Crystallization under external pressure

The influence of external pressure on the dynamics of crystallization is examined by considering a solid–liquid two-phase system in a cylinder closed by a piston. The dynamic equations are derived using three methods, namely, Rational Thermodynamics (Liu’s procedure), the Matrix Model, and the general equation for the nonequilibrium reversible-irreversible coupling (GENERIC) formalism. The constitutive relation for the multiphase pressure on the piston is identical for all three methods, whereas some aspects of the result for the phase change dynamics differ. The rational thermodynamics treatment constrains the phase change dynamics of only those structural variables that enter into the dissipation inequality, whereas the other two formalisms make statements about the phase change of all structural variables. Nevertheless, all three methods show, first, that the phase change happens instantaneously at constant volume and, second, where morphological detail can be built into the model without violating thermodynamic principles. It is discussed how an appropriate choice of the morphological variables allows one to incorporate impingement of crystals and depletion zone effects, as well as to distinguish crystal shapes.     

The mathematical representation of driven thermodynamic systems

A general framework for the treatment of driven systems in nonequilibrium thermodynamics is discussed for two selected theories and a simple model system. The framework is based upon the of modeling and control of general physical systems proposed by van der Schaft and co-workers. The crucial concept is the notion of a Dirac structure representing the dynamical equations of motion as well as the power conserving interconnection structure of the system. We applied the framework to two existing theories and a very simple model system. The two selected theories are the “General Equation for the Non-Equilibrium Reversible-Irreversible Coupling” (GENERIC) formalism of Grmela and Öttinger and the Matrix model of Jongschaap; the model system is a viscous gas in a cylinder and an externally driven piston. It is shown that the new approach provides not only a common framework for both theories, but also useful extensions, in particular, an extended GENERIC treatment of driven systems.     

Structure of saturation jumps in nonequilibrium displacement from porous media

The nonequilibrium displacement characteristics are investigated experimentally and by numerical calculation of the nonequilibrium percolation equations under various conditions of wetting of the porous medium by the displacing and displaced fluids. A scheme for calculating the disequilibrium parameter is proposed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 97–104, November–December, 1990.     

Periodicity and chaos in a flexible crank-rocker mechanism

In this paper we introduce a crank-rocker mechanism at which the rocker is flexible. Using Hamilton’s principle we obtain the governing equations of motion for the elastic mode of the rocker. By applying the Bubnov–Galerkin global averaging method, we reduce the governing equations of motion to an ordinary differential equation which is Duffing’s oscillator with time varying coefficients. Through the application of Banach’s fixed-point theorem we predict the periodic solutions. Then we study the geometrical features of the motion near the 1 : 1, 1 : 2 and 2 : 1 commensurabilities. It is also shown that homoclinic and heteroclinic orbits can exist for the system.     

On multiscale significance of Rice’s normality structure

The normality structure proposed by [Rice, J.R., 1971. Inelastic constitutive relations for solids: an integral variable theory and its application to metal plasticity. J. Mech. Phys. Solids 19, 433–455.] provides a minimal framework of multiscale thermodynamics. As shown in this paper, Rice’s multiscale thermodynamic formalism is exactly consistent with Ziegler’s essential notion [Ziegler, H., 1977. An Introduction to Thermomechanics, North-Holland, Amsterdam.] that the entire constitutive response is determined by the knowledge of two scalar potential functions: an energy function and a dissipation function. In Rice’s multiscale thermodynamic formulation, the variational equation relating macroscale and microscale thermodynamic fluxes and forces plays a central role and ensures the equality between microscale and macroscale dissipation rate. The variational equation can be further reformulated into a principle of maximum equivalent dissipation. Based on the variation equation, the transformation from microscale to macroscale is characterized by two linear transformations with the same corresponding matrix.     

