Slow motions of a rigid sphere in incompressible viscoelastic fluids

Problems concerned with the force of resistance and the moment of forces acting from the side of a viscoelastic fluid on a sphere moving with acceleration are considered in a linear formulation. Fairly simple relations are obtained for a fluid with a single relaxation time or a single after-action time. A discussion of the asymptotic expressions is given for a fluid with a large number of times.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 9–16, March–April, 1976.     

Slow nonstationary vertical motions of a die on the surface of an elastic half-space

We consider the dynamic contact problem on vertical motions of an absolutely rigid body on an elastic half-space. We assume that the contact region does not vary during the motion and there is no friction under the die bottom. We construct an approximate solution of the problem under the assumption that the variation in the contact pressure under the die bottom on the time interval in which the Rayleigh wave runs the distance equal to the contact area diameter is small. Computational formulas are obtained for the cases of circular and elliptic dies.     

Self-similarity of nonstationary boundary-layer motions

In this paper we show that problems concerning the development of a boundary layer on a semi-infinite plate when the outer flow speed is of the form U = (1 + ct)^b a, and on a cylinder when the outer flow speed has the forms U = ctx^m and U = (1 + ct)^b ax^m, are self-similar. We present the results of numerical calculations for various values of , b, and m. We consider the problem of a stepwise nonstationary heating of a plate, impulsively set into motion in an incompressible fluid; we show that this problem is self-similar and obtain its solution numerically.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 122–125, July–August, 1975.     

One-dimensional nonstationary motions of a liquid

An effective method is developed for solving the problem of the nonstationary motion of a liquid with plane, cylindrical, and spherical symmetry [1]. It is based on the idea of dividing the region of disturbed motion into two parts and using matched asymptotic expansions. Solutions are presented to typical problems associated with the motion of a piston, and these make it possible to obtain the solution to problems of an explosion in a liquid, oscillations of a bubble, and so forth. It is also shown that the well-known solutions for such problems given, for example, in the book of Naugol'nykh and Roi on the basis of the acoustic approximation with allowance for nonlinear terms are incorrect.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 3–8, March–April, 1980.     

Forced axisymmetric motions of viscoelastic cylindrical shells

The isothermal response of a viscoelastic cylindrical shell, of finite length, to arbitary axisymmetric surface forces, initial conditions, and boundary conditions is considered within the linear theory of thin shells. The problem is formulated with the effects of shear deformation and rotatory inertia included; the viscoelastic properties are assumed to be isotropic and homogeneous. The response is first found formally in terms of a causal Green's function. It is then shown that when Poisson's ratio is constant, the causal Green's function can be expanded in a series of orthonormal spatial eigenfunctions of an associated elastic shell eigenvalue problem. The resulting solution for the general problem is an eigenfunction series with Laplace transformed time-dependent coefficients. The general solution is applied to predicting the motion of a uniform, simply-supported cylindrical shell, initially quiescent, which is subjected to a step pressure moving with constant velocity. For this example, the relaxation function of the shell material in uniaxial extension is taken to be that of a standard linear solid. The motions predicted by simpler shell models, namely, shells with bending only and without bending, are also considered for comparison. Here, the absolute values of the Fourier coefficients in the shell displacement series go to zero faster than the inverse of the first or second power of positive integers when bending is excluded or included, respectively. Numerical results are presented for a moderately long and relatively thick, nearly elastic, cylindrical shell.     

Twisted films of Newtonian and viscoelastic liquids

Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 170–178, May–June, 1988.     

Slow viscoelastic radial flow between parallel disks

A perturbation analysis is presented for the steady-state radial flow of a third-order fluid between two parallel disks. The results include previous perturbation analyses in which various other rheological models were used. The pressure drop needed to maintain the radial flow is less than that for the Newtonian creeping-flow solution because of fluid inertia and shear-thinning viscosity, whereas the normal stresses have the opposite effect. Possible use of the “radial flow viscometer” for experimental evaluation of second-order constants is also discussed. Finally, molecular stretching in the flow system is examined using the elastic dumbbell model for a polymer molecule.     

