Effect of internal viscosity and elasticity of ellipsoidal macromolecules on the rheological behavior of dilute polymer solutions. rheological equations of state

The structural-continuum approach is used to obtain the Theological equations of state for dilute polymer solutions whose macromolecules can be modeled by an ellipsoid of rotation with internal viscosity and elasticity.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 94–98, May–June, 1976.     

Influence of the elasticity of polymer solutions on the reduction of resistance

The Toms effect is examined from the point of view of the viscoelastic properties of solutions of polymers. It is shown that the magnitude of the reduction of resistance depends on the set of the relaxation times of the macromolecules and the scales of the turbulent vortices.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 63–68, January–February, 1972.     

Anomalous readings of Pitot tubes in suspensions of solid particles and in solutions of polymers,lowering the turbulent friction

An anomalous lowering of the readings of Pitot tubes in flows of suspensions of solid particles of equal densities has been observed in a liquid. A universal dependence has been obtained, connecting the defect in the readings of Pitot tubes with their diameter, with the diameter of the particles of the suspensions, with the velocity of the flow, and with the volumetric concentration of the particles. The article presents new data on anomalous readings of Pitot tubes in polymer solutions, lowering the turbulent friction resistance. The character of this phenomenon is analogous to the lowering of the readings in suspensions. It is postulated that the anomalous readings of Pitot tubes in polymer solutions are connected with elasticoviscous associates of macromolecules of the polymer and molecules of the solvent. Experimental data are used to calculate the effective dimensions and volumetric concentration of the associates in the solutions.Moscow. Translated from Izvestiya Akademii Nauk SSSR. Mekhanika Zhidkosti i Gaza, No. 2, pp. 48–54, March–April, 1972.     

Dynamics of nanofibres conveyed by low Reynolds number flow in a microchannel

In this paper we aim to create an experimental and numerical model of nano and micro filaments suspended in a confined Poiseuille flow. The experimental data obtained for short nanofibres will help to elucidate fundamental questions concerning mobility and deformation of biological macromolecules due to hydrodynamic stresses from the surrounding fluid motion. Nanofibres used in the experiments are obtained by electrospinning polymer solutions. Their typical dimensions are 100–1000 μm (length) and 0.1–1 μm (diameter). The nanofibre dynamics is followed experimentally under a fluorescence microscope. A precise multipole expansion method of solving the Stokes equations, and its numerical implementation are used to construct a bead-spring model of a filament moving in a Poiseuille flow between two infinite parallel walls. Simulations show typical behaviour of elongated macromolecules. Depending on the parameters, folding and unfolding sequences of a flexible filament are observed, or a rotational and translation motion of a shape-preserving filament. An important result of our experiments is that nanofibres do not significantly change their shape while interacting with a micro-flow. It appeared that their rotational motion is better reproduced by the shape-preserving Stokesian bead model with all pairs of beads connected by springs, omitting explicit bending forces.     

Molecular theories of nonlinear viscoelasticity of polymers

It is emphasized that considerable advances have been made recently in the development of the molecular theories of nonlinear viscoelasticity of concentrated solutions and melts of linear polymers. The new ideas in this exceptionally important field of the rheology of polymers are analysed. The methods by which the constraints (entanglements) imposed on the motion of macromolecules by the polymer environment are taken into account are also considered in the paper. The most detailed discussion is devoted to the model of topological constraints in the form of a tube and to the self-consistent theory of anisotropic micro-viscoelasticity which takes into account the relaxation nature of the interaction of macromolecules with their surrounding medium as well as the anisotropy of their mobility.Invited paper, presented at the XII-th All-Union Symposium on Rheology held in Riga (USSR), December 7–9, 1982.     

Existence of Solutions for Steady Detonation of Gas Suspensions

The existence of solutions of the traveling–wave type is studied for a system of equations that describes a one–dimensional motion of a suspension of evaporating particles in a viscous and heat–conducting chemically reacting gas. Using topological methods, it is shown that solutions corresponding to weak, strong, and Chapman—Jouguet detonation exist under certain restrictions on energy release and mass transfer.     

