Partially invariant solutions for a submodel of radial motions of a gas

All partially invariant solutions in terms of the group of extensions for a model of radial motions of an ideal gas are found. The solutions are obtained by the method of separation of variables in an equation containing functions of one variable but different functions of different independent variables. The solutions predict different continuous unsteady convergence or expansion of the gas under the action of a piston with a point sink or source. If the sink or source affects all particles simultaneously, a collapse or an explosion occurs. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 5, pp. 26–34, September–October, 2007.     

One class of partially invariant solutions of the Navier-Stokes equations

A family of partially invariant solutions of the Navier-Stokes equations of rank 2 and defect 2 is considered. These solutions describe the three-dimensional unsteady motions of a viscous incompressible fluid in which the vertical velocity component and the pressure are independent of the horizontal coordinates. In particular, they can be interpreted as flows in a horizontal layer, one boundary of which is the free surface. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 2, pp. 24–33, March–April, 1999.     

Acceleration of convergence by shifting the spectrum of implicit finite difference operators associated with the equations of gas dynamics

Eigensystem analysis techniques are applied to finite difference formulations of the Navier-Stokes equations in one dimension. Spectra of the resulting implicit difference operators are computed. The largest eigenvalues are calculated by using a combination of the Frechet derivative of the operators and Arnoldi's method. The accuracy of Arnoldi's method is tested by comparing the rate of convergence of the iterative method with the dominant eigenvalue of the original iteration matrix. On the basis of the pattern of eigenvalue distributions for various flow configurations, a shifting of the implicit operators in question is devised. The idea of shifting is based on the power method of linear algebra and is very simple to implement. This procedure has improved the rates of convergence of CFD codes (developed at NASA Ames Research Center) by 20%–50%. The sensitivity of the computed solution with respect to the shift is also studied. Finally, an adaptive shifting of the spectrum together with Wynn's acceleration algorithm are discussed. It turns out that the shifting process is a preconditioner for Wynn's method.     

Regular, partially invariant solutions of rank 1 and defect 1 of equations of plane motion of a viscous heat-conducting gas

A system of the Navier-Stokes equations of two-dimensional motion of a viscous heat-conducting perfect gas with a polytropic equation of state is considered. Regular, partially invariant solutions of rank 1 and defect 1 are studied. A sufficient condition of their reducibility to invariant solutions of rank 1 is proved. All solutions of this class with a linear dependence of the velocity-vector components on spatial coordinates are examined. New examples of solutions that are not reducible to invariant solutions are obtained. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 6, pp. 23–33, November–December, 2006.     

Differentially invariant solutions of equations of plane steady flows of a viscous heat-conducting perfect gas with a polytropic equation of state

A system of Navier-Stokes equations for two-dimensional steady flows of a viscous heatconducting perfect gas with a polytropic equation of state is considered. Differentially invariant solutions of this system are studied. Bases of differential invariants and operators of invariant differentiation are constructed for all subgroups of the admitted group. Examples of new differentially invariant solutions are obtained.     

Regular partially invariant solutions of defect 1 of the equations of ideal magnetohydrodynamics

All irreducible regular partially invariant submodels with one noninvariant function for the equations of ideal magnetohydrodynamics are constructed. The submodels are completed to involution, and partially integrated. The submodels specify Ovsyannikov vortex type motion or motion with homogeneous deformation in some spatial directions. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 2, pp. 5–15, March–April, 2009.     

Exact solutions for evolutionary submodels of gas dynamics

A new class of exact solutions with functional arbitrariness describing the motion of a polytropic gas is constructed on the basis of invariant submodels of rank two of the evolutionary type. In the solutions obtained, the velocity is a linear function of some spatial coordinates. These solutions describe continuous gas dispersion and the motion with density collapse at a finite time.     

Invariant and partially invariant solutions of integro-differential equations for linear thermoviscoelastic aging materials with memory

A linear thermoviscoelastic model for homogeneous, aging materials with memory is established. A system of integro-differential equations is obtained by using two motions (a one-dimensional motion and a shearing motion) for this model. Applying the group analysis method to the system of integro-differential equations, the admitted Lie group is determined. Using this admitted Lie group, invariant and partially invariant solutions are found. The present paper gives a first example of application of partially invariant solutions to integro-differential equations.     

Systematization of relaxation gas dynamics equations. A review

The phenomenological theory of relaxation gas dynamics equations is outlined for laminar flows of multicomponent reacting gases in an approximation analogous to the Navier-Stokes approximation. A system of general equations of relaxation gas dynamics including the level kinetics equations for all excited internal degrees of freedom is formulated on the basis of notions of continuum mechanics. A procedure of going over to particular cases characterized by certain relations between the relaxation times is described and examples of the corresponding closed systems of gas dynamics equations including systems containing the balance equations of the level or mode approximations for the vibrational energy levels of molecules of a gas mixture are given. A method of constructing a database of the models of the rate constants of physicochemical processes as coefficients in the source terms of the balance equations is considered.     

A fully implicit scheme for simulating ionized gas flows using the gas dynamics electrodynamics coupled system

The flow of ionized gases under the influence of electromagnetic fields is governed by the coupled system of the compressible flow equations and the Maxwell equations. In this system, coupling of the flow with the electromagnetic field is obtained through nonlinear and stiff source terms, which may cause difficulties with the numerical solution of the coupled system. The discontinuous Galerkin finite element method is used for the numerical solution of this system. For the magnetic field vector, discontinuous Galerkin discretization is performed using a divergence‐free vector base for the magnetic field to preserve zero divergence in the element and retain the implicit constraint of a divergence‐free magnetic field vector down to very low level both globally and locally. To circumvent difficulties resulting from the presence of the stiff source terms, implicit time marching is used for the fully coupled system to avoid wrong wave shapes and propagation speeds that are obtained when the coupling source terms are lagged in time or by using splitting iterative schemes. Numerical solutions for benchmark problems computed on collocated meshes for the flow and electromagnetic field variables with this fully coupled monolithic approach showed good agreement with other numerical solutions and exact results. Copyright © 2014 John Wiley & Sons, Ltd.     

Functional analysis of integral transport equations in linear rarefied gas dynamics

Summary A study is proposed on the functional properties of the solutions of an interesting class of linear integral equations governing linear problems in rarefied gas dynamics. The analysis is carried out through a systematic study of the integral operator generated by the kernel of the equations themselves.

---

Sommario In questa nota vengono studiate le proprietà funzionali di alcune equazioni integrali lineari che regolano problemi basilari della dinamica dei gas rarefatti. Gli operatori integrali, relativi alle operazioni stesse, sono analizzati in dettaglio.

     

Group classification and invariant solutions for the equations of flow and heat transfer of a viscoplastic medium

Exact solutions without major restrictions on the properties of the material are needed in research on the flow (especially viscosity) of metals at high temperatures under nonisothermal conditions. Often the shear resistance is governed mainly by the temperature and the deformation rate. Here are examined the group properties of the equations of flow and heat transfer of a medium whose shear resistance is a function of temperature and rate of shear deformation. The properties specific to metals are not used, so the results are applicable to a variety of media.