PSI - Issue 65

E. Yu. Prosviryakov et al. / Procedia Structural Integrity 65 (2024) 177–184 E. Yu. Prosviryakov, O. A. Ledyankina, L. S. Goruleva / Structural Integrity Procedia 00 (2024) 000–000

178

2

al.(2024), Ershkov et al.(2023)]. An important special case for the exact integration of the equations of motion is creeping flows [Privalova et al. (2018)]. In this case, the Reynolds number takes a value close to zero, which allows integrating the Stokes equations instead of the Navier-Stokes equations. For the exact integration of the Navier Stokes equations, the Lin-Sidorov-Aristov ansatz [Drazin et al. (2006)), Aristov et al. (2009), Ershkov et al. (2021), Prosviryakov et al. (2024), Ershkov et al. (2023), Privalova et al. (2018)] is very often used. Recently, exact solutions have been proposed for fluid flows with energy dissipation [Goruleva et al. (2022)], for multilayer fluids [Burmasheva et al. (2022), Burmasheva et al. (2021)]. The article [Goruleva et al. (2023)] summarizes numerous studies on finding exact solutions for the equations of magnetohydrodynamics. In [Goruleva et al. (2023)] the exact solutions for three-dimensional Kolmogorov flows with the Rayleigh function are considered. When constructing the exact solutions in [Goruleva et al. (2022), Burmasheva et al.(2022), Burmasheva et al. (2021) Goruleva et al. (2023) Goruleva et al. (2023)], the Lin-Sidorov-Aristov class was used to describe nonlinear motions and Stokes flows. This article considers in detail an algorithm for constructing exact solutions for creeping flows for problems of magnetohydrodynamics of multilayer fluids. Stokes flows in [Prosviryakov et al. (2024)] have not been described previously with the Rayleigh function. When constructing the exact solutions, the methodology presented in the articles and reviews [Ershkov et al. (2021), Ershkov et al. (2023), Goruleva et al. (2022), Burmasheva et al. (2022), Burmasheva et al. (2021) Goruleva et al. (2023)] was used.

2. Equations of motion

To describe multilayer incompressible fluids or gases, we assume that the continuous flow consists of layers. In other words, a continuous medium is stratified discretely by density or by dissipative coefficients of viscosity or thermal conductivity (thermal diffusivity). In this case, the unsteady flow of a continuous medium, taking into account internal heat release (Rayleigh’s dissipative function), is described by a vector system of partial differential equations for the layer number k :

( ) k

 V



( ) k    ( ) k P

( ) k

V

t

,

( ) k

T



( ) k   

( ) k T Q

( ) k

t



,

( ) k  V

0



(1)

The system of equations in (1) consists of vector and scalar equations. System (1) is formed of the Navier-Stokes equation, the Fourier equation (energy equation), and the continuity equation. Further, the study and finding of exact solutions will be carried out in a three-dimensional rectangular Cartesian coordinate system. In the equations of motion of a multilayer fluid (1), the following notations are introduced:                                   1 2 3 1 1 2 3 2 1 2 3 3 1 2 3 , , , , , , , , , , , , , , V k k k k k k k k k k k k txxx V txxx V txxx V txxx  ( ) k  is the thermal diffusivity coefficient;  is the Hamiltonian,  is the Laplacian for the Cartesian rectangular coordinate system. Differentials are defined for each layer. We take into account that the fluid layers do not mix and have a constant thickness k h In the thermal conductivity equation (1), the ( ) k Q function determines the conversion of mechanical energy into thermal energy. The Rayleigh dissipative function is determined by the formula V is the velocity vector 1, k n  ; ( ) k P is pressure divided by the constant density of the liquid layer ( ) k  ; ( ) k  is kinematic (molecular) viscosity;