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We explored here the case of three-dimensional non-stationary flows of helical type for the incompressible couple stress fluid with given Bernoulli-function in the whole space (the Cauchy problem). In our presentation, the case of non-stationary helical flows with constant coefficient of proportionality α between velocity and the curl field of flow is investigated. In the given analysis for this given type of couple stress fluid flows, an absolutely novel class of exact solutions in theoretical hydrodynamics is illuminated. Conditions for the existence of the exact solution for the aforementioned type of flows were obtained, for which non-stationary helical flow with invariant Bernoulli-function satisfying to the Laplace equation was considered. The spatial and time-dependent parts of the pressure field of the fluid flow should be determined via Bernoulli-function if components of the velocity of the flow are already obtained. Analytical and numerical findings are outlined, including outstanding graphical presentations of various types of constructed solutions, in order to elucidate dynamic snapshots that show the timely development of the topological behavior of said solutions. [ABSTRACT FROM AUTHOR]

The present review analyzes classes of exact solutions for the convection and thermal diffusion equations in the Boussinesq approximation. The exact integration of the Oberbeck–Boussinesq equations for convection and thermal diffusion is more difficult than for the Navier–Stokes equations. It has been shown that the exact integration of the thermal diffusion equations is carried out in the Lin–Sidorov–Aristov class. This class of exact solutions is a generalization of the Ostroumov–Birikh family of exact solutions. The use of the class of exact solutions by Lin–Sidorov–Aristov makes it possible to take into account not only the inhomogeneity of the pressure field, the temperature field and the concentration field, but also the inhomogeneous velocity field. The present review shows that there is a class of exact solutions for describing the flows of incompressible fluids, taking into account the Soret and Dufour cross effects. Accurate solutions are important for modeling and simulating natural, technical and technological processes. They make it possible to find new physical mechanisms of momentum transfer for the design of new types of equipment. [ABSTRACT FROM AUTHOR]

The flow of a viscous incompressible fluid outflowing from a uniformly moving point source is considered. An exact solution to the problem is found in the way that the velocity decreases inversely with the radial coordinate. It is shown that a spherical volume of fluid is carried away by the source, the radius of which is inversely proportional with respect to the velocity of motion. In this case, a cylindrical discontinuity arises in the region of forming a wake behind the body, the dimensions of which are determined by the magnitude of the external pressure and do not depend on the velocity of the source. The obtained solutions are governed by hydrodynamical fields of flows which can be recognized as special invariants at symmetry reduction. [ABSTRACT FROM AUTHOR]

We explore the ansatz of derivation of non-stationary creeping flows for the Navier–Stokes equations in the case of incompressible flow of Newtonian fluids. In general case, such a solution should be obtained from the mixed system of 2 Riccati ordinary differential equations (in regard to time-parameter t ). But we have found an elegant way to simplify it to the proper analytical presentation of creeping flow as a solution of only one Riccati ordinary differential equation, which nevertheless has no analytical solution in general case. It was a motivation to obtain the elegant approximate solution for creeping flow: - curl-free velocity field consists of the constant component U in Ox -direction, non-stationary (but negligible) component V in Oy -direction, the constant (but negligible) component W in Oz -direction. Additional component of flow velocity field (with variable curl) consists of the appropriate harmonic functions , not depending on time-parameter t . Also the suggested ansatz is illuminated by showing (on the example of creeping flow past a sphere) how it can be used to find specific solutions of current technical interest, or that it can be used to some significant advantage over some of the many existing analytical, semi-analytical, and/or numerical techniques for solution of creeping flows. [ABSTRACT FROM AUTHOR]

In fluid mechanics, there is an essential deficiency of the analytical solutions of Navier–Stokes equations for 3D case of non-stationary flow. We explore the ansatz of derivation of non-stationary solution for the Navier–Stokes equations in the case of incompressible flow, which was suggested earlier. In general case, such a solution should be obtained from the mixed system of 2 Riccati ordinary differential equations (in regard to the time-parameter t ). But we find an elegant way to simplify it to the proper analytical presentation via exact solutions. The results are substantiated with illustrative examples, including numerical computations and appropriate figures. [ABSTRACT FROM AUTHOR]

In fluid mechanics, a lot of authors have been executing their researches to obtain the analytical solutions of Navier–Stokes equations, even for 3D case of compressible gas flow. But there is an essential deficiency of non-stationary solutions indeed. In our derivation, we explore the case of non-stationary helical flow of the Navier–Stokes equations for incompressible fluids at any given initial conditions for velocity fields ( it means an open choice for the space part of a solution ). Such a non-stationary helical flow is proved to be decreasing exponentially in regard to the time-parameter, the extent of time-dependent exponential component is given by the coefficient of kinematic viscosity, multiplied by the square of the coefficient of proportionality between the vorticity and velocity field. [ABSTRACT FROM AUTHOR]