Instability of plane Couette flow of viscoelastic liquids

A review of our work on the stability of plane Couette flow of a viscoelastic liquid is given. The first part of the review is based on the assumption of a “short memory” of the fluid. The Reynolds-Orr energy criterion intimates the possibility of instability at very low Reynolds numbers. A linear stability analysis for disturbances in the flow plane shows that beyond the stability limit given by the energy criterion there are always disturbances which grow with time. A critical assessment of the short memory theory shows the severe limitations of its applicability.In the second part of the paper, the assumption of short memory is dropped. The stability of plane Couette flow with respect to special disturbances perpendicular to the flow plane is investigated for a Maxwell fluid. The flow is unstable if the product of Reynolds number and Weissenberg number is higher than a certain limit, which has the value one for a simple Maxwell fluid. This result can also be interpreted as follows: The flow becomes unstable if the velocity at the boundary walls is higher than the shear wave velocity of the fluid.     

Theoretical description of transient shear flow of viscoelastic liquids

A constitutive equation presented earlier has been generalized to describe the transient shear flow behaviour of polymer liquids. The mathematical structure and properties of such a generalization have been discussed. The practical expressions for the generalized relaxation modulus have been proposed.     

Stability of viscoelastic flow around periodic arrays of cylinders

Low Reynolds number flow of Newtonian and viscoelastic Boger fluids past periodic square arrays of cylinders with a porosity of 0.45 and 0.86 has been studied. Pressure drop measurements along the flow direction as a function of flow rate as well as flow visualization has been performed to investigate the effect of fluid elasticity on stability of this class of flows. It has been shown that below a critical Weissenberg number (We_c), the flow in both porosity cells is a two-dimensional steady flow, however, pressure fluctuations appear above We_c which is 2.95±0.25 for the 0.45 porosity cell and 0.95±0.08 for the higher porosity cell. Specifically, in the low porosity cell as the Weissenberg number is increased above We_c a transition between a steady two-dimensional to a transient three-dimensional flow occurs. However, in the high porosity cell a transition between a steady two-dimensional to a steady three-dimensional flow consisting of periodic cellular structures along the length of the cylinder in the space between the first and the second cylinder occurs while past the second cylinder another transition to a transient three-dimensional flow occurs giving rise to time- dependent cellular structures of various wavelengths along the length of the cylinder. Overall, the experiments indicate that viscoelastic flow past periodic arrays of cylinders of various porosities is susceptible to purely elastic instabilities. Moreover, the instability observed in lower porosity cells where a vortex is present between the cylinders in the base flow is amplifieds spatially, that is energy from the mean flow is continuously transferred to the disturbance flow along the flow direction. This instability gives rise to a rapid increase in flow resistance. In higher porosity cells where a vortex between the cylinders is not present in the base flow, the energy associated with the disturbance flow is not greatly changed along the flow direction past the second cylinder. In addition, it has been shown that in both flow cells the instability is a sensitive function of the relaxation time of the fluid. Hence, the instability in this class of flows is a strong function of the base flow kinematics (i.e., curvature of streamlines near solid surfaces), We and the relaxation time of the fluid.     

Elongational flow behavior of viscoelastic liquids: Modelling bubble dynamics with viscoelastic constitutive relations

Summary Earlier parts of this series have described a technique based on the collapse of single bubbles in the fluids for studying the elongational rheology of viscoelastic solutions and melts of moderate viscosities ( ₀ > 10²p) at relatively high strain rates . The present paper describes the modelling of bubble collapse with both rate and integral type constitutive relations using a body coordinate system. Predictions of the stress at the bubble wall as a function of time during collapse from a BKZ model and a modified corotational Maxwell model compared favorably with experimental data for two polymer solutions, 1% polyacrylamide in water/glycerine and 2% hydroxypropyl cellulose in water.

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Zusammenfassung In vorangehenden Veröffentlichungen dieser Reihe wurde eine Methode beschrieben, mit Hilfe derer man aus dem Zerfall von einzelnen Blasen in einer Flüssigkeit auf die Dehn-Rheologie viskoelastischer Lösungen und Schmelzen mittlerer Viskosität ( ₀ > 10² P) bei relativ hohen Dehngeschwindigkeiten schließen kann. Die vorliegende Untersuchung beschreibt Modelle des Blasenzerfalls mit Hilfe von Stoffgleichungen sowohl vom rate- als auch vom Integral-Typ, wobei ein körperfestes Koordinatensystem benutzt wird. Die Voraussagen der Spannung an der Blasenwand als Funktion der Zeit während des Zerfalls bei Verwendung eines BKZ- und eines modifizierten korotatorischen Maxwell-Modells zeigen eine recht gute Übereinstimmung mit experimentellen Werten, die an zwei Polymerlösungen, nämlich einer 1%igen Polyacrylamid-Lösung in einer Wasser-Glycerin-Mischung und einer 2%igen wäßrigen Hydropropylcellulose, erhalten worden sind.

