scholarly journals Exact Solutions to the Navier–Stokes Equations with Couple Stresses

Symmetry ◽  
2021 ◽  
Vol 13 (8) ◽  
pp. 1355
Author(s):  
Evgenii S. Baranovskii ◽  
Natalya V. Burmasheva ◽  
Evgenii Yu. Prosviryakov
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Incompressible Fluid ◽  
Exact Solutions ◽  
Stokes Equations ◽  
Viscous Incompressible Fluid ◽  
Navier Stokes ◽  
Shear Stresses ◽  
Overdetermined System ◽  
Navier Stokes Equations

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.

Uniqueness of the exact solutions of the Navier—Stokes equations having null nonlinearity

2006 ◽  
Vol 136 (6) ◽  
pp. 1303-1315 ◽  
Author(s):  
Sun-Chul Kim ◽  
Hisashi Okamoto
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Exact Solutions ◽  
Euler Equations ◽  
Stokes Equations ◽  
Elliptic Partial Differential Equations ◽  
Navier Stokes ◽  
Classical Solutions ◽  
Overdetermined System ◽  
Navier Stokes Equations

We consider an overdetermined system of elliptic partial differential equations arising in the Navier–Stokes equations. This analysis enables us to prove that the well-known classical solutions such as Couette flows and others are the only solutions that satisfy both the stationary Navier–Stokes and Euler equations.

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

2020 ◽  
pp. 29-46
Author(s):  
N. V. Burmasheva ◽  
◽  
E. Yu. Prosviryakov ◽  
Keyword(s):  
Incompressible Fluid ◽  
Compatibility Condition ◽  
Large Scale ◽  
Stokes Equations ◽  
Viscous Incompressible Fluid ◽  
Navier Stokes ◽  
Arbitrary Form ◽  
Overdetermined System ◽  
Navier Stokes Equations ◽  
Fluid Medium

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.

Exact Solutions of the Unsteady Navier-Stokes Equations

1989 ◽  
Vol 42 (11S) ◽  
pp. S269-S282 ◽  
Author(s):  
C. Y. Wang
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Exact Solutions ◽  
Stokes Equations ◽  
Nonlinear Partial Differential Equations ◽  
Similarity Solutions ◽  
Navier Stokes ◽  
Partial Differential ◽  
Navier Stokes Equations

The unsteady Navier-Stokes equations are a set of nonlinear partial differential equations with very few exact solutions. This paper attempts to classify and review the existing unsteady exact solutions. There are three main categories: parallel, concentric and related solutions, Beltrami and related solutions, and similarity solutions. Physically significant examples are emphasized.

scholarly journals Stability Analysis of Symmetrical Rotors Partially Filled with a Viscous Incompressible Fluid

2001 ◽  
Vol 7 (5) ◽  
pp. 301-310 ◽  
Author(s):  
Zhu Changsheng
Keyword(s):  
Stability Analysis ◽  
Incompressible Fluid ◽  
Stokes Equations ◽  
Viscous Incompressible Fluid ◽  
Damping Ratio ◽  
Navier Stokes ◽  
Fluid Viscosity ◽  
Navier Stokes Equations ◽  
Rotor Systems ◽  
The Stability

On the basis of the linearized fluid forces acting on the rotor obtained directly by using the two-dimensional Navier-Stokes equations, the stability of symmetrical rotors with a cylindrical chamber partially filled with a viscous incompressible fluid is investigated in this paper. The effects of the parameters of rotor system, such as external damping ratio, fluid fill ratio, Reynolds number and mass ratio, on the unstable regions are analyzed. It is shown that for the stability analysis of fluid filled rotor systems with external damping, the effect of the fluid viscosity on the stability should be considered. When the fluid viscosity is included, the adding external damping will make the system more stable and two unstable regions may exist even if rotors are isotropic in some casIs.

Time delay and Lagrangian approximation for Navier–Stokes flow

Analysis ◽  
2015 ◽  
Vol 35 (4) ◽  
Author(s):  
Nazgul Asanalieva ◽  
Carolin Heutling ◽  
Werner Varnhorn
Keyword(s):  
Fluid Flow ◽  
Time Delay ◽  
Incompressible Fluid ◽  
Stokes Flow ◽  
Stokes Equations ◽  
Viscous Incompressible Fluid ◽  
Navier Stokes ◽  
Incompressible Fluid Flow ◽  
Navier Stokes Equations

AbstractWe consider the nonstationary nonlinear Navier–Stokes equations describing the motion of a viscous incompressible fluid flow for

Flow of a viscous incompressible fluid after a sudden point impulse near a wall

2009 ◽  
Vol 629 ◽  
pp. 425-443 ◽  
Author(s):  
B. U. FELDERHOF
Keyword(s):  
Boundary Condition ◽  
Incompressible Fluid ◽  
Stokes Equations ◽  
Viscous Incompressible Fluid ◽  
Slip Boundary Condition ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Slip Boundary ◽  
Short Time ◽  
Short Times

The flow of a viscous incompressible fluid generated by a sudden impulse near a wall with no-slip boundary condition is studied on the basis of the linearized Navier–Stokes equations. It turns out that the flow differs significantly from that for the perfect slip boundary condition, except far from the wall and at short times. At short time the flow is irrotational and can be described by a potential which varies with the square root of time. Correspondingly the pressure disturbance is quite large at short times. It shows an oscillation at later times if the impulse is directed parallel to the wall and decays monotonically for impulse perpendicular to the wall.

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