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The paper addresses the problem of modeling a fracture propagation in linear elastic porous media driven by injection of non-Newtonian power-law fluid. The model involves the lubrication theory equation expressing the mass conservation of fluid and a nonlocal singular integral relation for the fracture aperture as a functional of the fluid pressure. We assume only the viscosity-dominated case in which the fracture toughness is neglected. This allows considering the fracture as an opened part of a preexisting closed fracture of larger length. The numerical method consists in solving equations of the model over the whole length of the preexisting fracture without distinguishing the tip region of the opened part. The problem is solved via the finite element method. The weak formulation of the solution allows pressure singularity with the required asymptotics near the fracture tip. Comparison of the numerical and the exact self-similar solutions in the case of a constant flow rate, zero fluid leakoff, and zero fracture toughness reveals the accuracy of the approximate solution to at least O(h1/2), where h is the maximal linear size of the grid cells. As an illustration, we also demonstrate the numerical experiments with periodic fluid injection and variable fluid efficiency.

Exact solutions of the equations of ideal magnetohydrodynamics describing the class of unsteady flows of an electrically conducting fluid with a constant total pressure are constructed. The solutions are written in the Lagrange coordinate system; arbitrariness in its choice was used to parameterize magnetic field lines. The wide functional arbitrariness the solutions provide a significant variation in the picture of the described fluid motions. An example of unsteady flow of an ideal electrically conducting fluid in a cylindrical channel with fixed magnetic tubes is given.

Incompressible stationary flows of ideal plasma are studied. By introduction of curvilinear system of coordinates in which streamlines and magnetic force lines form a family of coordinate surfaces, MHD equations are partially integrated and brought to a certain convenient form. It is demonstrated that the admissible group of Bogoyavlenskij's symmetry transformations (Bogoyavlenskij, 2001 [1] ) performs as a scaling transformation (the scaling factor is an arbitrary function of one variable) for the curvilinear coordinates. Analytic description of stationary flows with constant total pressure is given. It is shown, that contact magnetic surfaces of such flows are translational surfaces, i.e. are swept out by translating one curve rigidly along another curve. Examples of solutions with constant total pressure possessing an arbitrariness in one function of two variables and three functions of one variable are given.

The physical properties of ideal plasma flow described by the Ovsyannikov plane vortex are studied. The particle trajectories and magnetic lines are shown to be plane curves, and an algorithm for describing the motion in three-dimensional space is proposed. Some exact solutions of the submodel are obtained and studied.

We give a description of the ideal plasma flow, which is governed by an exact partially invariant solution of the magnetohydrodynamics equations. The solution generalizes known one-dimensional flow with spherical waves. The generalization consists in addition of the special tangent vector components of the velocity and the magnetic field at any plasma particle. In the special case of zeroth tangential component the solution coincides with the classical one-dimensional one. This paper describes a three-dimensional picture of the plasma flow, governed by the obtained solution.

We investigate properties of plasma flow, governed by an exact reduction of the ideal magnetohydrodynamics equations which was obtained in our recent article. The reduction is called a singular vortex. It gives a description of a three-dimensional nonstationary plasma flow in terms of solution of reduced system of PDEs with two independent variables. In this paper we investigate symmetry properties of the reduced system and construct its invariant solutions. Also we give a description of the physical features of plasma motion, governed by the singular vortex. Namely, we prove that in the observed class of solutions the particles trajectories as well as magnetic field lines are flat curves. The position and orientation of the plane containing the particle's trajectory depends on the initial position of the particle. We describe the shape of trajectories and magnetic field lines for the cases of stationary and self-similar plasma flow.

A full description of irrotational ideal gas motions is given, in which pressure depends only on time. It is shown that under such motions the dependence of velocity vector on spatial coordinates is linear and of a special kind.

An exact solution is constructed, which describes a gas glow in a strip between a rectilinear source and sink. With time, the strip turns and expands. In the case of consistent boundary conditions, the flow in the strip is continuous. If the consistency constraints are violated, a shock wave is formed inside the strip.

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.

An invariant submodel of the two-dimensional equations of gas dynamics, constructed on an operator which is a combination of the time-shift, rotation and projective operators, is investigated using the PODMODELI program [1]. A canonical form of the submodel is constructed and a preliminary analysis of it is carried out (the group property, the hyperbolicity region and the first integrals). The self-similar solution of the submodel is investigated in detail. It determines the solutions of the submodel in question with closed invariant streamlines. Using a hierarchy of submodels, first integrals are obtained in the “second-level” submodel. A qualitative description of the nature of the motion is given (the contact characteristics and the particle trajectories). It is shown that the solution possesses discrete symmetry - invariance under rotation around the origin of coordinates by an angle that is a multiple of 2π/N, with a certain natural N. It is pointed out that for certain values of the parameters, solutions of this type describe the gas motion with vacuum regions. The features of the flows obtained are illustrated by examples — the exact solutions of the gas — dynamic equations, which describe the expansion of a gas to a vacuum in an infinite time.