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Nomenclature a material constant - b material constant - g metric tensor, space coordinates - m material constant - n material constant - p pressure - P _G pressure within bubble - P _R pressure outside bubble at the wall - P pressure far away from the bubble - R bubble radius - dR/dt - R ₀ initial bubble radius - t time - u velocity - U potential function - Y R/R ₀ Greek symbols covariant body metric tensor - surface tension - rate of deformation matrix, II -second invariant of - strain rate - ₀ zero shear rate viscosity - _e elongational viscosity - _ef effective viscosity - ¹, ², ³ coordinates in body system - ^{1 1}/R ₀ ³ - body stress tensor - density - space stress tensor - relaxation time - _ef effective relaxation time - bubble pressure function, defined in eq. [19] - vorticity tensor With 11 figures and 1 table     

On some nearly viscometric flows of viscoelastic liquids

Superposition of oscillatory shear imposed from the boundary and through pressure gradient oscillations and simple shear is investigated. The integral fluid with fading memory shows flow enhancement effects due to the nonlinear structure. Closed-form expressions for the change in the mass transport rate are given at the lowest significant order in the perturbation algorithm. The elasticity of the liquid plays as important a role in determining the enhancement as does the shear dependent viscosity. Coupling of shear thinning and elasticity may produce sharp increases in the flow rate. The interaction of oscillatory shear components may generate a steady flow, either longitudinal or orthogonal, resulting in increases in flow rates akin to resonance, and due to frequency cancellation, even in the absence of a mean gradient. An algorithm to determine the constitutive functions of the integral fluid of order three is outlined.Nomenclature A _n Rivlin-Ericksen tensor of order . - A _k Non-oscillatory component of the first order linear viscoelastic oscillatory velocity field induced by the kth wave in the pressure gradient - d Half the gap between the plates - e _x, e _z Unit vectors in the longitudinal and orthogonal directions, respectively - G(s) Relaxation modulus - G History of the deformation - Stress response functional - I() Enhancement defined as the ratio of the frequency dependent part of the discharge to the frequencyindependent part of it at the third order - I ^*() Enhancement defined as the ratio of the increase in discharge due to oscillations to the total discharge without the oscillations - k Power index in the relaxation modulus G(s) - k _i ^–1 Relaxation times in the Maxwell representation of the quadratic shear relaxation modulus (s ₁, s ₂) - m _i ^–1, n _i ^–1 Relaxation times in the Maxwell representations of the constitutive functions ₁(s ₁,s ₂,s ₃) and ₄ (s ₁, s ₂,s ₃), respectively - P Constant longitudinal pressure gradient - p Pressure field - _mx ,⁽³⁾ _nz ,⁽³⁾ Mean volume transport rates at the third order in the longitudinal and orthogonal directions, respectively - ₀,⁽³⁾, ₁,⁽³⁾ Frequency independent and dependent volume transport rates, respectively, at the third order - s = t- Difference between present and past times t and      

On instabilities of single-integral constitutive equations for viscoelastic liquids

Various types of instabilities are exposed in this paper for time-strain separable single-integral viscoelastic constitutive equations (CE's). They were distinguished into two groups and defined as Hadamard and dissipative type of instabilities. As for the Hadamard-type, previously obtained criteria are found to be necessary only. They are necessary and sufficient only for thermodynamic stability. Improved, stricter Hadamard stability criteria are described briefly in this paper, and then applied to study of stability of several CE's. It is shown that the Currie potential with the K-BKZ equation and the model proposed by Papanastasiou et al. are Hadamard unstable. In the case of dissipative stability, the necessary and sufficient condition for stress boundedness in any regular flow with a given history, is proved. Then, this criterion was applied to the neoHookean, Mooney, and Yen and McIntire specifications of the general K-BKZ model, to exhibit unbounded solutions. In addition, Larson-Monroe potential which is later proved to be Hadamard unstable but satisfies the above criterion of boundedness, is shown to have unstable decreasing branch in steady simple shear flow. At present, to the authors' knowledge, there is no viscoelastic single-integral CE of factorable type proposed in the literature which can satisfy all the Hadamard and dissipative stability criteria.     

