PSI - Issue 65

E.Yu. Prosviryakov et al. / Procedia Structural Integrity 65 (2024) 185–190 E.Yu. Prosviryakov, O.A. Ledyankina, L.S. Goruleva / Structural Integrity Procedia 00 (2024) 000–000

186

2

convection with complex topology [Burmasheva et al. (2020), Aristov et al. (2016), Aristov et al. (2015), Prosviryakov et al. (2018)]. Of particular interest are the study of the shear flow of a vertically vortexed liquid [Burmasheva et al. (2020)], the study of dynamic equilibria [Prosviryakov et al. (2018)], and the modeling of steady and unsteady flows of the Ostroumov-Birikh type [Aristov et al. (2016), Aristov et al. (2015)]. The Lin-Sidorov Aristov family of exact solutions is used for describing flows with a dissipative Rayleigh function [Goruleva et al. (2022)] and for mathematical modeling of large-scale flows with a free (unknown) boundary [Ershkov et al. (2023)]. The scientific papers [Zubarev et al. (2019), Prosviryakov et al. (2019), Privalova et al. (2022), Prosviryakov et al. (2023)] provide an extension of the exact solutions announced in [Aristov et al. (2016), Burmasheva et al. (2020), Aristov et al. (2016), Aristov et al. (2015), Prosviryakov et al. (2018)]. This paper considers exact solutions for Stokes-type slow (creeping) convective flows of inhomogeneous liquids (solutions). The announced exact solutions are important for modeling inhomogeneous convective flows in a force field. The system of constitutive relations used to describe the convective flow of a viscous incompressible fluid includes three equations, namely the Navier-Stokes equation in the Boussinesq approximation, the concentration equation, and the incompressibility equation:

V



C V g F ,

P     

(1)

t



C



d C

 

,

(2)

t



0   V .

(3)

 



Here, 

, , , , , x y z t x y z V V V  V is the velocity vector and its projection on the axes of the Cartesian coordinate system; P is the deviation of pressure from hydrostatic, related to the constant average density of the fluid  ; g is the free-fall acceleration vector;  is the diffusion coefficient of volumetric expansion of the fluid; C is a deviation from the average concentration;     , , , , , x y z x y z t F F F  F is the vector of the density of mass forces;  , d are the coefficients of kinematic viscosity and the coefficient of diffusion of a dissolved substance in a liquid, respectively.

2. Class of solutions quadratic in two coordinates

Let us generalize the well-known Lin–Sidorov–Aristov family [Zubarev et al. (2019), Prosviryakov et al. (2019), Privalova et al. (2022), Prosviryakov et al. (2023)] and define a new class of exact solutions. The family under consideration includes velocity components that linearly depend on the horizontal (longitudinal) coordinates. In this case, the functions of pressure and concentration are presented in the form of a quadratic dependence on the same coordinates:

2 y V U UxUyU UxyU       , 2 x

x

0

1

2

3

4

5

2

2

2 y V V VxVyV VxyV       , 2 x

y

0 1

2

3

4

5

2

2

0 2 z V W Wx W y    , 1