On the overall elastic moduli of composites with spherical coated fillers

A micromechanical approach is presented to estimate the overall linear elastic moduli of three phase composites consisting of two phase coated spherical particles randomly dispersed in a homogeneous isotropic matrix. The theoretical method is based on Eshelby’s equivalent inclusion method and its recent extension by Shodja and Sarvestani [J. Appl. Mech. 68 (2001) 3] to evaluate the local field variables in case of double (multi) inhomogeneities. Using Tanaka–Mori theorem [J. Elasticity 2 (1972) 199] and a decomposition of Green’s function integral equation, the pair-wise average phase values of stress and strain in two interacting coated particles are estimated. Following Ju and Chen [Acta Mech. 103 (1994) 103; Acta Mech. 103 (1994) 123] the ensemble phase volume average of stress and strain fields can be evaluated within a representative volume element containing a finite number of coated particles. Comparisons with classical bounds are presented to illustrate the accuracy of the proposed method.     

Calculation of accelerated motion of fluid in a tube

The results of calculation of accelerated flow of a fluid in a tube are compared with known experiments [1] in the laminar regime. The difference method was used to obtain a solution for unsteady axisymmetric flow that becomes steady over the length of the tube; this case was calculated earlier by Gromeka in the form of a series. An expression is derived for the coefficient of friction as a function of the Reynolds number Re and the acceleration of the fluid. The comparison reveals agreement between the results with an error not worse than 37%. However, the calculation gives a coefficient of friction proportional to Re to the power –1.5, whereas the experiment [1] reveals a weaker dependence proportional to Re to the power –1.15.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 158–160, September–October, 1981.     

Existence and nonuniqueness of local separation zones in viscous jets

The plane steady problem of the flow of a viscous wall jet past a smoothed break in the contour of a body is considered. For convenience, the flow in the neighborhood of the junction between two flat plates inclined at an angle to each other is chosen for study. As a result of the small extent of the region investigated the flow field is divided into two layers: the main part of the jet, which undergoes inviscid rotation, and a thin sublayer at the wall, which ensures the satisfaction of the no-slip condition. Particular interest attaches to the flow regime in which the solution in the sublayer satisfies the Prandtl boundary layer equations with a given pressure gradient. A similar problem was studied in [1–4]. The present case is distinguished by the structure of the free interaction region in a small neighborhood of the point of zero surface friction stress. By means of the method of matched asymptotic expansions, applied to the analysis of the Navier-Stokes equations, it is established that the interaction mechanism is that described in [5–7]. As a result, an integrodifferential equation describing the behavior of the surface friction stress function is obtained. A numerical solution of this equation is presented. The range of plate angles on which solutions of the equation obtained exist and, therefore, flows of this general type are realized is determined. The essential nonuniqueness of the possible solutions is established, and in particular attention is drawn to the possible existence of six permissible friction distributions.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 38–45, January–February, 1986.The author wishes to thank V. V. Sychev and A. I. Ruban for their useful advice and discussion of the results.     

Lie Symmetry Analysis of Differential Equations in Finance

Lie group theory is applied to differential equations occurring as mathematical models in financial problems. We begin with the complete symmetry analysis of the one-dimensional Black–Scholes model and show that this equation is included in Sophus Lie's classification of linear second-order partial differential equations with two independent variables. Consequently, the Black–Scholes transformation of this model into the heat transfer equation follows directly from Lie's equivalence transformation formulas. Then we carry out the classification of the two-dimensional Jacobs–Jones model equations according to their symmetry groups. The classification provides a theoretical background for constructing exact (invariant) solutions, examples of which are presented.     

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.     

Flow of a viscous fluid in an annular gap with intensive blowing from the walls

Self-similar solutions are obtained in [1, 2] to the Navier-Stokes equations in gaps with completely porous boundaries and with Reynolds number tending to infinity. Approximate asymptotic solutions are also known for the Navier-Stokes equations for plane and annular gaps in the neighborhood of the line of spreading of the flow [3, 4]. A number of authors [5–8] have discovered and studied the effect of increase in the stability of a laminar flow regime in channels of the type considered and a significant increase in the Reynolds number of the transition from the laminar regime to the turbulent in comparison with the flow in a pipe with impermeable walls. In the present study a numerical solution is given to the system of Navier-Stokes equations for plane and annular gaps with a single porous boundary in the neighborhood of the line of spreading of the flow on a section in which the values of the local Reynolds number definitely do not exceed the critical values [5–8]. Generalized dependences are obtained for the coefficients of friction and heat transfer on the impermeable boundary. A comparison is made between the solutions so obtained and the exact solutions to the boundary layer equations.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 21–24, January–February, 1987.     

