PSI - Issue 40

Larisa S. Goruleva et al. / Procedia Structural Integrity 40 (2022) 171–179 Larisa S. Goruleva, Evgeniy Yu. Prosviryakov / Structural Integrity Procedia 00 (2019) 000 – 000

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necessary to have a stock of exact solutions. Over the entire time of analytical and numerical integration of the incompressible fluid motion equations, hydrodynamists are discussing the necessity of finding new exact solutions (Ershkov et al., 2021; Aristov et al., 2009; Drazin e al., 2006; Pukhnachev et al., 2006). After publication of a new exact solution, it is thoroughly studied by the scientific community. It was noted in (Broman et al., 2010) that mathematicians, mechanical scientists, and physicists understand and interpret an exact solution of the Navier – Stokes equations differently. At first, the reviewers and authors of other papers state shortcomings, e.g. narrow applicability, too severe mathematical restrictions in the formalization of the physical process, neglect of some terms, and a number of other objections (Ershkov et al., 2021). Some time passes, and exact solutions find their applications. They are applicable to testing computer programs and codes, they can act as test problems to verify numerical methods, and above all they are used for the description of new physical mechanisms to be implemented in engineering (Ershkov et al., 2021; Shtern, 2012; Shtern, 2018). The following applications of exact solutions of the Navier – Stokes equations for incompressible media can be mentioned: the Poiseuille formula (Ershkov et al., 2021; Aristov et al., 2009; Drazin e al., 2006; Pukhnachev et al., 2006), which is now indispensable for echocardiography (cardiac ultrasound); the Taylor – Couette exact solution, which grounded the use of the Couette viscometer measuring dynamic (kinematic) fluid viscosity (Ershkov et al., 2021; Aristov et al., 2009; Drazin e al., 2006; Pukhnachev et al., 2006) ; the von Kármán solution (Ershkov et al., 2021; Aristov et al., 2009; Drazin e al., 2006; Pukhnachev et al., 2006), which describes the Levich rotating disk electrode (Ershkov et al., 2021; Aristov et al., 2009; Drazin e al., 2006; Pukhnachev et al., 2006). As the scientific community members become accustomed to a new exact solution, they begin to feel that the obtained formulas are final and that they can no longer be modified or generalized. Therefore, there often arises a discussion on whether it is expedient to find exact solutions to the Navier – Stokes equations. It is always preferable to have analytical formulas and to use them in applications. In this case, solution correctness can always be verified by direct substitution of the solution into the original equations. On the other hand, the computations are labor and time consuming. Besides, there is no theorem ensuring the existence of a solution, and the currently available regular methods of constructing a solution to the Navier – Stokes equations are rather poorly developed and sooner heuristic. This paper generalizes two classical exact solutions. The Couette exact solution describing steady-state inhomogeneous fluid flow, two-dimensional in velocities and three-dimensional in coordinates, was first discussed in (Aristov et al., 2014). Other boundary value and initial boundary value problems for isobaric stratified and shear flows were studied in (Aristov et al., 2015; Prosviryakov et al., 2020; Zubarev et al., 2019; Prosviryakov, 2019; Prosviryakov, 2017; Prosviryakov et al., 2018). The published exact solutions were characterized by the presence of the vertical component of the vortex vector without prerotation. In other words, the stratified flow becomes shear type without application of the Coriolis field. A problem statement for the description of gradient flows (Poiseuille fluid motion) was discussed in (Prosviryakov, 2016). It was shown in (Aristov et al., 2014; Aristov et al., 2015; Prosviryakov, 2016) that the discussed boundary value problems are applicable to modeling equatorial countercurrents and the motion of incompressible media in chemical engineering equipment and in aviation technology. The classical generalization of the Couette and Poiseuille flows in a horizontal infinite fluid layer was discussed in (Aristov et al., 2014; Aristov et al., 2015; Prosviryakov et al., 2020; Prosviryakov et al., 2018; Prosviryakov, 2016; Aleksenko et al., 2018), the lower boundary being assumed to be fixed. Nevertheless, for applications it is important to study fluid motion with the moving lower boundary of the fluid layer. This paper partially fills the gap in such research for inhomogeneous flows of a viscous incompressible fluid.

Nomenclature x , y , z

Cartesian coordinates , x y V V velocity vector components P hydrostatic pressure g free fall acceleration U , 1 U , V

exact solution components

layer thickness

h