Consider a fluid undergoing planar Couette flow. This flow is defined by the streaming velocity,

(2.65)

According to Curie's principle the only nonequilibrium flux that will be excited by such a flow is the pressure tensor. According to the constitutive relation equation (2.56) the pressure tensor is,

(2.66)

where η is the shear viscosity and γ is the strain rate. If the strain rate is time dependent then the shear stress, -P_xy = -P_yx = ηγ (t). It is known that many fluids do not satisfy this relation regardless of how small the strain rate is. There must therefore be a linear but time dependent constitutive relation for shear flow which is more general than the Navier-Stokes constitutive relation.

Poisson (1829) pointed out that there is a deep correspondence between the shear stress induced by a strain rate in a fluid, and the shear stress induced by a strain in an elastic solid. The strain tensor is, ∇ε where ε(r,t) gives the displacement of atoms at r from their equilibrium lattice sites. It is clear that,

(2.67)

Maxwell (1873) realised that if a displacement were applied to a liquid then for a short time the liquid must behave as if it were an elastic solid. After a Maxwell relaxation time the liquid would relax to equilibrium since by definition a liquid cannot support a strain (Frenkel, 1955).

It is easier to analyse this matter by transforming to the frequency domain. Maxwell said that at low frequencies the shear stress of a liquid is generated by the Navier-Stokes constitutive relation for a Newtonian fluid (2.66). In the frequency domain this states that,

(2.68)

where,

(2.69)

denotes the Fourier-Laplace transform of A(t).

At very high frequencies we should have,

(2.70)

where G is the infinite frequency shear modulus. From equation (2.67) we can transform the terms involving the strain into terms involving the strain rate (we assume that at t=0, the strain ε(0)=0). At high frequencies therefore,

(2.71)

The Maxwell model of viscoelasticity is obtained by simply summing the high and low frequency expressions for the compliances iω∕G and η^-1,

(2.72)

The expression for the frequency dependent Maxwell viscosity is,

(2.73)

It is easily seen that this expression smoothly interpolates between the high and low frequency limits. The Maxwell relaxation time τ_M = η∕G controls the transition frequency between low frequency viscous behaviour and high frequency elastic behaviour.

The Maxwell model provides a rough approximation to the viscoelastic behaviour of so-called viscoelastic fluids such as polymer melts or colloidal suspensions. It is important to remember that viscoelasticity is a linear phenomenon. The resulting shear stress is a linear function of the strain rate. It is also important to point out that Maxwell believed that all fluids are viscoelastic. The reason why polymer melts are observed to exhibit viscoelasticity is that their Maxwell relaxation times are macroscopic, of the order of seconds. On the other hand the Maxwell relaxation time for argon at its triple point is approximately 10^-12 seconds! Using standard viscometric techniques elastic effects are completely unobservable in argon.

If we rewrite the Maxwell constitutive relation in the time domain using an inverse Fourier-Laplace transform we see that,

(2.74)

In this equation η_M(t) is called the Maxwell memory function. It is called a memory function because the shear stress at time t is not simply linearly proportional to the strain rate at the current time t, but to the entire strain rate history, over times S where 0 ≤ s ≤ t. Constitutive relations which are history dependent are called non-Markovian. A Markovian process is one in which the present state of the system is all that is required to determine its future. The Maxwell model of viscoelasticity describes non-Markovian behaviour. The Maxwell memory function is easily identified as an exponential,

(2.75)

Although the Maxwell model of viscoelasticity is approximate the basic idea that liquids take a finite time to respond to changes in strain rate, or equivalently that liquids remember their strain rate histories, is correct. The most general linear relation between the strain rate and the shear stress for a homogeneous fluid can be written in the time domain as,

(2.76)

There is an even more general linear relation between stress and strain rate which is appropriate in fluids where the strain rate varies in space as well as in time,

(2.77)

We reiterate that the differences between these constitutive relations and the Newtonian constitutive relation, equations (2.56b), are only observable if the strain rate varies significantly over either the time or length scales characteristic of the molecular relaxation for the fluid. The surprise is not so much the validity of the Newtonian constitutive relation is limited. The more remarkable thing is that for example in argon, the strain rates can vary in time from essentially zero frequency to 10¹²Hz, or in space from zero wavevector to 10^-9m^-1, before non-Newtonian effects are observable. It is clear from this discussion that analogous corrections will be needed for all the other Navier-Stokes transport coefficients if their corresponding thermodynamic fluxes vary on molecular time or distance scales.