Exact Solutions to the Navier–Stokes Equations with Couple Stresses

This article discusses the possibility of using the Lin–Sidorov–Aristov class of exact solutions and its modifications to describe the flows of a fluid with microstructure (with couple stresses). The presence of couple shear stresses is a consequence of taking into account the rotational degrees of freedom for an elementary volume of a micropolar liquid. Thus, the Cauchy stress tensor is not symmetric. The article presents exact solutions for describing unidirectional (layered), shear and three-dimensional flows of a micropolar viscous incompressible fluid. New statements of boundary value problems are formulated to describe generalized classical Couette, Stokes and Poiseuille flows. These flows are created by non-uniform shear stresses and velocities. A study of isobaric shear flows of a micropolar viscous incompressible fluid is presented. Isobaric shear flows are described by an overdetermined system of nonlinear partial differential equations (system of Navier–Stokes equations and incompressibility equation). A condition for the solvability of the overdetermined system of equations is provided. A class of nontrivial solutions of an overdetermined system of partial differential equations for describing isobaric fluid flows is constructed. The exact solutions announced in this article are described by polynomials with respect to two coordinates. The coefficients of the polynomials depend on the third coordinate and time.

ON A BOUNDARY VALUE PROBLEM FOR ONE OVERDETERMINED SYSTEM IN A HALF-PLANE

This paper considers a boundary value problem for an overdetermined system of equations in a half-plane. This problem arises in particular when solving a stationary system of the two-velocity hydrodynamics with one pressure and homogeneous divergent conditions and the Dirichlet boundary conditions for two phase velocities, as well as in problems of electrodynamics. The generalized solution to a stationary system of the two-velocity hydrodynamics in the case of two-dimensional unbounded domains, for instance, in a half-plane, has a significant difference from the three-dimensional case. Namely, in the two-dimensional case for the velocities it is impossible to satisfy the pre-set conditions at infinity and the condition of boundedness at infinity is imposed. In this case, the medium is considered to be homogeneous, and the energy dissipation occurs due to the shear viscosities of the phases of the subsystems, and other effects are not discussed in this paper. The mass transfer occurs due to the mass force. With an appropriate choice of functional spaces, the existence and uniqueness of a generalized solution with an appropriate stability estimate has been proved.

On the construction of exact solutions of two-dimensional quasi-hydrodynamic system

Предложены новые методы построения точных решений квазигидродинамической системы для двумерных течений. Показано, что с любым гладким решением некоторой переопределенной системы дифференциальных уравнений в частных производных можно ассоциировать общее точное решение квазигидродинамической системы и системы Навье-Стокса. Любая собственная функция двумерного оператора Лапласа также порождает общее решение указанных систем. Приведены примеры решений как в нестационарном, так и в стационарном случае. Обсужден принцип суперпозиции векторных полей скорости жидкости для конкретных течений. New methods for constructing exact solutions of the quasi-hydrodynamic system for two-dimensional flows are proposed. It is shown that with any smooth solution of some overdetermined system of partial differential equations one can associate common exact solution of the quasi-hydrodynamic system and the Navier-Stokes system. Any eigenfunction of the two-dimensional Laplace operator also generates common solution to these systems. Examples of solutions are given in both the non-stationary and stationary cases. The principle of superposition of the fluid velocity vector fields for specific flows is discussed.

Compressive Covariance Sensing-Based Power Spectrum Estimation of Real-Valued Signals Subject to Sub-Nyquist Sampling

In this work, an estimate of the power spectrum of a real-valued wide-sense stationary autoregressive signal is computed from sub-Nyquist or compressed measurements in additive white Gaussian noise. The problem is formulated using the concepts of compressive covariance sensing and Blackman-Tukey nonparametric spectrum estimation. Only the second-order statistics of the original signal, rather than the signal itself, need to be recovered from the compressed signal. This is achieved by solving the resulting overdetermined system of equations by application of least squares, thereby circumventing the need for applying the complicated ℓ 1 -minimization otherwise required for the reconstruction of the original signal. Moreover, the signal need not be spectrally sparse. A study of the performance of the power spectral estimator is conducted taking into account the properties of the different bases of the covariance subspace needed for compressive covariance sensing, as well as different linear sparse rulers by which compression is achieved. A method is proposed to benefit from the possible computational efficiency resulting from the use of the Fourier basis of the covariance subspace without considerably affecting the spectrum estimation performance.