Self-similar solutions of three-dimensional laminar magnetohydrodynamic boundary-layer equations

Self-similar solutions of three-dimensional boundary-layer equations of an incompressible fluid in ordinary hydrodynamics were considered in [1–3] et al. The present work looks for self-similar solutions of three-dimensional magnetohydrodynamic boundary-layer equations.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, Vol. 9, No. 4, pp. 10–17, July–August, 1968.     

Molecular motions in macromolecules under mechanic and electric force fields

General aspects of the dynamics of macromolecules are reviewed on the basis of a parallel analysis of the material response under mechanical and electric force fields.Long-range (normal and segmental dynamics) as well as short-range motions are studied on the basis of experimental data, correlation function, and memory equation techniques.Accordingly, several methods to describe the relaxational behavior of macromolecules and their pendant molecular groups are proposed.Presented as a keynote lecture at the 4th European Rheology Conference, September 4–9, 1994, Seville, Spain     

Nonequilibrium stretching dynamics of dilute and entangled linear polymers in extensional flow

We propose an extension of the FENE-CR model for dilute polymer solutions [M.D. Chilcott, J.M. Rallison, Creeping flow of dilute polymer solutions past cylinders and spheres, J. Non-Newtonian Fluid Mech. 29 (1988) 382–432] and the Rouse-CCR tube model for linear entangled polymers [A.E. Likhtman, R.S. Graham, Simple constitutive equation for linear polymer melts derived from molecular theory: Rolie–Poly equation, J. Non-Newtonian Fluid Mech. 114 (2003) 1–12], to describe the nonequilibrium stretching dynamics of polymer chains in strong extensional flows. The resulting models, designed to capture the progressive changes in the average internal structure (kinked state) of the polymer chain, include an ‘effective’ maximum contour length that depends on local flow dynamics. The rheological behavior of the modified models is compared with various results already published in the literature for entangled polystyrene solutions, and for the Kramers chain model (dilute polymer solutions). It is shown that the FENE-CR model with an ‘effective’ maximum contour length is able to describe correctly the hysteretic behavior in stress versus birefringence in start-up of uniaxial extensional flow and subsequent relaxation also observed and computed by Doyle et al. [P.S. Doyle, E.S.G. Shaqfeh, G.H. McKinley, S.H. Spiegelberg, Relaxation of dilute polymer solutions following extensional flow, J. Non-Newtonian Fluid Mech. 76 (1998) 79–110] and Li and Larson [L. Li, R.G. Larson, Excluded volume effects on the birefringence and stress of dilute polymer solutions in extensional flow, Rheol. Acta 39 (2000) 419–427] using Brownian dynamics simulations of bead–spring model. The Rolie–Poly model with an ‘effective’ maximum contour length exhibits a less pronounced hysteretic behavior in stress versus birefringence in start-up of uniaxial extensional flow and subsequent relaxation.     

One-Dimensional Steady Infiltration with Water Extraction

Exact solutions of the one-dimensional steady infiltration with water extraction are obtained for a particular but realistic form of the soil–water conductivity. The results are compared with numerical solutions for slightly different forms of the soil–water conductivity function, showing the sensitivity of the shape of the profiles on the exact representation of the soil–water conductivity. The exact solutions are an extension of Warrick's (1974) solution for the case where the soil can become saturated.     

Asymptotic Representations of Solutions of an Ordinary Differential Equation of the <Emphasis Type="Italic">n</Emphasis>th Order with Exponential Nonlinearity

We establish necessary and sufficient conditions for the existence of one class of solutions of a differential equation with exponential nonlinearity. Asymptotic representations of these solutions are obtained.__________Translated from Neliniini Kolyvannya, Vol. 7, No. 4, pp. 562–573, October–December, 2004.     

On Global Solutions of Linear Difference Equations with Deviating Argument

We present conditions under which global solutions of linear difference equations with deviating argument are solutions of ordinary difference equations.__________Translated from Neliniini Kolyvannya, Vol. 8, No. 1, pp. 29–45, January–March, 2005.     