On a class of constitutive equations for viscoelastic liquids

Simple rheological equations that describe non-linear viscoelastic phenomena in polymeric liquids have long attracted the attention of many rheologists. Although there are many ways of deriving such equations, only one concept is considered here. This concept is based on the introduction of an internal parameter, the recoverable strain tenson, and arises from a special kinematic study together with the formalism of irreversible thermodynamics. The main part of the paper sets out the theory for a single mode but a multimode extension is demonstrated towards the end and is compared with experimental data. Finally some of the problems that remain unsolved in the theory are discussed.The aim of this paper is to acquaint rheologists with the author's views as the model rheological equations he developed have recently been discussed in the literature without his participation.     

Singular limit of equations for linear viscoelastic fluids with periodic boundary conditions

We consider a one-parameter family of problems, governing, for any fixed parameter, the motion of a linear viscoelastic fluid in a two-dimensional domain with periodic boundary conditions. The asymptotic behavior of each problem is analyzed, by proving the existence of the global attractor. Moreover, letting the parameter go to zero, since the memory effect disappears, we obtain a limiting problem, given by the Navier-Stokes equations. For any fixed parameter, we construct an exponential attractor. The resulting family is robust, meaning that these exponential attractors converge, in an appropriate sense, to an exponential attractor of the limiting problem.     

On pulsatile flow of non-Newtonian liquids

Rheologica Acta - We consider the behaviour of non-Newtonian liquids as they are made to flow through straight pipes of circular cross section under the action of a pressure -gradient which...     

Oscillatory flow of linear viscoelastic fluids in thin-walled elastico-viscous tubes

Rheologica Acta - The oscillatory flow of a linear visco-elastic fluid in a thin-walled elastico-viscous tube is investigated for the case of long waves. It is found that the elasticity of the...     

On the large deflections of linear viscoelastic beams

The quasi-static response and the stored and dissipated energies due to large deflections of a slender inextensible beam made of a linear viscoelastic material and subjected to a time-dependent inclined concentrated load at the free end are investigated. The beam cross-section is considered prismatic, the self-weight is disregarded and the material is initially stress free. The set of four first-order non-linear partial integro-differential equations obtained from geometrical compatibility, equilibrium of forces and moments, and linear viscoelastic constitutive relation is numerically solved using a one-parameter shooting method combined with a fourth-order Runge-Kutta algorithm. An analytical expression is derived to divide the energy supplied by the external load into conserved and dissipated parts. For the case study presented, a three-parameter solid linear viscoelastic constitutive model is employed and a step load is applied. The variables are made non-dimensional, so four parameters govern the problem: the ratio between the final and initial relaxation moduli, the load magnitude, the angle of inclination and the unloading time. A finite-element model is also performed to compare and validate the analytical and numerical formulations. Results are presented for encastré curvature and tip displacement versus time, geometrical configuration, load versus tip displacement, total work done by the external force, stored and dissipated energies versus time, energy per unit length along arc length for three times and versus time for two beam sections.     

On the stability of viscoelastic fluid flow

Summary In continuation of two foregoing investigations where stability of rectilinear flows of viscoelastic fluids between two plane parallel plates was analysed, the present paper discusses the analogous problem for flow in a cylindrical tube (Poiseuille flow) and a cylindrical annulus. The linearized disturbance equations for steady instabilities of cellular type are derived taking into account viscoelasticity according to second-order theory as well as inertial forces. ForPoiseuille flow analytical solutions are obtained, whereas for flow in an annulus a technique of numerical integration is utilized. It is found that curves of neutral stability are similar to the ones for plane channel flow giving a hint in this ease also that possibly two different types of instabilities may occur simultaneously.

