A class of exact solutions of the Navier-Stokes equation

The unsteady problem of a viscous incompressible flow in free space is investigated, and a class of exact solutions of the Navier-Stokes equation is given for a general initial condition. This flow field represents several shear layers superimposed on an irrotational, three-dimensional straining flow. This solution incorporates the three main features of vortex motion, i.e., stretching, convection and viscous diffusion of vorticity. The solution is exemplified for several kinds of initial condition. One of them represents a flow approaching a steady state in which the above three effects are brought to an equilibrium. Another solution shows collision of two shear layers in various arrangements. Two parallel shear layers merge into a single layer, while two antiparallel shear layers in which the vortices in each layer are in the opposite directions disappear as pair cancellation, and two layers merge like vectors by oblique collision. It is also shown that N shear layers, in general, merge like vectors.     

Obtaining the exact solutions of the Navier-Stokes equations

The Navier-Stokes equations of two-dimensional, steady, incompressible flow are studied using the unknown streamlines as one of the coordinates. The solution of the equations for in-plane motion is obtained, which corresponds to the well known solution of Couette flow, radial flow, spiral flow and the flow between two concentric cylinders. Possibilities of obtaining other solutions by this approach are also discussed.     

Some unsteady exact pseudo-plane solutions for the Navier-Stokes equations

Unsteady pseudo-plane and plane motions for Navier-Stokes equations have been studied, which are generalized Beltrami flows, related to separable stream functions. All the pseudo-plane motions of the first kind of such a type, a wide class of plane universal solutions, which seems to be new, and a class of pseudo-plane motions of the second kind, dependent on an arbitrary function of time, with a primitive steady plane solution which is the celebrated Kelvin's cat's eye vortices are formed.

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Sommario Si studiano moti non stazionari pseudopiani e piani dell'equazioni di Navier Stokes. Questi moti sono flussi generalizzati di Beltrami con una particolare dipendenza della funzioni di corrente dalle coordinate spaziali in modo separato. Nell'ambito di questa famiglia di flussi vengono esplicitamente trovati tutti i moti pseudopiani di primo tipo e vengono anche individuate un'ampia classe di soluzioni universali piane e una classe di moti pseudopiani del secondo tipo che possiedono come flussoprimitivo i famosi vortici adocchi di gatto di Kelvin.

     

A class of exact solutions of the ideal plasticity equations

A class of exact solutions is formulated for the ideal plasticity equations in the case of plane deformation. The solutions describe the plastic state of various wedges, notched half-planes, and domains in the form of funnels produced by detonation, the plastic state of domains having a horn configuration, etc. Many of the solutions have natural boundaries comprising envelopes of slip lines. The equations for the boundaries, slip lines, and stresses are presented in explicit form.     

Several exact solutions of the simplified Navier-Stokes equations (SNSE)

The Simplified Navier-Stokes equations (SNSE) and their exact solutions for the flow near a rotating disk and the flow in the vicinity of a stagnation point for both two- and three-dimensional flows are presented in this paper. The analysis shows that in the aforementioned cases the exact solutions of the inner-outer-layer-matched SNSE^[4] are completely consistent with those of the complete Navier-Stokes equations (CNSE) and that the exact velocity solutions of D-SNSE^[1,3] agree with those of CNSE, however, the exact pressure solutions of D-SNSE do not agree with those of CNSE. The maximum relative pressure errors between the exact solutions of D-SNSE and CNSE can be as high as a hundred per cent.     

A class of exact solutions for N-dimensional incompressible magnetohydrodynamic equations

In this paper, a sufficient and necessary condition is presented for existence of a class of exact solutions to N-dimensional incompressible magnetohydrodynamic(MHD)equations. Such solutions can be explicitly expressed by appropriate formulae. Once the required matrices are chosen, solutions to the MHD equations are directly constructed.     

On some new exact solutions of incompressible steady state Navier-Stokes equations

In the present work, Lie symmetries are constructed for the steady state incompressible Navier-Stokes equations in two dimensional space. The Navier-Stokes equations are transformed into a fourth order quasilinear partial differential equation through well known approach of stream function. Seven similarity solutions are obtained using one-parameter Lie group of transformations with commuting infinitesimal operators. Among the seven solutions so obtained, three are new and analysed physically. Also, the remaining four solutions are reported in Polyanin and Zaitsev (Handbook of nonlinear partial differential equations, Chapman and Hall/CRC Press, Boca Raton, pp. 607–608, 2004) and interpreted hydrostatically.     

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.     

A class of exact solutions of the dynamical equations for rigid-viscoplastic bodies

Summary We here discuss the class of plane, irrotational flows of a rigid-perfectly plastic or of a rigid-viscoplastic body. Such flows turn out to be of a limited variety: they are either familiar hydrodynamical flows or are geometrically similar to a model flow whose velocity field can be represented in parametric form in terms of the complex error function. Qualitative features of such flows are discussed as well as the motivation for considering them.

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Zusammenfassung Betrachtet wird die Klasse der ebenen, wirbelfreien Strömungen in einem starr-idealplastischen oder starrviskoplastischen Körper. Solche Strömungen sind entweder von bekanntem Typ oder sind einer Modellströmung ähnlich, deren Geschwindigkeitsfeld eine parametrische Darstellung besitzt, die durch die komplexe Fehlerfunktion beschreibbar ist. Sowohl die qualitativen Hauptmerkmale als auch die Bedeutung solcher Strömungen werden besprochen.

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This paper summarizes research described in more detail in (1).     

An exact solution of the Navier-Stokes equations

Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 168–169, July–August, 1989.     

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.     

A new method for obtaining the exact solutions of the Navier-Stokes (NS) equations and its applications

In this paper we propose a new method for obtaining the exact solutions of the Mavier-Stokes (NS) equations for incompressible viscous fluid in the light of the theory of simplified Navier-Stokes (SNS) equations developed by the first author^[1,2], Using the present method we can find some new exact solutions as well as the well-known exact solutions of the NS equations. In illustration of its applications, we give a variety of exact solutions of incompressible viscous fluid flows for which NS equations of fluid motion are written in Cartesian coordinates, or in cylindrical polar coordinates, or in spherical coordinates. The project supported by National Natural Science Foundation of China.     

An absence of a certain class of periodic solutions in the Navier-Stokes equations

In the case of the 3D Navier-Stokes equations, it is proved that there exists a constant>0 with the following property: Every time-periodic solution with a period smaller than is necessarily a stationary solution. An explicit formula for is also provided.     

Spatial local solutions of the Navier-Stokes equations

This paper considers solutions of the Navier-Stokes equations polynomial in the coordinates, which. are called local solutions. For an incompressible fluid, all higher-order terms (sums of higher-order. monomials) of degree 2 are found and it is proved that nontrivial axisymmetric higher-order terms. of degree higher than 2 do not exist. Nonsolenoidal axisymmetric solutions are listed, which can be. treated as steady-state barotropic gas flows in a potential external-force field. All elliptic vortices. generalizing the well-known Kirchhoff solution are calculated. All solutions of degree 3 with the. higher-order term of partial form are found. Some of these solutions break down in a finite time. regardless of the value and sign of viscosity. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 2, pp. 109–119, March–April, 2009.