Asymptotic solution of the axisymmetric problem of ideal fluid flow in the neighborhood of cusped cavities

A denumerable set of axisymmetric solutions of the equations of motion of an ideal fluid in the neighborhood of a cavity closure point, including an infinitely remote point, is obtained. This family of solutions is similar to the well-known family of corresponding two-dimensional solutions (1), (2).Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 84–87, July–August, 1990.     

Some self-similar solutions of Grad's equations

It is shown that the self-similar solutions of the Navier-Stokes and Burnett equations found earlier by the authors [1–9] can be extended to the case of two-dimensional flows of a weakly rarefied gas described by Grad's equations. Examples are given of numerical realization of self-similar solutions for flow in an expanding planar channel. It is found that there are appreciable differences between the behavior of the self-similar solutions of the Navier-Stokes, Burnett, and Grad equations in the neighborhood of a channel wall.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 88–94, May–June, 1982.     

On Asymptotics of Solutions of Semiexplicit Systems of Differential Equations

We study the problem of the existence of analytic solutions of a certain semiexplicit system of differential equations and obtain sufficient conditions for the existence of analytic solutions of the Cauchy problem in the neighborhood of a singular point.__________Translated from Neliniini Kolyvannya, Vol. 8, No. 1, pp. 132–144, January–March, 2005.     

Nonuniqueness of the solution of the problem of viscous interaction on an axisymmetric body

The article discusses solutions of the equations of the hypersonic boundary layer on an axisymmetric offset slender body (with a power exponent equal to 3/4), taking account of interactions with a nonviscous flow. It is shown that, in this case, the equations of the boundary layer have solutions differing from the self-similar solution corresponding to flow around a semi-infinite body. The solutions obtained are analogous to solutions for a strong interaction on a plate with slipping and triangular vanes [1–4], but are obtained over a wide range of values of the parameter of viscous interaction. An asymptotic solution is given to the problem with the approach to zero of the interaction parameter.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 41–47, September–October, 1973.The authors thank V. V. Mikhailova for discussion of the work and useful advice.     

Chaotic and periodic solutions of the problem of fluid convection in a closed channel

Exact periodic solutions of the nonlinear dissipative system describing the convection of a viscous heat-conducting fluid in a closed channel are shown to exist and are investigated. The question of the compatible existence of stationary, chaotic and periodic solutions is examined.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.6, pp. 36–42, November–December, 1992.     

Integro-Differential Inclusions with Hukuhara Derivative

For an integro-differential inclusion with Hukuhara derivative, we introduce the notion of quasisolution and present conditions under which the set of quasisolutions coincides with the set of ordinary solutions. We also prove theorems on the relaxation of ordinary solutions and the compactness of the set of these solutions.__________Translated from Neliniini Kolyvannya, Vol. 8, No. 1, pp. 80–88, January–March, 2005.     

Propagation of solitary waves over a semi-infinte underwater trough

Self-similar solutions describing the incidence of a uniform solitary wave on a semi-infinite linear trough are obtained on the basis of the nonlinear ray method [1]. Previously, in investigating the incidence of a wave on a trough [2] the conditions at the discontinuities present in the solutions were derived on the assumption that they are of low intensity. In the present study the use of the conditions at the discontinuities obtained by investigating soliton interaction [3–5] has made it possible to construct a series of new solutions and take into account wave reflection effects and the formation of a shadow zone beyond the trough. The types of solutions that occur are established in terms of the relations between the wave parameters and the relative depth of the trough. To ensure that self-similar solutions exist for all values of the parameters it was necessary to introduce a type of discontinuity not previously encountered [5–7].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 102–107, July–August, 1987.The author wishes to thank A. G. Kulikovskii and A. A. Barmin for discussing the work.     

Invariant solutions of the Navier-Stokes equations

We examine certain invariant solutions of the Navier-Stokes equations. We prove theorems concerning the existence of solutions of boundary-value problems of the corresponding S/H systems.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 6, pp. 56–64, November–December, 1972.In conclusion the author thanks V. V. Pukhnachev for a discussion of the results and for his advice.