$, the unnormalized axial tensor grows exponentially but $B\to \mathrm{diag}(1,0,0)$; then stretch remains bounded only if $s{\tau }_2<1$. Orientation normalization and stretch relaxation therefore impose different thresholds.

Linearizing the differential pom-pom model about its relaxed state leaves no first-order stretch perturbation in incompressible flow. Its anisotropic orientation obeys $\dot{a}+a/{\tau }_1=A_1$, producing the displayed relaxation modulus. In a slow-flow expansion, $A=I+{\tau }_1{A}_1+{\tau }_1^2(2A_1^2-A_2)+O(3)$. Trace normalization and the first stretch correction affect only isotropic stress at this order, giving the displayed second-order fluid coefficients. Linear small-amplitude response allows arbitrary frequency, whereas this local second-order expansion also requires slow time dependence.

An ideal dashpot resists relative motion with force proportional to velocity. Maintaining a displacement rate $\dot{x}$ requires force $\mu \dot{x}$ and dissipates power $\mu {\dot{x}}^2$ for $\mu >0$. Connecting it in parallel or series with a spring gives different constitutive laws.

The Kelvin-Voigt model puts a spring and dashpot in parallel. Their common displacement makes their forces add. With the convention $e^{i\omega t}$, the complex stiffness is $k+i\omega \mu$, giving storage stiffness $k$ and loss stiffness $\omega \mu$. Under a force step, $x=(F_0/k)(1-e^{-t/\tau })$ with $\tau =\mu /k$. The series spring–dashpot is a Maxwell fluid analogue and has different relaxation and creep behavior.

Under harmonic stress, strain amplitude is ${\sigma }_0/\sqrt{E^2+{\omega }^2{\eta }^2}$ and its lag is $\arctan (\omega \eta /E)$. Low frequencies are nearly elastic; high-frequency stress-controlled deformation is small and nearly a quarter cycle behind stress.

For $E\epsilon +\eta \dot{\epsilon }={\sigma }_0\Theta (t)$ with initially zero strain, $\epsilon =({\sigma }_0/E)(1-e^{-t/\tau })$ and $\tau =\eta /E$. Strain is continuous at load application, approaches a bounded value and recovers exponentially after unloading.

For a rod with unit mass density and elastic modulus, the Kelvin-Voigt model gives ${\varphi }_{tt}-{\varphi }_{xx}-\gamma {\varphi }_{txx}=0$, $\gamma >0$. For the operator with the opposite overall sign, $-{\partial }_t^2+{\partial }_x^2+\gamma {\partial }_t{\partial }_x^2$, the inverse under the $e^{-i\omega t}$ Fourier transform convention is the displayed expression. Its poles are ${\omega }_{\pm }=-i\gamma k^2/2\pm \sqrt{k^2-{\gamma }^2{k}^4/4}$. With ${\Omega }_k=\sqrt{k^2-{\gamma }^2{k}^4/4}$ real, the retarded fundamental solution in spatial Fourier variables is ${\tilde{G}}_R(k,t)=-\Theta (t)e^{-\gamma k^{2}t/2}\sin ({\Omega }_{k}t)/{\Omega }_k$. Its initial derivative is $-1$, fixed by the leading coefficient $-{\partial }_t^2$. At a repeated root take the continuous limit; for imaginary ${\Omega }_k$ the sine becomes a hyperbolic sine.

For a quasistatic cantilever of Kelvin-Voigt material, the bending moment is $M=EI{y}_{xx}+{\eta }_{s}I{y}_{xxt}$. A tip load gives $M=F(t)(\ell -x)$, so the free-tip displacement obeys the displayed equation with $\tau ={\eta }_s/E$. A step load gives a finite asymptotic deflection. If inertia is retained, the interior equation is $\rho A{y}_{tt}+EI{y}_{xxxx}+{\eta }_{s}I{y}_{xxxxt}=0$, with the corresponding moment and shear conditions at the tip.

