Corotational Maxwell Fluid Research Articles

Series expansion for normal stress differences in large-amplitude oscillatory shear flow from Oldroyd 8-constant framework

The Oldroyd 8-constant framework for continuum constitutive theory contains a rich diversity of popular special cases for polymeric liquids. In this paper, we focus on the normal stress difference responses to large-amplitude oscillatory shear (LAOS) flow. The nonlinearity of the polymeric liquids, triggered by LAOS, causes these responses at even multiples of the test frequency. We call responses at a frequency higher than twice the test frequency higher harmonics. The normal stress difference responses for the Oldroyd 8-constant framework has recently yielded to the exact analytical solution. However, in its closed form, Bessel functions appear 24 times, each within summations to infinity. In this paper, to simplify the exact solution, we expand it in a Taylor series. We truncate the series after its 17th power of the shear rate amplitude. Our main result reduces to the well-known expression for the special cases of the corotational Jeffreys and corotational Maxwell fluids. Whereas these special cases yielded to the Goddard integral expansion (GIE), the more general Oldroyd 8-constant framework does not. We use Ewoldt grids to show our main result to be highly accurate for the corotational Jeffreys and corotational Maxwell fluids. For these two special cases, our solutions agree closely with the exact solutions as long as Wi/De<3310. We compare our main result, for the special case of the Johnson–Segalman fluid, with measurements on dissolved polyisobutylene in the isobutylene oligomer. For this, we use the Spriggs relations to generalize our main result to multimode, which then agrees closely with the measurements.

Power series for normal stress differences of polymeric liquids in large-amplitude oscillatory shear flow

Exact solutions for normal stress differences in polymeric liquids subjected to large-amplitude oscillatory shear flow (LAOS) contain many Bessel functions, each appearing in infinite sums. For the simplest relevant model of a polymeric liquid, the corotational Maxwell fluid, Bessel functions appear 38 times in the exact solution. By relevant, we mean that higher harmonics are predicted in LAOS. By contrast, approximate analytical solutions for normal stress differences in LAOS often take the form of the first few terms of a power series in the shear rate amplitude, and without any Bessel functions at all. Perhaps the best example of this, from continuum theory, is the Goddard integral expansion (GIE) that is arrived at laboriously. There is thus practical interest in extending the GIE to an arbitrary number of terms. However, each term in the GIE requires much more work than its predecessor. For the corotational Maxwell fluid, for instance, the GIE for the normal stress differences has yet to be taken beyond the fifth power of the shear rate amplitude. In this paper, we begin with the exact solution for normal stress difference responses in corotational Maxwell fluids, then perform an expansion by symbolic computation to confirm up to the fifth power, and then to continue the GIE. In this paper, for example, we continue the GIE to the 41st power of the shear rate amplitude. We use Ewoldt grids to show that our main result is highly accurate. We also show that, except in its zero-frequency limit, the radius of convergence of the GIE is infinite. We derive the pattern for the common denominators of the GIE coefficients and also for every numerator for the zeroth harmonic coefficients. We also find that the numerators of the other harmonics appear to be patternless.

Series expansion for shear stress in large‐amplitude oscillatory shear flow from oldroyd 8‐constant framework

AbstractWhen polymeric liquids undergo large‐amplitude oscillatory shear flow, the shear stress responds as a Fourier series, the higher harmonics of which are caused by the fluid nonlinearity, and the first harmonic of which is a nonlinear function of both the frequency and the shear rate amplitude. The Oldroyd 8‐constant framework for continuum constitutive theory contains a rich diversity of popular special cases for polymeric liquids. The shear stress response for the Oldroyd 8‐constant framework has recently yielded to exact analytical solution. However, in its closed form, Bessel functions appear 24 times, each within infinite summations. In this paper, to simplify the exact solution, we expand it in a Taylor series. We truncate the series after its 16th power of the shear rate amplitude. Our main result reduces to the well known expression for the corotational Jeffreys and corotational Maxwell fluids. Whereas these special cases yielded to the Goddard integral expansion (GIE), the more general Oldroyd 8‐constant framework does not. We use Ewoldt grids to show our main result to be highly accurate, for the corotational Jeffreys and corotational Maxwell fluids. For these two special cases, our solutions agree closely with the exact solutions so long as . We apply our main result for the special case of the Johnson‐Segalman fluid to describe the measured frequency sweep and shear rate amplitude sweep responses of molten atactic polystyrene. For this, we use the Spriggs relations to generalize our main result to multimode, which then agrees closely with the measured responses.