Forward roll coating flows of viscoelastic liquids

Roll coating is distinguished by the use of one or more gaps between rotating cylinders to meter and apply a liquid layer to a substrate. Except at low speed, the two-dimensional film splitting flow that occurs in forward roll coating is unstable; a three-dimensional steady flow sets in, resulting in more or less regular stripes in the machine direction. For Newtonian liquids, the stability of the two-dimensional flow is determined by the competition of capillary and viscous forces: the onset of meniscus nonuniformity is marked by a critical value of the capillary number. Although most of the liquids coated industrially are non-Newtonian polymeric solutions and dispersions, most of the theoretical analyses of film splitting flows relied on the Newtonian model. Non-Newtonian behavior can drastically change the nature of the flow near the free surface; when minute amounts of flexible polymer are present, the onset of the three-dimensional instability occurs at much lower speeds than in the Newtonian case.Forward roll coating flow is analyzed here with two differential constitutive models, the Oldroyd-B and the FENE-P equations. The results show that the elastic stresses change the flow near the film splitting meniscus by reducing and eventually eliminating the recirculation present at low capillary number. When the recirculation disappears, the difference of the tangential and normal stresses (i.e., the hoop stress) at the free surface becomes positive and grows dramatically with fluid elasticity, which explains how viscoelasticity destabilizes the flow in terms of the analysis of Graham [M.D. Graham, Interfacial hoop stress and instability of viscoelastic free surface flows, Phys. Fluids 15 (2003) 1702–1710].     

Wave motions in viscoelastic tubes. Forced oscillations

A characteristic of small blood and lymphatic vessels is the capacity of the wall to change its rheological properties and lumen by active contraction of the annular muscle cells contained in it [1–3]. A model of flow in the vessels taking this feature into account has been proposed in [4, 5], where a linear stability analysis is also given. A consequence of wall activity is the existence of auto-oscillatory flow conditions [6–8], which have also been discovered in the numerical solutions of the corresponding problems [9, 10]. Up to the present time flows have only been studied under steady conditions at the ends of the vessel and in the environment. The wall of an actual blood vessel is subject to various actions, frequently of a periodic nature: pressure pulsations at entry and rhythmically changing external forces applied from the surrounding tissues. Data exist on the sensitivity of vessels to transient actions [11–13], in particular on the relationship of their hydraulic resistance to frequency and amplitude of the action. There has been frequent discussion of the hypothesis that bv contraction of muscles in its walls or by external compression the vessel can act as a valveless pump [14, 15]. Within the framework of the quasione-dimensional approximation given below [4] the movement of liquid along a viscoelastic tube in the presence of small amplitude periodic external actions has been studied. A general solution of the problem has been constructed and concrete examples are given illustrating the features of forced wave motions in a tube having passive and active properties.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No, 4, pp. 94–99, July–August, 1984.     

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.     

Secondary flows of viscoelastic liquids in straight tubes

The unsteady flow of the Green–Rivlin fluids in straight tubes of arbitrary cross-section driven by a pulsating pressure gradient is investigated. The non-linear constitutive structure defined by a series of nested integrals over semi-infinite time domains is perturbed simultaneously with the boundary of the base flow through a novel approach to domain mapping. The dominant primary component of the flow, the longitudinal field, and the much weaker transversal field arise at the first and the second orders of the analysis, respectively. The secondary field is driven by first-order terms stemming from the linearly viscoelastic longitudinal flow at the first order. The domain mapping technique employed yields a continuous spectrum of unconventional closed cross-sectional shapes. We present longitudinal velocity profiles and transversal time-averaged, mean secondary flow streamline patterns for a specific fluid and for representative cross-sectional shapes in the spectrum the triangular, square and hexagonal shapes.     

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      

Bulk rheology of dilute suspensions in viscoelastic liquids

A theoretical relation is derived for the bulk stress in dilute suspensions of neutrally buoyant, uniform size, spherical drops in a viscoelastic liquid medium. This is achieved by the classic volume-averaging procedure of Landau and Lifschitz which excludes Brownian motion. The disturbance velocity and pressure fields interior and exterior to a second-order fluid drop suspended in a simple shear flow of another second-order fluid were derived by Peery [9] for small Weissenberg number (We), omitting inertia. The results of the averaging procedure include terms up to orderWe ². The shear viscosity of a suspension of Newtonian droplets in a viscoelastic liquid is derived as $$\eta _{susp} = \eta _0 \left[ {1 + \frac{{5k + 2}}{{2(k + 1)}}\varphi - \frac{{\psi _{10}^2 \dot \gamma ^2 }}{{\eta _0^2 }}\varphi f_1 (k, \varepsilon _0 )} \right],$$ whereη ₀, andω ₁₀ are the viscosity and primary normal stress coefficient of the medium,ε ₀ is a ratio typically between ?0.5 and ?0.86,k is the ratio of viscosities of disperse and continuous phases, and \(\dot \gamma \) is the bulk rate of shear strain. This relation includes, in addition to the Taylor result, a shear-thinning factor (f ₁ > 0) which is associated with the elasticity of the medium. This explains observed trends in relative shear viscosity of suspensions with rigid particles reported by Highgate and Whorlow [6] and with drops reported by Han and King [8]. The expressions (atO (We ²)) for normal-stress coefficients do not include any strain rate dependence; the calculated values of primary normal-stress difference match values observed at very low strain rates.     

