Search

A flat strong discontinuity in a polymer fluid is investigated in the rheological Vinogradov−Pokrovsky model. Relationships for a strong discontinuity are given. The existence of discontinuous stationary solutions of the problem without rotation of the velocity vector at the discontinuity and with rotation (rotational discontinuity) is substantiated. Examples of numerical stationary solutions are given.

Under consideration is some mathematical model describing magnetohydrodynamic motion of viscoelastic polymeric fluid in the cylindrical near-axial zone of a swirl chamber. The absence of steady-state solutions is proved for the cylindrical zone with a fixed lateral boundary. The solutions are also considered in the case of near-axial zone with a free lateral boundary.

Steady-state solutions for the problem of magnetohydrodynamic motion of an incompressible polymer fluid over an infinite rotating disk are considered. The representation of the solution used here is similar to the Von Karman self-similar solution for a viscous fluid. Examples of the numerical steady-state solutions for various values of the problem parameters are presented.

We study linear stability of a steady state for a generalization of the basic rheological Pokrovskii–Vinogradov model which describes the flows of melts and solutions of an incompressible viscoelastic polymeric medium in the nonisothermal case under the influence of a magnetic field. We prove that the corresponding linearized problem describing magnetohydrodynamic flows of polymers in an infinite plane channel has the following property: For some values of the conduction current which is given on the electrodes (i.e. at the channel boundaries), there exist solutions whose amplitude grows exponentially (in the class of functions periodic along the channel).

The paper considers stationary solutions for the motion problem of a finite layer of an incompressible polymer fluid over an infinite rotating disk. An approximate stationary solution representation is used, similar to the self-similar Karman solution for a viscous fluid. Examples of stationary numerical solutions are given for various values of the problem’s parameters.

We study the linear stability of a resting state for a generalization of the basic rheological Pokrovski–Vinogradov model for flows of solutions and melts of an incompressible viscoelastic polymeri...

We study a new rheological model describing flows of melts and solutions of incompressible viscoelastic polymeric media in an external uniform magnetic field in the presence of a temperature drop and a conduction current. We find an asymptotic representation of the spectrum of the linear problem resulting from the linearization of the initial boundary value problem in an infinite plane channel about a Poiseuille-type flow. For this Poiseuille-type flow we also find the parameter domain of linear Lyapunov's stability., Comment: arXiv admin note: substantial text overlap with arXiv:1912.13195

Equations describing the flow of an incompressible viscoelastic polymer fluid in the presence of a magnetic field are considered. The symmetrization of this system of equations is discussed.

Basing on the rheological mesoscopic Pokrovskii–Vinogradov model, the equations of nonrelativistic magneto-hydrodynamics, and the heat conduction equation with dissipative terms, we obtain a closed coupled system of nonlinear partial differential equations that describes the flow of solutions and melts of linear polymers. We take into account the rheology and induced anisotropy of a polymeric fluid flow as well as mechanical, thermal, and electromagnetic impacts. The parameters of the equations are determined by mechanical tests with up-to-date materials and devices used in additive technologies (as $$3D$$ printing). The statement is given of the problems concerning stationary polymeric fluid flows in channels with circular and elliptical cross-sections with thin inclusions (some heating elements). We show that, for certain values of parameters, the equations can have three stationary solutions of high order of smoothness. Just these smooth solutions provide the defect-free additive manufacturing. To search for them, some algorithm is used that bases on the approximations without saturation, the collocation method, and the relaxation method. Under study are the dependences of the distributions of the saturation fluid velocity and temperature on the pressure gradient in the channel.

A linear problem of small perturbations of the flow of immiscible polymer fluids in a flat channel is formulated. For this problem, the existence of solutions unlimitedly increasing with time, the presence of which means linear instability of the state of rest of a polymer fluid with a strong discontinuity, is studied.

We study a generalization of the Pokrovski–Vinogradov model for flows of solutions and melts of an incompressible viscoelastic polymeric medium to nonisothermal flows in an infinite plane channel under the influence of magnetic field. For the linearized problem (when the basic solution is an analogue of the classical Poiseuille flow for a viscous fluid described by the Navier–Stokes equations) we find a formal asymptotic representation for the eigenvalues under the growth of their modulus. We obtain a necessary condition for the asymptotic stability of the Poiseuille-type shear flow. For analysis we use new result, that generalizes Birkhoff theorem on the case, when the coefficient matrix of the eigenvalue itself has zero with multiplicity greater than one as an eigenvalue. We also get the necessary condition for Lyapunov stability of the shear Poiseuille-type flow as a result of acquired representation.

A new rheological model, an extension of the Pokrovskii-Vinogradov rheological model, describing the flows of melts and solutions of incompressible viscoelastic polymeric media in external uniform magnetic field in the presence of a temperature drop and conduction current is studied. An asymptotic representation of the linear problem spectrum resulting from the linearization of the initial boundary value problem in an infinite plane channel about a Poiseuille-type flow is obtained. For this Poiseuille-type flow the parameter domain of linear Lyapunov’s stability is determined.

