Stretching a plane surface in a viscoelastic fluid with prescribed skin friction

A model of forced convection flow due to stretching surface is derived to represent the physical system with prescribed skin friction. To achieve the similar solutions, the partial differential equations are reduced into ordinary differential equations. The analytic solutions of the resulting problems have been obtained by a homotopy analysis method. The convergence of the developed series solution is seen. Finally, the results of velocity, temperature, the stretching velocity, and Nusselt number are analyzed. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

On the plane deformation of fibre-reinforced viscoelastic plates

The realization method of elastic solutions is used to solve the problem of bending of a viscoelastic plate reinforced unidirectionally by elastic fibres. Numerical computations are carried out for three kinds of external load. The plane deformation of this plate is discussed.     

Vibration of a viscoelastic rectangular orthotropic plate on a viscoelastic foundation

The integrodifferential equation describing the vibration of a viscoelastic rectangular orthotropic plate resting on a viscoelastic foundation with two foundation moduli is investigated. The solution of the starting equation is obtained by the method of straight lines [1] in conjunction with the averaging method [2]. By setting c₁=0 or c₂=0 or their combinations in the solutions obtained we obtain various particular cases that are generalized in the present investigation.Institute of Cybernetics and Computer Center, Academy of Sciences of the Uzbek SSR, Tashkent. Institute of Mechanics and Earthquake Resistance of Structures, Academy of Sciences of the Uzbek SSR, Tashkent. Translated from Mekhanika Polimerov, No. 6, pp. 1087–1094, November–December, 1970.     

Solution of the problem of the plane deformation of a tube made of physically nonlinear quadratic viscoelastic material

An exact solution is obtained for the problem of the plane deformation of a tube of quadratically nonlinear viscoelastic material characterized by four kernels. For given boundary pressures the solution contains functions determined from simple creep and relaxation tests for stresses and strains lying in the region of linearity of the relation. In accordance with the method of approximations for nonlinear materials [9], an approximation is given for a series of convolution integrals of increasing multiplicity in transforms and inverse transforms together with an error estimate.Moscow Institute of Electronic Engineering. Translated from Mekhanika Polimerov, No. 1, pp. 117–123, January–February, 1973.     

Drawing a viscoelastic extrudate

The problem of the shape of a drawn viscoelastic extrudate of arbitrary cross section is investigated. The solution of the problem is represented as a superposition of the nonstationary extension under constant load and the motion determined by the throughput of the process. As an example, calculations are made for a circular polyethylene extrudate.Mekhanika Polimerov, Vol. 3, No. 2, pp. 326–333, 1967     

The Stokes Problem in a Ball and in the Exterior of a Ball

We analyze the formula for the solution to the Stokes problem in a ball and in the exterior of a ball which was derived in the previous paper by the author published in 2002. The main result is pointwise estimates for the kernels of potentials in this formula. Bibliography: 4 titles.     

RETRACTED ARTICLE: Effect of electromagnetic field on the stability of viscoelastic fluid film flowing down an inclined plane

The stability of thin electrically conducting viscoelastic fluid film flowing down on a non-conducting inclined plane in the presence of electromagnetic field is investigated under induction-free approximation. Surface evolution equation is derived by long-wave expansion method. The stabilizing role of Hartman number M (magnetic field) and the destabilizing role of the viscoelastic property \({\varGamma}\) and the electric parameter E on such fluid film are established through the linear stability analysis of the surface evolution equation. Investigation shows that at small values of Hartman number (M), the influence of electric parameter (E) on the viscoelastic parameter \({(\varGamma)}\) is insignificant, while for large values of M, E introduces more destabilizing effect at low values of \({\varGamma}\) than that at high values of \({\varGamma }\). An interesting result also perceived from our analysis is that the stabilizing effect of Hartman number (M) is decreasing with the increase of the values of \({\varGamma}\) and E, even it gives destabilizing effect after a certain high value of the electric field depending on the high value of \({\varGamma}\). The weakly nonlinear study reveals that the increase of \({\varGamma}\) decreases the explosive and subcritical unstable zones but increases the supercritical stable zone keeping the unconditional zone almost constant.     

Investigation of the vibration of viscoelastic pipes in a viscoelastic soil

The vibration equation for a viscoelastic pipe laid in a viscoelastic soil is obtained and solved. A specific problem is examined using the results of an experimental investigation. The time dependence of the pipe displacements relative to the soil is constructed.M. T. Urazbaev Institute of Structural Mechanics and Seismic Stability, Academy of Sciences of the Uzbek SSR, Tashkent. Translated from Mekhanika Polimerov, No. 6, pp. 1112–1115, November–December, 1973.     