Microcrack interaction brittle rock subjected to uniaxial tensile loads

A micromechanics-based model is proposed to describe unstable damage evolution in microcrack-weakened brittle rock material. The influence of all microcracks with different sizes and orientations are introduced into the constitutive relation by using the statistical average method. Effects of microcrack interaction on the complete stress–strain relation as well as the localization of damage for microcrack-weakened brittle rock material are analyzed by using effective medium method. Each microcrack is assumed to be embedded in an approximate effective medium that is weakened by uniformly distributed microcracks of the statistically-averaged length depending on the actual damage state. The elastic moduli of the approximate effective medium can be determined by using the dilute distribution method. Micromechanical kinetic equations for stable and unstable growth characterizing the ‘process domains’ of active microcracks are taken into account. These ‘process domains’ together with ‘open microcrack domains’ completely determine the integration domains of ensemble averaged constitutive equations relating macro-strain and macro-stress. Theoretical predictions have shown to be consistent with the experimental results.     

The turbulent boundary layer on the moving surface of a cylindrical body

A study is made of the problem of a two-dimensional turbulent boundary layer on the moving surface of a cylindrical body (a Rankine oval with a relative elongation of four) moving at constant velocity in an incompressible fluid. For the numerical simulation of the turbulent flow of the fluid, the boundary layer is divided into exterior and interior regions in accordance with a two-layer model, using different expressions for the coefficients of turbulent transfer for each region. A study was nade of the development of the boundary layer on the body at different speeds of the body surface and different Reynolds numbers. The following integral characteristics were found by numerical calculation: the work of friction as the body is displaced; the work expended on the movement of its surface; and, for a flow regime with separation, the work of the pressure force. In this case the following model of separation flow is assumed: beyond the singular point in the solution of the boundary layer equations that indicates the appearance of a region of reverse flow, the pressure and friction stress on the wall are constant and are determined by their values at the singular point.Translated from Izvestiya Akademii Nauk SSSH, Mekhanika Zhidkosti i Gaza, No. 5, pp. 61–67, September–October, 1984.Finally, the author would like to thank G. G. Chernyi and Yu. D. Shevelev for useful discussions and for their interest in this work.     

Calculation of nonequilibrium distribution functions and dissociation rate of diatomic molecules in a boundary layer

The effect of the nonequilibrium velocity and rotation distributions of the dissociating molecules on the population of the upper vibrational levels and the dissociation rate, when the reaction kinetics are strongly influenced by the motion of the gas, is analyzed by solving the Boltzmann equation by a method permitting the calculation of the distribution function under conditions of strong disequilibrium.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 165–172, March–April, 1989.     

The damping factor in the theory of the Prandtl mixing length

An introduction to the classical Prandtl equation of the additional factor which takes account of the effect of molecular velocity -the damping factor — enables us to describe the process of flow in the transition sector of the boundary region of a turbulent boundary layer by relying on the model of interaction of molecular and molar momentum exchange between the layers of averaged turbulent motion. The application of not only the universal forms of physical variables but also special conservative variables, the connection between which does not depend on the conditions on the outer edge of the boundary layer, widens the opportunities for applying the method to the boundary region of the gradient boundary layer. The comparatively laborious nature of the calculation from the conservative equations brought about the introduction of much simpler but at the same time fairly accurate approximation equations.The content of the article was delivered at a session of the Sixth All-Union Congress on Theoretical and Applied Mechanics, Tashkent, 24–30 September, 1986.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 46–53, September–October, 1987.     