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Zusammenfassung Im Anschluß an zwei vorangegangene Untersuchungen, in denen die Stabilität von geradlinigen Strömungen viskoelastischer Flüssigkeiten zwischen zwei ebenen, parallelen Platten behandelt worden war, wird hier das analoge Problem für die Strömung in einem Kreiszylinder (Poiseuille-Strömung) und in einem Ringspalt analysiert. Die Störungsgleichungen für stationäre Störströmungen vom Zellulartyp werden unter Zugrundelegung einer Theorie zweiter Ordnung für die Viskoelastizität und Einbeziehung der Trägheit in linearisierter. Form abgeleitet. Für diePoiseuille-Strömung lassen sich analytische Lösungen finden, wohingegen für die Ringspalt-Strömungen die Lösungen durch numerische Integration gewonnen werden. Die Indifferenzkurven sind von ähnlicher Gestalt wie bei der Strömung durch den ebenen Kanal und weisen ebenfalls auf die Möglichkeit eines gleichzeitigen Vorhandenseins zweier verschiedener Störströmungen hin.

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The main results of this paper were presented as the second part of a talk entitled On stability of rectilinear flows of viscoelastic fluids at the annual meeting of German Rheologists held in Bad Münster a. Stein from 28–30 May 1969.     

On the stability of viscoelastic fluid flow

Summary This paper supplements the earlier ones of this series, which dealt withstationary instabilities of cellular type and generalizes the case of planeCouette flow foroscillatory cellular disturbances, usually termed as over-stability. Using the well-known scheme of linearization the disturbance equations are derived and the analytical solutions have been given. By numerical evaluation of these, the curves of neutral stability are calculated explicitely for certain typical cases. It is found that, in general, the overstability mode is higher than the stationary mode but that both can come close to each other if certain conditions are satisfied.

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Zusammenfassung Die vorliegende Veröffentlichung ergänzt die vorangehenden dieser Serie, welchestationäre Instabilitäten vom Zellulartyp zum Gegenstand haben, indem sie für den Fall der ebenenCouette-Strömungoszillierende Zellularströmungen (overstability) in die Untersuchung einbezieht. Unter Verwendung des wohlbekannten Linearisierungsverfahrens werden die Störungsgleichungen abgeleitet und analytisch gelöst. Aus diesen Lösungen werden für einige typische Fälle die Indifferenzkurven numerisch bestimmt. Man findet ganz allgemein, daß die Indifferenzkurven der oszillierenden Störungen höher liegen als diejenigen der stationären Störungen, daß beide jedoch unter gewissen Bedingungen sehr nahe beieinander liegen können.

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The main results of this paper were presented as the first part of a talk entitled Stationäre und periodische Instabilitäten in Strömungen viskoelastischer Flüssigkeiten vomCouette- undPoiseuille-Typ at the annual meeting of German Rheologists held in Berlin from May 11–13, 1970.     

On the stability of viscoelastic fluid flow

Summary The stability of a viscoelastic fluid flowing through a plane parallel channel, in the presence of constant pressure gradient is analyzed. The linearized disturbance equations for steady instabilities of cellular type are derived (taking inertia fully into account but viscoelasticity as in second-order theory) and solved using an approximation method as well as direct integration procedure. It is found that, as also in the case of planeCouette flow, such types of instabilities should be expected only for materials for which second normal-stress difference is positive. The curves of neutral stability possess more complicated nature than those forCouette flow, and may give a hint to the simultaneous existence of two types of disturbances with different cell widths, provided the parameter representing the ratio of inertial to elastic forces lies in a certain range.

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Zusammenfassung Es wird die Stabilität der Strömung einer viskoelastischen Flüssigkeit durch einen ebenen, parallelen Kanal bei Anwesenheit eines konstanten Druckgradienten untersucht. Bei Erfassung der Viskoelastizität durch eine Theorie zweiter Ordnung und vollständiger Berücksichtigung der Trägheit werden die linearisierten Störungsgleichungen abgeleitet und sowohl mittels einer Näherungsmethode als auch durch ein direktes Integrationsverfahren gelöst. Man findet, daß genauso wie im Fall der ebenenCouette-Strömung Instabilitäten eines derartigen Typs nur dann zu erwarten sind, wenn die zweite Normalspannungsdifferenz des Materials positiv ist. Die Indifferenzkurven sind von komplizierterer Form als bei derCouette-Strömung und deuten darauf hin, daß u. U. zwei Arten von Störungen mit verschiedener Zellgröße gleichzeitig auftreten können, wenn der Parameter, welcher das Verhältnis von Trägheits- und elastischen Kräften bezeichnet, in einem gewissen Wertebereich liegt.

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Herrn Professor Dr. K.Hansen zum 60. Geburtstag gewidmet

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The main results of this paper were presented as the first part of a talk entitled On stability of rectilinear flows of viscoelastic fluids at the annual meeting of German Rheologists held in Bad Münster a. Stein from 28–30 May 1969.