Maxwell fluids

An example describing the application of this algorithm to the finite element modelling of free surface flow of a Maxwell fluid is given in Chapter 5. 

The Maxwell model is also called Maxwell fluid model. Briefly it is a mechanical model for simple linear viscoelastic behavior that consists of a spring of Young s modulus (E) in series with a dashpot of coefficient of viscosity (ji). It is an isostress model (with stress 5), the strain (f) being the sum of the individual strains in the spring and dashpot. This leads to a differential representation of linear viscoelasticity as d /dt = (l/E)d5/dt + (5/Jl)-This model is useful for the representation of stress relaxation and creep with Newtonian flow analysis. 

One of the mechanisms of drag reduction is that transmission of eddies can be damped by the viscoelastic properties of fluids. The transfer process of an isolated eddy in Maxwell fluids with viscoelastic properties was studied, and the expressions describing such phenomena were obtained [1103]. The results of the study showed that eddy transmission was damped significantly with an increase of the viscoelastic properties of the fluids. 

The simplest model that can show the most important aspects of viscoelastic behaviour is the Maxwell fluid. A mechanical model of the Maxwell fluid is a viscous element (a piston sliding in a cylinder of oil) in series with an elastic element (a spring). The total extension of this mechanical model is the sum of the extensions of the two elements and the rate of extension is the sum of the two rates of extension. It is assumed that the same form of combination can be applied to the shearing of the Maxwell fluid. 

In Chapter 1 it was pointed out that the Maxwell fluid is a very simple model of the first order effects observed with viscoelastic liquids. The equation of a Maxwell fluid is... 

It is interesting to consider the response of a Maxwell fluid to an arbitrary shear rate history. Denoting the shear rate as y(t), an arbitrary function of time, the equivalent of equation 3.83 is... 

So if we substitute the complex stress and strains into the constitutive equation for a Maxwell fluid the resulting relationship is given by Equation (4.21) ... 

Figure 3.10 Basic mechanical elements for solids and fluids a) dash pot for a viscous response, b) spring for an elastic response, c) Voigt or Kelvin solid, d) Maxwell fluid, and e) the four-parameter viscoelastic fluid...

When a spring and a dash pot are connected in series the resulting structure is the simplest mechanical representation of a viscoelastic fluid or Maxwell fluid, as shown in Fig. 3.10(d). When this fluid is stressed due to a strain rate it will elongate as long as the stress is applied. Combining both the Maxwell fluid and Voigt solid models in series gives a better approximation for a polymeric fluid. This model is often referred to as the four-parameter viscoelastic model and is shown in Fig. 3.10(e). Atypical strain response as a function of time for an applied stress for the four-parameter model is found in Fig. 3.12. 

Using the upper-convective Maxwell fluid eqn. (9.166) it reduces to... 

Together with Eq. 3.3-17, Eq. 3.3-16 is the White-Metzner constitutive equation, which has been used frequently as a nonlinear viscoelastic model. Of course, for small deformations, X(i) = dx/dt, and the single Maxwell fluid equation (Eq. 3.3-9) is obtained. 

Figure 3.33 Dynamic rheogram for a solution of 100mM CPyCI and 60 mM NaSal. The solution behaves like a Maxwell fluid with a single relaxation time.

Vilchis et al. [81] presented a new idea to achieve better control of the particle size distribution by the synthesis in situ of a water-soluble copolymer of acrylic acid-styrene as suspension stabilizer without additional inorganic phosphate. Publications describe increasing the particle formation by using a physical (population balance, Maxwell fluid, power law viscosity, compartment mixing) modeling approach [22,60,98,105]. 

This equation is derived by integrating Eq.( 11-29) with boundary condition)/ = 0, T = To at r = 0. Although the model has some elastic character the viscous response dominates at all but short times. For this reason, the element is known as a Maxwell fluid. 

The analysis of the corner singularity is delicate. We refer to the recent works of Hinch [36] and Renardy [37,38], who have contructed a matched asymptotic expansion for the steady solution to a Maxwell fluid flow near the corner. 

In a recent work [42], Renardy characterizes a set of inflow boundary conditions which leads to a locally well-posed initial boundary value problem for the two-dimensional flow of an upper-convected Maxwell fluid transverse to a domain bounded by parallel planes. 

In [62] Renardy proves the linear stability of Couette flow of an upper-convected Maxwell fluid under the 2issumption of creeping flow. This extends a result of Gorodtsov and Leonov [63], who showed that the eigenvalues have negative real parts (I. e., condition (S3) holds). That result, however, does not allow any claim of stability for non-zero Reynolds number, however small. Also it uses in a crucial way the specific form of the upper-convected derivative in the upper-convected Maxwell model, aind does not generalize so far to other Maxwell-type models. 

It is worth noticing that Tlapa and Bernstein [71] have proven that the Squire theorem holds true for the PoiseuiUe flow of m upper-convected Maxwell fluid. It means that any instability, which may be present for three dimensional disturbances, is also present for two dimensional ones at a lower value of the Reynolds number. This property is not true, in general, for non-Newtonian fluids [72]. 

In [80], as in previous works (e.g. [81]) on viscoelastic flows, Chen assumes disturbances of the form for a steady Couette flow of two upper-convected Maxwell fluids, and... 

In [91], Chen studies the long wave asymptotics of the concentric Poiseuille flow of two upper-converted Maxwell fluids under axisymmetric perturbations. He concludes that stability can generally be achieved by increasing the volume of the more elastic component , in agreement with the lubrication effect. The corresponding short wave asymptotic study of the same flow is done in [92]. In the case of coextrusion flows a study for arbitrary wavelengths and low Reynolds numbers is done in [93]. 

M. Renardy, An alternative approach to inflow boundary conditions for Maxwell fluids in three space dimensions, J. Non-Newtonian Fluid Mech., 36 (1990) 419-425. 

M. Renardy, Existence of steady flows for Maxwell fluids with traction boundary conditions on open boundaries, Z. Angew. Math. Mech. 75 (1995) 153-155. 

M. Renardy, Initial value problems with inflow boundaries for Maxwell fluids, SIAM J. Math. Anal., (1996) to appear. 

M. Renardy, A rigorous stability proof for plane Couette flow of an uppe-convected Maxwell fluid at zero Reynolds number, Eur. J. Mech. B, 11 (1992) 511-516. 

M. Renardy and Y. Renardy, Linear stability of plane Couette flow of an upper-convected Maxwell fluid, J. Non-Newtonian Fluid Mech., 22 (1986) 23-33. 

These difficulties can be understood by the following not rigorous computation. Let us consider in this domain, the flow of a Maxwell fluid of relaxation time X. The velocity field U and the stress tensor o are solutions of the following equations (as commonly accepted, inertia and mass forces are neglected) ... 

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