Analysis of diffusion-induced bubble growth in viscoelastic liquids

Transport models of diffusion-induced bubble growth in viscoelastic liquids are developed and evaluated. A rigorous model is formulated that can be used to describe bubble growth or collapse in a non-linear viscoelastic fluid, and takes into account convective and diffusive mass transport as well as surface tension and inertial effects. Predictions for bubble growth dynamics demonstrating the importance of fluid elasticity are presented. These predictions indicate that for diffusion-induced bubble growth in viscoelastic liquids, the lower bound for growth rate is given by growth in a Newtonian fluid and the upper bound by diffusion-controlled growth. The influence of non-linear fluid rheology on bubble growth dynamics is examined and found to be relatively minor in comparison to fluid elasticity. It is shown how previously published models employing various approximations can be derived from the rigorous model. Comparisons of predicted bubble growth dynamics from the rigorous and approximate models are used to establish the ranges of applicability for two commonly-used approximations. These comparisons indicate that models using a thin boundary layer approximation have a rather limited range of applicability. An analysis of published experimental bubble growth data is also carried out using appropriate transport models.     

Nonequilibrium effects in the filtration of viscoelastic liquids

An experimental investigation was made of nonequilibrium effects with the filtration of viscoelastic petroleums and polymer solutions. A scheme is proposed to explain the effects obtained. A comparison between theoretical dependences and experimental data confirms the proposed scheme of the phenomenon. The filtration of viscoelastic liquids has recently attracted considerable attention. The interest in it is connected with the discovery of well-expressed elastic properties in several high-viscosity petroleums, as well as with the need to explain the laws governing the movement of solutions and polymer melts in a porous medium, with application to industrial chemical equipment and to the injection of polymer solutions into oil-bearing strata with the aim of increasing the yield of petroleum. Among articles devoted to the filtration of viscoelastic liquids, there should be mentioned the fundamental investigation of Marshall and Metzner [1], me theoretical article of Wissler [2], and me review of Savins [3]. Specifically, it is shown in [1] that, as a result of elastic effects, the filtration resistance in a steady-state flow rises in accordance with the relationship 1+A(V/r)² where V is the filtration rate; r is the mean pore radius; is the relaxation time; A is a constant on the order of magnitude of 10. In [1] this increase in the resistance is explained qualitatively by the action of specific, for elastic liquids, normal stresses in contracting and then expanding pore channels. A corresponding calculation (for a flat channel and a determined nonlinear model of a viscoelastic liquid) was made in [2] using the method of perturbation theory.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 76–83, September–October, 1973.     

On periodic flow of non linear viscoelastic liquids

Summary After brief reference to energetical principles in order to find suitable constitutive equations for non-linear viscoelastic fluids, the behaviour of stress components is examined, for a simple shear motion with sinusoidal time time varying displacements of a liquid with an oscillating fading memory.

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Sommario Dopo aver richiamato brevemente un possibile procedimento per la determinazione della equazione costitutiva per liquidi viscoelastici nel caso di grandi deformazioni, si esamina il comportamento delle componenti del tensore degli sforzi nel caso di un moto periodico a filetti rettilinei e paralleli, supposto che il liquido abbia una memoria debole e oscillante nel tempo.

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Research supported by C.N.R., Gruppo nazionale per la fisica matematica.     

Slow flow of a viscoelastic fluid through a contraction

Summary The creeping flow of a viscoelastic fluid through a tapered contraction is analysed by using a perturbation method. Finite element method has been applied to solve the vorticity transport and stream function equations.For a contraction ratio of 2:1 results for the stream function, the velocity distribution and the stress components along the tube axis are presented and discussed. The results indicate that for a viscoelastic fluid there is a tendency of circular motion in the contraction zone.

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Zusammenfassung Mit Hilfe einer Störungsrechnung wird die schleichende Strömung einer viskoelastischen Flüssigkeit durch eine Rohrverengung berechnet. Die dabei auftretenden Randwertaufgaben werden mit der Methode der finiten Elemente gelöst.Für ein Kontraktionsverhältnis von 2:1 werden Ergebnisse für die Stromfunktion, die Geschwindigkeitsverteilung und den Verlauf der Reibungsspannungen entlang der Rohrachse angegeben und diskutiert. Die Ergebnisse für das Geschwindigkeitsfeld zeigen, daß in der Kontraktionszone eine Sekundärströmung erzeugt wird, die aus zwei entgegengesetzt drehenden Ringwirbeln besteht. Da diese der Primärströmung zu überlagern ist, besteht bei einer viskoelastischen Flüssigkeit im Unterschied zu einer newtonschen in der Kontraktionszone die Tendenz zu einer Zirkulationsströmung.

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With 9 figures