PSI - Issue 40

Natalya V. Burmasheva et al. / Procedia Structural Integrity 40 (2022) 75–81 Natalya V. Burmasheva at al. / Structural Integrity Procedia 00 (2022) 000 – 000

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1. Introduction Stratified viscous fluids are a working model representation for describing the flows of viscous incompressible fluids stratified by some physical parameters. An example of such fluids is seawater, the density and the salinity of which change with depth. With a decrease in density, the property of the instability of a stratified fluid is more and more manifested, i.e. mixing begins. As a result, intense vertical transfer of the media, heat, and momentum is induced. A stable stratified fluid, on the contrary, causes a decrease in the vertical exchange. This and many other effects associated with stratified fluids are taken into account when these fluids are used in various technical devices. For example, they are used in the oil industry, when a mixture of water and oil is pumped through pipes over long distances. In this case, as a rule, the boundary between two adjacent fluid layers is deformed. However, in some situations (at low flow velocities, where stabilizing gravity is dominant due to the finite difference in the layer densities), this boundary can be considered as a plane. The article discusses the selection of boundary conditions at the common boundary between the two adjacent layers in a two-layer fluid.

Nomenclature V ( i )

velocity field at the i -th layer dynamic viscosity coefficient



dynamic viscosity coefficient for the i -th layer

 i

density



density at the i -th layer Hamilton operator

 i

 

Laplace operator

i , j , k

unit vectors of axes Ox , Oy , Oz

U ( i ) , u ( i ) , V ( i )

exact solution components for the i -th layer

thickness of the i -th layer stress tensor at the i -th layer

h i

 ( i )

hydrostatic pressure

p

2. Formulation of the problem For simplicity, we consider the isothermal flow of a viscous two-layer fluid in an extended horizontal layer. We will assume that the boundary between the two layers is plane and non-deformable and that the fluid is stratified in terms of density and viscosity, i.e. the density and viscosity of the two-layer fluid are assumed to be piecewise constant functions. The thicknesses h 1 , h 2 of the two layers are generally also different. Despite the seeming simplicity of the described model representation, the results of analyzing the properties of such a fluid can be extended to a larger number of layers and to more physically complex flows (convective, thermal diffusion, etc.). Thus the system of Navier-Stokes equations in the case under study will take the form         i i i i i ,      V V V , (1)

  i   V = . 0

(2)

Here,





  i

  i

  i

  i

, y z V V V V = , x