Exact solution for intrinsic nonlinearity in oscillatory shear from the corotational Maxwell fluid

The ratio of the shear stress amplitudes of higher harmonics to the first, called relative intensities, is a widely established way of characterizing materials and of representing large-amplitude oscillatory shear flow (LAOS) measurements. If we divide these relative intensities by appropriate powers of strain amplitude, we define intrinsic nonlinearities. In their limits of vanishing strain amplitude, these relative intensities give relative intensity parameters, mQ0, where m is the number of the harmonic. Here, we arrive at the first exact analytical solution for the mth intrinsic nonlinearity for the shear stress of a corotational Maxwell fluid and, for example, evaluate it for the first three harmonics (third, fifth and seventh). We then use the Spriggs relations to generalize these relative intensities and corresponding intrinsic nonlinearities to multimode. For consistency, we check the expansion of our results against the well-known first term in the expansion in powers of the strain amplitude [Appl. Rheol., 26, 53809 (2016)]. We also introduce an expansion of the intrinsic nonlinearity in the shear rate amplitude. Finally, we compare our generalized results, third-to-first (I3/1) and fifth-to-first (I5/1) relative intensities, with measurement on molten 1,4-cis-polyisoprene.

Thermodynamic instability of polymeric liquids in large-amplitude oscillatory shear flow from corotational Maxwell fluid

When polymeric liquids are subjected to oscillatory shear flow, if the amplitude of the shear rate is high enough, the shear stress response will become aperiodic. This has been variously attributed to fracture, common line ingress, slip, shear-banding and phase change, however, the underlying causes for these is unclear. In this paper we explore the creation of new thermodynamic phases as the trigger for these phenomena. Specifically, we examine two thermodynamic instability criteria that have been suggested for large-amplitude oscillatory shear flow (LAOS). One of these criteria is based on non-equilibrium thermodynamics (the Ziegler criterion), and the other, on equilibrium thermodynamics (the free energy criterion). The advent of exact solutions for stress responses to LAOS provokes this investigation. We use one such exact solution to evaluate these criteria for the simplest relevant constitutive model, the corotational Maxwell fluid. By relevant, we mean at least predicting higher harmonics in LAOS. We find that, in steady shear flow, the Ziegler instability arises when the dimensionless shear rate exceeds the square root of the golden ratio, Applying our results to instability measurements on dissolved polybutadiene, we find the Ziegler criterion to be useful at low frequency, and the free energy criterion to be useful elsewhere.

Power series for shear stress of polymeric liquid in large-amplitude oscillatory shear flow

Exact solutions for shear stress in a polymeric liquid subjected to large-amplitude oscillatory shear flow (LAOS) contain many Bessel functions. For the simplest of these, for instance, the corotational Maxwell fluid, in the closed form for its exact solution, Bessel functions appear 42 times, each of which is inside a summation. Approximate analytical solutions for shear stress in LAOS often take the form of the first few terms of a power series in the shear rate amplitude, and without any Bessel functions at all. There is thus practical interest in extending the Goddard integral expansion (GIE), to an arbitrary number of terms. In continuum theory, these truncated series are arrived at laboriously using the GIE. However, each term in the GIE requires much more work than its predecessor. For the corotational Maxwell fluid, for instance, the GIE for the shear stress has yet to be taken beyond the sixth power of the shear rate amplitude. In this paper, we begin with the exact solution for shear stress responses in corotational Maxwell fluids, and then perform an expansion by symbolic computation to confirm up to the sixth power, and to then continue the GIE. In this paper for example, we continue the GIE to the 40th power of the shear rate amplitude. We use Ewoldt grids to show our main result to be highly accurate. We also show the radius of convergence of the GIE to be infinite.

Normal stress differences from Oldroyd 8-constant framework: Exact analytical solution for large-amplitude oscillatory shear flow

The Oldroyd 8-constant framework for continuum constitutive theory contains a rich diversity of popular special cases for polymeric liquids. In this paper, we use part of our exact solution for shear stress to arrive at unique exact analytical solutions for the normal stress difference responses to large-amplitude oscillatory shear (LAOS) flow. The nonlinearity of the polymeric liquids, triggered by LAOS, causes these responses at even multiples of the test frequency. We call responses at a frequency higher than twice the test frequency higher harmonics. We find the new exact analytical solutions to be compact and intrinsically beautiful. These solutions reduce to those of our previous work on the special case of the corotational Maxwell fluid. Our solutions also agree with our new truncated Goddard integral expansion for the special case of the corotational Jeffreys fluid. The limiting behaviors of these exact solutions also yield new explicit expressions. Finally, we use our exact solutions to see how η∞ affects the normal stress differences in LAOS.

