Numerical results for the conservation of energy of Maxwell media for the Rayleigh–Stokes problem

This publication continues our studies of analytical solutions of the Rayleigh–Stokes problem for Maxwell fluids [J. Zierep, C. Fetecau, Energetic balance for the Rayleigh–Stokes problem of a Maxwell fluid, Int. J. Eng. Sci. 45 (2007) 617–627]. We start from the Fourier sine transform. The numerical result is given and discussed for the velocity u, the power of the wall shear stresses L, the dissipation Φand the boundary layer thickness δ. These new results are important for nature and technology.     

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.     

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.     

Near crack line elastic–plastic analysis for a infinite plate loaded by two pairs of point tensile forces

The near crack line analysis method is used to investigate a center crack loaded by two pairs of point tensile forces in an infinite plate in an elastic–perfectly plastic solid, and the analytical solutions are obtained in this paper. These solutions include: the elastic–plastic stress field near the crack line, the law that the length of the plastic zone along the crack line is varied with an external loads and the bearing capacity of an infinite plate with a center crack. The results of this paper are sufficiently precise near the crack line because the assumptions of the small scale yielding theory have not been used and no other assumptions have been taken.     

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.     

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.     

Response of flexure–torsion coupled composite thin-walled beams with closed cross-sections to random loads

A study of the flexure–torsion coupled random response of the composite beams with solid or thin-walled closed-sections subjected to various types of concentrated and distributed random excitations is dealt with in this paper. The effects of flexure–torsion coupling, shear deformation and rotary inertia are included in the present formulations. The random excitations are assumed to be stationary, ergodic and Gaussian. Analytical expressions for the displacement response of the composite beams are obtained by using normal mode superposition method combined with frequency response function method. The present method can produce the effective solutions for the composite Timoshenko beams with circumferentially antisymmetric (CAS) configuration and more general beam assemblages of connected beams. The influences of flexure–torsion coupling, shear deformation and rotary inertia on the random response of an appropriately chosen composite beam from the literature are demonstrated and discussed.     

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.     

Conditions for the absolute instability of solutions of differential-difference equations

Conditions for the absolute instability of solutions of differential-difference equations are established.Translated from Neliniini Kolyvannya, Vol. 7, No. 3, pp. 430–436, July–September, 2004.     

Transverse-roll global modes in a Rayleigh–Bénard–Poiseuille system with streamwise variable heating

The effect of a spatially inhomogeneous heating of the bottom wall in Rayleigh–Bénard–Poiseuille convection is studied for slow streamwise variations of the temperature profile. The problem is defined by the constant Reynolds number of the Poiseuille through flow, assumed to be low (typically 10), the constant Prandtl number, and the spatial evolution of the Rayleigh number , assumed to be subcritical everywhere except in a limited region around its single maximum . In this initial study, all spanwise inhomogeneities such as side walls or spanwise variable heating are neglected to obtain two-dimensional (transverse roll) global mode solutions by means of WKBJ asymptotics. The resulting frequency selection yields, at leading order, a global mode frequency equal to the local absolute frequency ω^t at the streamwise location where the Rayleigh number is maximum, with higher-order corrections for non-parallelism. These allow the determination of critical values of for global instability as a function of the profile of the local Rayleigh number and of Prandtl and Reynolds numbers.     

Group Stratification and Exact Solutions of the Equation of Transonic Gas Motions

Exact solutions of the Kármán–Guderley equation that describes spatial gas flows in the transonic approximation are considered. A group stratification of the equation with respect to the infinite-dimensional part of the admissible group is constructed. New invariant and partly invariant solutions are obtained. The possibility of existence of solutions continuous in the entire space is analyzed for invariant submodels with one independent variable. A solution of the Kármán–Guderley equation of the double-wave type is constructed.     

Rheological equations of state of weak solutions of polymers as rigid ellipsoidal macromolecules

Rheological equations of state are obtained for weak solutions of polymers as rigid ellipsoidal macromolecules, taking into account the rotational Brownian motion of the macromolecules, their energy, and the external force fields (electric and magnetic). As an example the effect of the inertia of the macromolecules on the rheological behavior of the solutions is examined.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 2, pp. 125–129, March–April, 1972.