PSI - Issue 40

Natalya V. Burmasheva et al. / Procedia Structural Integrity 40 (2022) 82–89 Natalya V. Burmasheva, Evgeniy Yu. Prosviryakov / Structural Integrity Procedia 00 (2019) 000 – 000

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2

other words, for such flows, the third projection of the velocity vector is assumed to be zero. In terms of physics, we speak, e.g., of the flows of viscous fluids in long pipes (with the diameter much smaller than the length) or of the flows of such fluids in narrow slits with a slowly varying gap width. In this paper, the shear isobaric steady flow of viscous fluids in an extended horizontal layer is modeled on the basis of a class of exact solutions to the Navier-Stokes equations. It is assumed that one of the surfaces bounding the layer under consideration is hydrophobic, i.e. the condition of fluid slip is implemented at this solid boundary. For a given boundary condition, exact solutions are obtained that describe the flow corresponding to it; the properties of this solution are analyzed depending on the value of the slip length, which characterizes the degree of roughness of the hydrophobic surface.

Nomenclature V

velocity field

Hamilton operator

 

Laplace operator i , j , k unit vectors of axes Ox , Oy , Oz U , u , V exact solution components  slip length N outer normal h layer thickness  kinematic viscosity coefficient  dynamic viscosity coefficient p hydrostatic pressure  stress tensor

2. Introduction We consider the steady isobaric isothermal flow of a viscous incompressible fluid in an extended horizontal layer of a given thickness h . The characteristics of the fluid (kinematic viscosity  and dynamic viscosity  ) are assumed to be known. The equations of the model describing this motion are as follows: ( ) ,  =   V V V , (1)

0   V = .

(2)

Here,

( ) , , x y z V V V V =

is a velocity field;

  

  







( ) , V

x y z V V V x y + +  

 =

z



is a convective derivative;