Stress relaxation is decay of stored stress after a deformation is imposed and then held fixed. It probes a material's memory through its relaxation modulus. An instantaneous generalized Newtonian fluid has no stored shear stress once the shear rate is zero and cannot describe a nontrivial relaxation tail.

A linear viscoelastic fluid has causal stress response linear in the deformation history, often expressed as ${\sigma }^d(t)=2{\int }_0^{\infty }G(s)E(t-s)ds$. The approximation requires small deformation and rotation over the relevant memory times. High frequency is allowed at sufficiently small amplitude. An integrable decaying relaxation modulus gives a finite zero-shear viscosity; an instantaneous solvent term may be written separately.

A Maxwell fluid branch connected in parallel with a Newtonian dashpot. Branches share strain and their stresses add. With $\tau ={\eta }_M/G$, the displayed modulus uses the $e^{i\omega t}$ convention. Its low-frequency viscosity is ${\eta }_M+{\eta }_s$ and static shear modulus zero, so the material ultimately flows.

In time-translation-invariant linear response, each infinitesimal past strain step contributes its amplitude times the elapsed-time relaxation modulus. Adding those causal contributions gives the displayed stress-history integral for shear. An equivalent formulation convolves the creep compliance with stress increments. This constitutive law assumes sufficiently small strains about a specified equilibrium or relaxed state; nonlinear deformation, aging and evolving material structure can invalidate that simple superposition.

A small step of shear stress $s_{0}H(t)$ produces shear strain $\gamma (t)=s_{0}J(t)$ from a relaxed state. This time-dependent deformation under held stress is creep. The causal constitutive convolution gives $\hat{s}=p\hat{G}\hat{\gamma }$, proving the displayed Laplace transform relation. For a Maxwell fluid, $J(t)=1/G_0+t/(G_0\tau )$: an elastic jump followed by viscous flow.

For the convention $e^{i\omega t}$, complex viscosity is the ratio of oscillatory shear stress amplitude to shear rate amplitude. A causal relaxation modulus gives the displayed transform. Its associated complex shear modulus is $G^{\ast }=i\omega {\eta }^{\ast }=G^{\prime }+i{G}^{\prime \prime }$, where $G^{\prime }$ is the storage modulus and $G^{\prime \prime }$ the loss modulus.

The relaxation modulus is the shear-stress response at lag s to a unit small step of shear strain. By linear superposition it is the causal memory kernel multiplying shear rate. Its decaying part represents stored stress that relaxes after deformation ceases.

For a linear viscoelastic fluid, write the solvent stress explicitly as $2{\mu }_{0}E(t)$, or use the displayed kernel with a causal endpoint delta satisfying ${\int }_0^{\infty }{\delta }_{+}(s)f(s)ds=f(0)$. If a symmetric delta is integrated with half its weight at the endpoint, its coefficient must instead be $2{\mu }_0$. Mixing those conventions loses a factor of two. The Oldroyd-B model has the displayed exponentially decaying polymer part after linearization.

The three viscometric functions describe steady homogeneous simple shear flow: its shear viscosity and the first and second normal-stress differences per squared shear rate. Take x as the flow direction, y as the gradient direction and z as the vorticity direction, so $N_1={\sigma }_{xx}-{\sigma }_{yy}$ and $N_2={\sigma }_{yy}-{\sigma }_{zz}$. Isotropic pressure drops out of these differences.

A corotational Jeffreys fluid obeys $\stackrel{\circ }{S}+S/\tau =2{\mu }_0\stackrel{\circ }{D}+2{\mu }_{1}D/\tau$, with deviatoric stress $S$, rate-of-strain tensor $D$, Jaumann derivative and relaxation time $\tau >0$. Put $A=S-2{\mu }_{0}D$. In the corotating-frame representation of a Jaumann derivative, ${\tilde{A}}^{\prime }+\tilde{A}/\tau =2({\mu }_1-{\mu }_0)\tilde{D}/\tau$. An integrating factor gives an exponentially weighted history transported between the rotating frames. The infinite-past formula selects histories with $e^{t/\tau }\tilde{A}(t)\to 0$ as $t\to -\infty$; finite initial data also contribute an exponentially decaying homogeneous term.

