Abstract
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.
Highlights
The overwhelming majority of studies dealing with fluid flows are based on the application of the conventional Navier–Stokes Equations supplemented by the incompressibility condition [1,2]
The Navier–Stokes Equations are derived from the postulates of the Newtonian mechanics of continua, each particle of which is viewed as a material point
In order to begin constructing exact solutions of system (2)–(5), we model unidirectional flows, i.e., we consider fluid flows with the velocity defined as follows: V = (Vx ( x, y, z, t), 0, 0)
Summary
The overwhelming majority of studies dealing with fluid flows are based on the application of the conventional Navier–Stokes Equations supplemented by the incompressibility condition [1,2]. The stress tensor becomes asymmetric in this case since additional stresses occur due to taking into account the deformation properties of the vortex velocities of elementary fluid volumes [9,10,11] These media are currently termed micropolar [9,10,11,12,13,14]. The Lin–Sidorov–Aristov family found in [18,19,20] is chosen as a basis for constructing exact solutions This class prescribes functional variable separation for the velocity field described by linear forms with respect to two (horizontal or longitudinal) coordinates x and y. In view of the relevance of the research and on account of the insufficient completeness of the exact integration of micropolar fluid motion equations, we construct new exact solutions to the generalized Navier–Stokes Equations for unidirectional, shearing, and three-dimensional flows
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