Linear Maxwell fluid

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.