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.

In a homogeneous Maxwell fluid, transverse momentum conservation and shear stress relaxation give $\tau u_{tt}+u_t=\nu u_{yy}$, where $\nu =\mu /\rho$ is the kinematic viscosity. A Fourier mode $e^{iky+\sigma t}$ has the displayed quadratic dispersion relation. For $4\tau \nu k^2>1$ its two exponents oscillate while decaying at rate $1/(2\tau )$; for smaller wavenumbers they decay without oscillating. At fixed $\nu$ and $k$, $\tau \to 0$ leaves the diffusion equation branch $\sigma \to -\nu k^2$ and removes the fast stress relaxation branch $\sigma \sim -1/\tau$.

For a series Maxwell model with $\tau =\eta /k$, $E^{\prime \prime }=k\omega \tau /(1+{\omega }^2{\tau }^2)$. At fixed strain amplitude the energy dissipated per cycle is $\pi E^{\prime \prime }{\epsilon }_0^2$ and peaks at $\omega \tau =1$. At fixed stress amplitude it is instead $\pi {\sigma }_0^2/(\eta \omega )$, which has no finite-frequency maximum. Specifying the imposed variable and whether loss means energy per cycle or power is essential.

For a no-slip channel with half-width $h$, a harmonic pressure gradient $\Delta p{e}^{i\omega t}$ and complex viscosity $\hat{\mu }=G_0\tau /(1+i\omega \tau )$, put $z^2=i\omega \rho h^2/\hat{\mu }$. Solving $\hat{\mu }{U}^{\prime \prime }-i\omega \rho U=\Delta p$ gives $U=-\Delta p[1-\cosh (zy/h)/\cosh z]/(i\omega \rho )$; integration gives the displayed flux per unit width. At low frequency it approaches $-2\Delta p{h}^3/(3G_0\tau )$. At fixed positive $h,\tau ,G_0,\rho$, the high-frequency leading flux is $-2\Delta ph/(i\omega \rho )$, although shear waves can persist in the local profile. If instead $\omega \tau \gg 1$ while $\rho {\omega }^2{h}^2/G_0\ll 1$, the negligible-inertia elastic limit is $Q\simeq -2i\omega \Delta p{h}^3/(3G_0)$. The last two limits impose different scale orderings.

For a half-space initially at rest and a wall given a step velocity $U$, momentum conservation and the Maxwell fluid stress law give $\tau u_{tt}+u_t=(G_0\tau /\rho )u_{yy}$. The displayed decaying Laplace transform solution obeys the wall condition. Inverting on a right-half-plane Bromwich contour yields a shear signal of speed $\sqrt{G_0/\rho }$. The limit $\tau \to 0$ at fixed $G_0\tau =\eta$ gives viscous error-function penetration; $\tau \to \infty$ at fixed $G_0$ gives a sharp elastic front.

For a quasistatic beam made of a two-element series Maxwell material, $\dot{M}+M/\tau =EI{\partial }_t{y}_{xx}$. A force step therefore produces an instantaneous elastic deflection followed by unbounded linear creep, as displayed. It cannot have the finite long-time plateau of a Kelvin-Voigt cantilever creep response; a solid with finite relaxed modulus requires an additional elastic element.

A prescribed local linear Maxwell force law filters a harmonic drag amplitude by $(1+i\omega \lambda {)}^{-1}$. For periodic states its time derivative has zero mean, so the mean force equals the mean driving force. Leading harmonic power is reduced by $[1+(\omega \lambda {)}^2{]}^{-1}$, as elastic storage changes the in-phase dissipative response.