Inhomogeneous Poiseuille flow

The paper presents an investigation of the isothermal steady flow of a viscous incompressible fluid in an extended flat layer using hydrodynamic equations. The bottom of the layer under consideration is limited by a stationary solid hydrophilic surface. At the upper boundary of the layer, the pressure field, which is inhomogeneous in both horizontal coordinates, and the velocity field are specified. These boundary conditions allow one to generalize the classical Poiseuille flow. The exact solution, satisfying the set boundary value problem, is described by a series of polynomials of different orders. The highest (fifth) degree of the polynomials corresponds to a homogeneous component of the horizontal velocity. Here, the pressure field depends only on the horizontal coordinates; the dependence is linear. The detailed analysis of the velocity field is carried out. The obtained results confirm that the determined exact solution can describe multiple stratification of the velocity field and the corresponding field of tangent stresses. The analysis of spectral properties of the velocity field is performed for a general case without specifying the values of physical constants that unambiguously identify the studied fluid. Therefore, the presented results are applicable to viscous fluids of various nature.