We study a new rheological model describing flows of melts and solutions of incompressible viscoelastic polymeric media in an external uniform magnetic field in the presence of a temperature drop and a conduction current. We derive an asymptotic representation of the spectrum of the linear problem resulting from the linearization of the initial boundary value problem in an infinite plane channel about a Poiseuille-type flow. For this Poiseuille-type flow, we also determine the parameter domain of linear Lyapunov stability.

The stability of the flow of an incompressible polymer fluid with a strong discontinuity when the fluid can flow through the discontinuity is discussed. Particular solutions of the linearized problem that increase with time are constructed numerically.

The temporally growing particular solutions of linearized equations of an incompressible polymeric liquid with strong discontinuities were constructed.

We study the motion of an incompressible polymeric fluid with volume charge and construct stationary solutions of the electrohydrodynamical equations.

The time-dependent mathematical model describing the vortex motion of an incompressible polymeric liquid is discussed. In the steady-state case certain particular solutions are found. In the case of the steady-state pressure along the axis of cylinder, a version of deriving this model for both fixed and free boundaries is given.

In this study, we consider a rheological Pokrovski–Vinogradov model of the flows of solutions and melts of an incompressible viscoelastic polymeric medium for a flow in an infinite plane channel with perforated walls. We prove the linear Lyapunov instability of the base solution with a constant flow rate in a perturbation class, which is periodic with respect to the variable and changing along the channel wall.

A problem of an incompressible polymer fluid flow past an infinite flat wedge is considered. The flow moves parallel to the plane of symmetry of the wedge and normal to the wedge rib. It is demonstrated that two surfaces of strong discontinuities are needed for the no-slip condition to be satisfied on the wedge surface. Steady solutions of the problem are studied, and the flow is shown to be asymmetric with respect to the plane of symmetry of the wedge.

Under study are the stationary solutions of the mathematical model describing nonisothermal electroconvection of a weakly conducting polymeric liquid in the channel between two coaxial cylinders.

We study a generalization of the Pokrovski–Vinogradov model for flows of solutions and melts of an incompressible viscoelastic polymeric medium to nonisothermal flows in an infinite plane channel under the influence of magnetic field. For the linearized problem (when the basic solution is an analogue of the classical Poiseuille flow for a viscous fluid described by the Navier–Stokes equations) we find a formal asymptotic representation for the eigenvalues under the growth of their modulus. We obtain a necessary condition for the asymptotic stability of the Poiseuille-type shear flow.

We formulate the problem for an ordinary differential equation describing the velocity profile of a polymeric fluid flowing between two coaxial cylinders. We discuss the results of numerical experiments for different values of the parameters characterizing the problem.

We study the linear stability of a resting state for a generalization of the basic rheological Pokrovski-Vinogradov model for flows of solutions and melts of an incompressible viscoelastic polymeric medium to the nonisothermal case under the influence of magnetic field. We prove that the corresponding linearized problem describing magnetohydrodynamic flows of polymers in an infinite plane channel has the following property: for certain values of the conduction current which is given on the electrodes, i.e. on the channel boundaries, the problem has solutions whose amplitude grows exponentially (in the class of functions periodic along the channel).

Some nonstationary mathematical model is constructed to describe the vortex motion of an incompressible polymeric liquid. In the stationary case, several partial solutions for this model are found. The derivation of a variant of this model is given for the case when the pressure along the axis is independent of time.

The feasibility of neutral stability for shock waves in the problem of a real gas flow about a wedge is studied by means of numerical experiments. A shock adiabat for a van der Waals gas is derived; that is, a relationship is found between the wedge and shock wave angles. A vast amount of numerical data is obtained, and some of them are summarized in tables. From these data, one can distinguish wedge flow regimes with neutrally stable shock waves.

A new rheological model describing flows of incompressible viscoelastic polymer fluid is studied. Lyapunov’s linear instability of the Poiseuille flow counterpart for the Navier-Stokes equations in a plane infinite channel is proven.

Some mathematical model is formulated and its stationary solutions are obtained which describe the flows of a weakly conducting polymeric liquid in a horizontal capacitor (a channel).

We study the problem for a supersonic stationary van der Waals gas flow onto a planar infinite wedge. For the case when the basic solution is a weak shock wave, i.e. the gas velocity behind the shock front is supersonic, we prove the local-in-time well-posedness of the corresponding nonstationary problem.