Order Reduction of Parametrically Excited Nonlinear Systems: Techniques and Applications

The basic problem of order reduction of nonlinear systems with time periodic coefficients is considered in state space and in direct second order (structural) form. In state space order reduction methods, the equations of motion are expressed as a set of first order equations and transformed using the Lyapunov–Floquet (L–F) transformation such that the linear parts of new set of equations are time invariant. At this stage, four order reduction methodologies, namely linear, nonlinear projection via singular perturbation, post-processing approach and invariant manifold technique, are suggested. The invariant manifold technique yields a unique ‘reducibility condition’ that provides the conditions under which an accurate nonlinear order reduction is possible. Unlike perturbation or averaging type approaches, the parametric excitation term is not assumed to be small. An alternate approach of deriving reduced order models in direct second order form is also presented. Here the system is converted into an equivalent second order nonlinear system with time invariant linear system matrices and periodically modulated nonlinearities via the L–F and other canonical transformations. Then a master-slave separation of degrees of freedom is used and a nonlinear relation between the slave coordinates and the master coordinates is constructed. This method yields the same ‘reducibility conditions’ obtained by invariant manifold approach in state space. Some examples are given to show potential applications to real problems using above mentioned methodologies. Order reduction possibilities and results for various cases including ‘parametric’, ‘internal’, ‘true internal’ and ‘true combination resonances’ are discussed. A generalization of these ideas to periodic-quasiperiodic systems is included and demonstrated by means of an example.     

Anisotropic percolatory-convective diffusion in underground leaching

The two-dimensional problem of the propagation of an impurity in underground flows is investigated with allowance for chemical reactions between the solvent and the ore-bearing matrix material. The flow velocity field generated by a pair of wells (source—sink) is assumed to be steady and known. A numerical method of solving the system of equations of convective diffusion and nonequilibrium sorption is discussed. Typical isoconcentration fields are presented. The effect of the influence parameters on the principal characteristics of the leaching process is analyzed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 96–101, January–February, 1987.     

Weakly rarefied gas flows between nonuniformly heated parallel plates

The aim of this research is to establish the validity of the predictions of the theory of slow nonisothermal flows, to study the limits of applicability (with respect to the Knudsen number) of the conclusions reached and to determine the effect of the Knudsen layers on these flows on the basis of a numerical investigation of slow nonisothermal weakly rarefied gas flow in a plane infinite channel with weakly nonequilibrium heating of the walls and a finite wall temperature difference. The gas flow is described by a relaxation transport equation. The results obtained show how quickly, as the Knudsen number decreases, the solutions of the transport equation outside the Knudsen layers tend to the solution of the equations of gas dynamics of slow nonisothermal flows (and not to the solution of the Navier-Stokes equations).Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 115–121, January–February, 1988.     

Evaluation of link-track performances using DEM

A two-dimensional discrete-element model for the interaction between link-track and soil is presented. The model was developed using commercial PFC2D code. Two different particles, sphere and clump of two spheres, were used to represent the soil. The soil parameters of the model were determined using Hertzian contact theory. Based on the model and soil parameters, simulations of biaxial tests and calculations of the internal angle of friction and cohesion were preformed. The simulation results showed that the internal angle of friction should not exceed the value of 0.65 when using the spherical particles. Based on the clumped particles model, simulations of shear tests with two grouser plates (lengths 100 and 150 mm) were performed under different soil conditions, normal pressures, and cleat heights. A curve fitting of the simulation results was performed using three semi-empirical models from Bekker, Janosi, and Wong for representing the shear stress–displacement relationship. The best fitting was achieved using Wong’s approach. The simulation results of the cleat effects were compared with Bekker’s grouser approach and McKyes’s formulation for soil–blade interaction. In most of the cases, the results of Bekker’s model were the lower bound and McKyes’s model, the upper bound of the DEM simulation results. The properties of the soil model for the DEM were determined using simulation results of shear tests by grouser plate. In the range investigated, the size of the shearing grouser plate is not significant in determining the soil model properties.     

A Knudsen layer in a flow with two-temperature relaxation

The boundary conditions are considered for the hydrodynamic equation of [1, 2] for highly nonequilibrium diatomic gas with vibrational relaxation.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 6, pp. 38–43, November–December, 1972.