Exact Solutions to the Oberbeck–Boussinesq Equations for Shear Flows of a Viscous Binary Fluid with Allowance Made for the Soret Effect

The paper considers an exact solution to the equations of thermal diffusion of a viscous incompressible fluid in the Boussinesq approximation with neglect of the Dufour effect for a steady shear flow. It is shown that the reduced system of constitutive relations is nonlinear and overdetermined. A nontrivial exact solution of this system is sought in the Lin–Sidorov–Aristov class. The resulting family of exact solutions allows one to describe steady-state inhomogeneous shear flows. This class generalizes the classical Couette, Poiseuille, and Ostroumov–Birikh solutions. It is demonstrated that the system of ordinary differential equations reduced within this class retains the properties of nonlinearity and overdetermination. A theorem on solvability conditions for the overdetermined system is proved; it is reported that, when these conditions are met, the solution is unique. The overdetermined system is solvable owing to the algebraic identity relating the horizontal velocity gradients, which are linear functions of the vertical coordinate. The constructive proof of the computation of hydrodynamic fields consists in the successive integration of the polynomials, the polynomial degree being dependent on the values of the boundary parameters.

Experimental Mueller matrix polarimetry with full Poincaré beams and a CCD camera

Recently, the use of full Poincaré beams for extracting the Mueller matrix of a sample has been proposed. These beams present all possible polarization states across their transverse section. By placing a CCD camera behind a simple polarization analyzer formed by a quarter wave phase plate and a linear polarizer, a polarization map of the beam cross section can be obtained. This polarization map is modified when a sample is inserted before the polarization state analyzer. Comparison of these two polarization maps allows to obtain the Mueller matrix of the sample. An overdetermined system of linear equations (thousands of equations) can be written from this comparison. Standard mathematical methods are used to find optimum solution of this overdetermined system of equations. Some experimental results will be presented to check the performance of the proposed polarimetric method.

Complete Riemann Solvers for the Hyperbolic GPR Model of Continuum Mechanics

In this paper, complete Riemann solver of Osher-Solomon and the HLLEM Riemann solver for unified first order hyperbolic formulation of continuum mechanics, which describes both of fluid and solid dynamics, are presented. This is the first time that these types of Riemann solvers are applied to such a complex system of governing equations as the GPR model of continuum mechanics. The first order hyperbolic formulation of continuum mechanics recently proposed by Godunov S. K., Peshkov I. M. and Romenski E. I., further denoted as GPR model includes a hyperbolic formulation of heat conduction and an overdetermined system of PDE. Path-conservative schemes are essential in order to give a sense to the non-conservative terms in the weak solution framework since governing PDE system contains non-conservative products, too. New Riemann solvers are implemented and tested successfully, which means it certainly acts better than standard local Lax-Friedrichs-type or Rusanov-type Riemann solvers. Two simple computational examples are presented, but the obtained computational results clearly show that the complete Riemann solvers are less dissipative than the simple Rusanov method that was employed in previous work on the GPR model.

EXISTENCE OF 3-WEAK SOLUTIONS FOR A NEW CLASS OF AN OVERDETERMINED SYSTEM OF FRACTIONAL PARTIAL INTEGRO-DIFFERENTIAL EQUATIONS

The paper deals with the existence of three different weak solutions of [Formula: see text] -Laplacian fractional for an overdetermined nonlinear fractional partial Fredholm–Volterra integro-differential system by using variational methods combined with a critical point theorem due to Bonanno and Marano.

Isothermal layered flows of a viscous incompressible fluid with spatial acceleration in the case of three Coriolis parameters

We study the solvability of the overdetermined system of Navier–Stokes equations, supple-mented by the incompressibility equation, which is used to describe isothermal large-scale shear flows of a rotating viscous incompressible fluid. Large–scale flows are studied in a thin-layer ap-proximation (the vertical velocity of the fluid is assumed to be zero). The rotation of a continuous fluid medium is described by three Coriolis parameters. The solution of the reduced system of Na-vier–Stokes equations is constructed in the Lin–Sidorov–Aristov class. In this case, both nonzero components of the velocity vector, the pressure and temperature fields are assumed to be full linear forms of two Cartesian coordinates, and the dependence on the third Cartesian coordinate has an arbitrary form (including non-polynomial). It is shown that the nonlinear overdetermined system of Navier–Stokes equations and of the incompressibility equation in the framework of the Lin–Sidorov–Aristov class reduces to the equivalent nonlinear overdetermined system of ordinary dif-ferential equations, in which the components of the hydrodynamic fields act as unknown functions. The compatibility condition for the equations of the resulting system is derived. It is shown that, if this compatibility condition is fulfilled, the system has a unique solution, and the spatial accelera-tions in both variables (the linearity with respect to them was postulated when choosing the solution class) prove to be constant functions. These results are a generalization of similar results obtained earlier in the study of solvability in the cases of one and two Coriolis parameters.