We study the stability problem for a stationary supersonic flow of inviscid non-heat-conducting gas in thermodynamical equilibrium moving onto a planar infinite wedge. As it is known, this problem has two solutions: a solution with a strong shock wave (when the velocity behind the front of the shock wave is subsonic) and a solution with a weak shock wave (when the velocity behind the front of the shock wave is supersonic). We consider the case of a weak shock wave and we prove that if the Lopatinski condition for the shock wave holds (in a weak sense), then the corresponding linearized initial boundary-value problem is well-posed. We thus find a classical solution to this problem. Unlike the case when the uniform Lopatinski condition holds, additional plane waves appear. For compactly supported initial data we show that the solution of the linearized problem converges to the zero solution as time tends to infinity. Therefore, for the case of a weak shock wave and when the Lopatinski condition holds in a weak sense, these results complete the proof of the well-known Courant–Friedrichs' conjecture that the strong shock wave solution is unstable whereas the weak shock wave solution is stable.

The equations describing flows of an incompressible viscoelastic polymer liquid are studied. Stationary solutions similar to the Poiseuille and Couette solutions for the system of the Navier-Stokes equations are constructed. Stationary discontinuous solutions of the polymer liquid equation are also considered.

By means of numerical calculations, the question is answered of the feasibility of the conditions of neutral stability for shock waves in the problem of a flow of real gases past a wedge.

The planar shock wave in a viscous gas which is treated as a strong discontinuity is unstable against small perturbations. As in the case of a planar shock wave we suggest such boundary conditions that the linear initial-boundary value problem on the stability of a curv ilinear shock wave (subject to these boundary conditions) is well-posed. We also propose a new effective computational algorith m for investigation the stability. This algorithm uses the nonstationary regularizat ion, the method of lines, the stabilizat ion method, the spline function technique and the sweep method. Applying it we succeed to obtain the stationary solution of the considered boundary-value problem justifying the stability of shock wave.

The method of dissipative energy integrals is used in studying the wellposedness of the mixed problem on the flow past a wedge in the linear formulation.

We consider the thermodynamical equilibrium state flow of an inviscid non-heat-conducting gas flowing around a plane infinite wedge, and study the stationary solution to this problem–the so-called strong shock wave; the flow behind the shock front is subsonic. We find the solution to the linear analog of the original mixed problem, prove that the solution trace on the shock wave is the superposition of the direct and reflected waves, and (the main point) justify the Lyapunov asymptotical stability of the strong shock wave provided that the uniform Lopatinsky condition is fulfilled. The initial data have a compact support, and the solvability conditions occur.

We discuss the possibility of considering the shock wave in a compressible viscous heat conducting gas as a strong discontinuity on which surface the generalized Rankine-Hugoniot conditions hold. The corresponding linearized stability problem for a planar shock lacks boundary conditions; i.e., the shock wave in a viscous gas viewed as a (fictitious) strong discontinuity is like undercompressive shock waves in ideal fluids and, therefore, it is unstable against small perturbations. We propose such additional jump conditions so that the stability problem becomes well-posed and its trivial solution is asymptotically stable (by Lyapunov). The choice of additional boundary conditions is motivated by a priori information about steady-state solutions of the Navier-Stokes equations which can be calculated, for example, by the stabilization method. The established asymptotic stability of the trivial solution to the modified linearized shock front problem can allow us to justify, at least on the linearized level, the stabilization method that is often used, for example, for steady-state calculations for viscous blunt body flows.

The stability of plane magnetohydrodynamic shock waves in a collisionless strongly magnetised plasma (described by Chew, Goldberger and Low equations) is studied. Stability results are obtained for the special cases of parallel and transversal fast shock waves by the test of fulfilment of the uniform Lopatinsky condition for corresponding linear stability problems. It is shown that fast parallel shock waves are always stable. The stability domain for fast transversal shock waves is found.

The stability of plane shock waves in Magnetohydrodynamics for an ideal medium is studied. Stability results are obtained for the special case of fast parallel shock waves in a polytropic gas. Linear stability is proved for a polytropic gas with arbitrary γ. The domain of structural (nonlinear) stability, where the uniform Lopatinsky condition is fulfilled for the stability problem, is found.

The stability of shock waves is discussed for a hydrodynamic model of motion of a continuous medium with a space electrical charge. The correctness of a mixed problem obtained by linearization of the hydrodynamic model and the equations of a strong discontinuity for electrohydrodynamic shock waves is proved. As is known, this indicates stability of this type of strong discontinuity in the model of a continuous medium considered.

This chapter is devoted to the issue of stability of strong discontinuities in fluids and magnetohydrodynamics (MHD) and surveys main known results in this field. All the main points in the stability analysis are demonstrated on the example of shock waves in ideal models of gas dynamics, relativistic gas dynamics, and MHD. Ideal MHD is a good example containing, besides shock waves, different other types of strong discontinuities. Other MHD discontinuities include contact, tangential, and rotational discontinuities. The issue of stability for all these MHD discontinuities is also examined in this chapter. The main attention is concentrated on the linearized stability analysis and the issue of uniform stability. The issue of structural (nonlinear) stability is briefly discussed for gas dynamical shock waves. Open problems and future directions are discussed in the end of the chapter.