Study of co-rotational Maxwell fluid in helical screw rheometer

In this paper, the steady flow of an incompressible, co-rotational Maxwell fluid in a helical screw rheometer is studied by ‘unwrapping or flattening’ the channel, lands, and the outside rotating barrel. The geometry is approximated as a shallow infinite channel, by assuming the width of the channel large as compared to the depth. The developed second order nonlinear coupled differential equations are transformed to a single differential equation. Using perturbation methods, analytical expressions are obtained for the velocity components in the x- and z-directions and the resultant velocity in the direction of the screw axis. Volume flow rates, shear and normal stresses, shear at wall, and forces exerted on fluid and average velocity are also calculated. The behavior of the velocity profiles are discussed with the help of graphs. We observe that the velocity profiles are strongly dependent on the non-dimensional parameter $(Wi)^{2}$ , the flight angle ϕ and the pressure gradients.

Viscous heating in large-amplitude oscillatory shear flow

When measuring rheological properties in oscillatory shear flow, one worries about experimental error due to the temperature rise in the sample that is caused by viscous heating. For polymeric liquids, for example, this temperature rise causes the measured values of the components of the complex viscosity to be systematically low. For such linear viscoelastic property measurements, we use an analytical solution by Ding et al. [J. Non-Newtonian Fluid Mech. 86, 359 (1999)10.1016/S0377-0257(99)00004-X] to estimate the temperature rise. However, for large-amplitude oscillatory shear flow, no such analytical solution is available. Here we derive an analytical solution for the temperature rise in a corotational Maxwell fluid (a model with just two parameters: η0 and λ) subject to large-amplitude oscillatory shear flow. This result can then be generalized to a superposition of corotational Maxwell models for a quantitative estimate of the temperature rise. We chose the corotational Maxwell model because, when generalized for multiple relaxation times, it gives an accurate prediction for molten plastics in large-amplitude oscillatory shear flow. We identify three relevant pairs of thermal boundary conditions: (i) both plates isothermal, (ii) with heat loss by convection from both plates, and (iii) one plate isothermal, the other with heat loss by convection. We find that the time-averaged viscous heating increases as an even power series of the dimensionless shear rate amplitude (Weissenberg number), and that it decreases with the dimensionless imposed frequency (Deborah number). We distinguish between the dimensionless time-averaged temperature rise, $\bar \Theta $Θ¯, and the oscillating part, $\tilde \Theta $Θ̃, where $\Theta \equiv \bar \Theta + \tilde \Theta $Θ≡Θ¯+Θ̃. We solve analytically for the $\bar \Theta $Θ¯ profile through the sample thickness for all three pairs of thermal boundary conditions. For the worst case, two adiabatic walls, we derive an expression for the oscillating part of the temperature rise, $\tilde \Theta $Θ̃. We find this $\tilde \Theta $Θ̃ to be a Fourier series of even harmonics whose contribution to the temperature rise can be as important as $\bar \Theta $Θ¯. If both plates are adiabatic, then the sample temperature rises without bound. Otherwise, it does not.

EXACT SOLUTION FOR STEADY PAINT FILM FLOW OF A PSEUDO PLASTIC FLUID DOWN A VERTICAL WALL BY GRAVITY

Here in this paper, the steady paint film flow on a vertical wall of a non-Newtonian pseudo plastic fluid for drainage problem has been investigated. The exact solution of the non-linear problem is obtained for the velocity profile. Also the average velocity, volume flux, shear stress on the wall, force to hold the wall in position and normal stress difference have been derived. We retrieve Newtonian case, when material constant μ₁ and relaxation time λ₁ equal zero. The results for co-rotational Maxwell fluid is also obtained by taking material constant μ₁ = 0. The effect of the zero shear viscosity η?, the material constant μ₁, the relaxation time λ₁ and gravitational force on the velocity profile for drainage problem are discussed and plotted.

Global generalized solutions for Maxwell-alpha and Euler-alpha equations

We study initial-boundary value problems for the Lagrangian averaged alpha models for the equations of motion for the corotational Maxwell and inviscid fluids in 2D and 3D. We show existence of (global in time) dissipative solutions to these problems. We also discuss the idea of dissipative solution in an abstract Hilbert space framework.

Large-amplitude oscillatory shear flow from the corotational Maxwell model

Using the single relaxation time corotational Maxwell fluid, we derive explicit analytical expressions for the first, third, and fifth harmonics of the alternating shear stress response in large-amplitude oscillatory shear (LAOS). We also derive corresponding expressions for the zeroth, second, and fourth harmonics of both the first and second normal stress differences. These harmonics are found to depend upon just two dimensionless groups: the Deborah and Weissenberg numbers, each of which causes non-Newtonian behavior. The form of the solution for the corotational Maxwell model in LAOS matches the forms of the analytical solutions for two molecular models for dilute solutions and one for concentrated solutions or melts. We also derive an analytical solution for the corotational Maxwell model after startup of LAOS. For this we find that both small and large amplitude cases approach a periodic limit cycle (alternance) at the same rate for both the shear stress response and for the normal stress differences. For molten high density polyethylene that is lightly filled with carbon black, we find good quantitative agreement with measured LAOS behavior when our analytical solution is superposed for multiple relaxation times.