Analysis of viscoelastic fluid flow with temperature dependent properties in plane Couette flow and thin annuli

The steady state flow in very thin annuli has been studied analytically for the case where the annular gap is much smaller than the radius of the inner cylinder and for the outer cylinder rotating at constant angular speed and the inner cylinder at rest. The cylinders were subjected to two different thermal boundary conditions. The exponential effect of temperature on the relaxation time and the viscosity coefficient was accounted into the governing differential equations using Nahme’s law. Effects of viscous dissipation as well as εDe² (viscoelastic index for SPTT constitutive equation) on the dimensionless velocity and temperature profiles have been investigated. Results show that while the properties of the fluid depend on temperature, the velocity and temperature profiles are different compared to those obtained with constant physical properties. The Nahme–Griffith number increases whereas εDe² as a viscoelastic index decreases when temperature dependent physical properties are considered. In addition, the results indicate that the viscous dissipation has a sensible effect on heat transfer and the Nusselt number decreases with an increase in the Nahme–Griffith number.     

Vibrations of a shock-absorbing device on a nonlinear viscoelastic base

A method of calculating a viscoelastic shock-absorbing device or structure by the method of averaging systems of integro-differential equations is considered. The principal quadratic theory of hereditary viscoelasticity is used as the physical relationship between forces and displacements of the shock-absorbing base or structure.Moscow Institute of Electronic Machine Construction. Translated from Mekhanika Polimerov, No. 4, pp. 730–737, July–August, 1972.     

Flutter of a viscoelastic plate

The method of averaging is used to investigate the flutter of a viscoelastic plate. It is shown that taking the viscous drag into account leads to a reduction of the critical velocity.Moscow Institute of Geodetic and Aerial Photography and Cartographic Engineers. Translated from Mekhanika Polimerov, No. 6, pp. 1077–1083, November–December, 1971.     

Webs and projective structures on a plane

We prove that there is a one-to-one correspondence between projective structures defined by torsion-free connections with skew-symmetric Ricci tensor and Veronese webs on a plane. The correspondence is used to characterise the projective structures in terms of second order ODEs.     

Quasianalytic classes of functions on a plane

Quasianalytic classes of functions in a Jordan domainG are defined. We consider classes of functions defined by conditions imposed on the decrease rate of the best uniform polynomial approximations and investigate the dependence of the quasianalyticity of these classes on the geometric structure of a domain.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 7, pp. 982–991, July, 1993.     

The problem of a viscoelastic cylinder rolling on a rigid half-space

The problem of a viscoelastic cylinder rolling on a rigid base, propelled by a line force acting at its centre, is solved in the noninertial approximation. The method used is based on a decomposition of hereditary integrals developed by the authors in previous work, and on the viscoelastic Kolosov-Muskhelishvili equations which are used to generate a Hilbert problem. In this formulation, the problem reduces to a nonsingular integral equation in space and time, which simplifies under steady-state conditions and for exponential decay materials, to algebraic form. There are also two subsidiary conditions.In the case of a standard linear model, explicit analytic results and numerical examples are given for the pressure function, for surface displacements, and also for hysteretic friction.     

Canonical frame of a curve on a conformal plane

We show how differential geometry of smooth curves on the conformal plane can be studied by Élie Cartan’s method of exterior forms and moving frames. We find the canonical form of the derivation equations of a curve (which is not a circle) in the case of a semi-isotropic frame. We give a new proof of the theorem that states that curves of constant (in particular, zero) conformal curvature are loxodromes. We integrate the system of structure equations of the isotropy subgroup of a point.     

Action of a flat stamp on a half-plane of viscoelastic material

A plane problem concerning the action of a flat stamp on a reinforced, linearly viscoelastic medium is considered. One of the main axes of elasticity is parallel to the boundary of the half-plane. The reinforcing material is supposed to be perfectly elastic, and the matrix material linearly viscoelastic. The solution of the problem is based on the assumption that up to the moment of application of the forces, the medium is stress-free and at rest. The formulation of the problem is given, and a method of solution presented. A numerical example is offered.Mekhanika Polimerov, Vol. 2, No. 5, pp. 688–692, 1966