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

173

3

kinematic (molecular) viscosity



2. Boundary value problem statement Consider the steady-state shear flows of a viscous incompressible fluid in an infinite horizontal thin layer. The medium motion is described by three angular moment equations (the Navier – Stokes equations) and the incompressibility equation (Prosviryakov et al., 2020; Prosviryakov, 2019; Prosviryakov, 2016):

2 2 V V V + +      , 2 2 2 2 x x x x y z        2 V V V + +      , 2 2 2 2 y y y x y z          2

x y V V  +



P



x V V x 

= − + 

x

y

x





y y V V  +



P



y

x V V x 

= − + 

y

y





y x V V x y    

,

0

+ =

P g z =



.

(1)



Here, x , y , z are the Cartesian coordinates, ( ) , , y V x y z are the velocity vector components,  is the kinematic (molecular) viscosity of the fluid, g is free fall acceleration, P is pressure divided by constant density. The equation system (1) is overdetermined since, for calculating three unknown functions, namely two velocity vector components x V and y V and pressure P , there are four equations. For the integration of system (1) for different cases, classes of exact solutions were proposed which enable the “excess” equation to be satisfied (Ershkov et al., 2021; Aristov et al., 2009; Drazin e al., 2006; Pukhnachev et al., 2006; Prosviryakov, 2019). Besides, transformations allowing the exact solutions of the Navier – Stokes equations (1) to be reproduced (replicated) were found in (Ershkov et al., 2021; Aristov et al., 2009; Drazin e al., 2006; Pukhnachev et al., 2006; Prosviryakov, 2019). Fluid motion in an infinite layer is induced by the displacement of the lower boundary 0 z = according to the law: ( ) , , 0 x V x y A By = + , ( ) , , 0 y V x y C = . (2) Formulas (2) specify the inhomogeneous (non-translational) motion of the layer boundary. In other words, the simultaneous setting of the translational and rotational motions caused by various inhomogeneities is implemented. Assume that the velocity on the free rigid upper boundary z h = of the layer is constant. Without limiting the generality of the reasoning, we further assume the velocity on this boundary to be zero: ( ) , , 0 x V x y h = , ( ) , , 0 y V x y h = . (3) In physical terms this means that, in fact, the fluid is contained between two infinitely long plane rigid plates. Let us stipulate at once that, strictly speaking, to set pressure on the free boundary z h = , we use solid lid approximation with specified zero velocities (Aristov et al., 2014; Prosviryakov et al., 2018). The kinematic and dynamic boundary condition is identically met in this case, with the fluid pressure specified on the free boundary as: ( ) 1 2 , , P x y h S S x S y = + + . (4) ) , , x V x y z and (