Dynamic modelling of melt spinning

Dynamics of melt spinning is reconsidered to determine essential effects in modelling of the process. Basic dynamic equations are reformulated with heat production due to viscous friction in the melt, as well as for non-isochoric deformation. Viscoelastic effects are accounted for by applying co-rotational Maxwell fluid model. Different cooling (heating) zones are taken into account with different cross-blow velocity and temperature of the cooling medium. Crystallization is included in the set of basic equations, and specific effects of crystallization on spinning are calculated. Example computations of velocity, temperature and stress profiles are performed for melt spinning of PET fibres with extreme spinning conditions.

“Necking” in high-speed spinning revisited

The conditions required for “necking” in melt-spinning are discussed. The adopted criterion defines necking as a filament-radius profile R(x) exhibiting two inflexion points ( R = 0 ), one characterized by a negative, the other one by a positive, third gradient R'. Analysis of the dynamics of spinning of a non-isothermal corotational Maxwell fluid leads to the conclusion that the conditions of necking require either long relaxation times (Deborah number > 1 2 ), or a large, positive tension gradient F possibly controlled by inertia. In inertialess Newtonian filaments, the necking condition can also be satisfied by local reduction of viscosity. The above three mechanisms of necking, viscoelastic, inertial, and viscous, are analyzed and discussed.

The mechanisms of ‘neck-like’ deformation in high-speed melt spinning. 1. Rheological and dynamic factors

In the search for mechanisms responsible for abrupt (‘neck-like’) deformation of polymer jets in high-speed fibre spinning, two groups of factors are analyzed. Kinematics of inertialess jets from Newtonian, Reiner-Rivlin, ‘power-law’ and co-rotational Maxwell fluids are studied. It is shown that the main factor responsible for S-shaped velocity profiles and concentrated deformation (‘necking’), is a positive viscosity gradient along the fluid jet. Elasticity, and non-linear (elongation-rate dependent) viscosity do not admit ‘necking’ under isothermal conditions, and reduce the necking effects produced by temperature-controlled viscosity increase. A similar study of Newtonian jets with inertial, surface tension, gravity and air drag effects, confirmed the controlling role of the viscosity gradient. Inertia, surface tension and air drag slightly reduce necking in non-isothermal jets; gravity exerts a small positive effect. Necking effects modelled by exponentially varying viscosity cannot quantitatively reproduce the experimental observations on high-speed fibre spinning. An additional source of viscosity gradient — stress-induced crystallization — will be discussed in a separate paper.

Reentrant corner singularities in non-newtonian flow. Part I: theory

Biorthogonal series expansions are used to study the creeping flow of a corotational Maxwell fluid in plane reentrant and non-reentrant sectors. The governing equations are formulated in terms of stream function and Airy stress function only, the radial parts of which are determined by an infinite system of fourth-order nonlinear ordinary differential equations which are seen to be singular perturbations of the linear second-order equations for Stokes flow. Exact formal solutions in logarithmic series are derived, but do not readily enable the asymptotics of the corner singularities to be studied. Instead, approximate methods based on Picard linearization and dominant component analysis are used to decouple the system, whence the singularities are studied in terms of generalized hypergeometric equations and Meijer G-functions. Certain conditions on the roots of associated quartic equations are derived, the satisfaction of which would indicate the existence of lip vortices in a reentrant sector and the integrability of stress in a neighbourhood of the corner.

Analysis of Isothermal Fiber Spinning with the Doi‐Edwards Constitutive Equation

The isothermal spinline velocity profile is computed using the Doi‐Edwards constitutive equation, and the profile is found to be nearly the same as that for a corotational Maxwell fluid, for which an analytic solution exists. Both models predict limited spinnability, i.e., solutions exist for short spinlines and low draw ratios only. This behavior contrasts to that of the upper‐convected Maxwell model, for which threadlines of any length may be spun. The spinning behavior of the Doi‐Edwards model is dominated not by the distribution of relaxation times, but by the elongational strain measure.

Elongational flow behavior of viscoelastic liquids: Part I. Modeling of bubble collapse

AbstractA spherically collapsing gas bubble will create a uniaxial extension of the fluid surrounding it. This flow may be readily analyzed, and it is possible to relate observed experimental parameters such as the pressure inside the bubble and the bubble collapse rate to the predictions of viscoelastic constitutive relations. This study compares the predictions of the Newtonian fluid and a viscoelastic model to observed bubble collapse in three fluids: one Newtonian fluid and two viscoelastic polymer solutions. Good agreement between data and the model was observed for the Newtonian experiments. The viscoelastic solutions, however, yielded rates of collapse which were greater than the Newtonian predictions based on equal zero‐shear viscosity. Subsequent modeling with a modified corotational Maxwell fluid yielded good agreement with the data for the transient elongational response in the viscoelastic solutions.