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The paper presents a family of exact solutions to the Navier-Stokes equation system used to describe inhomogeneous unidirectional flows of a viscous fluid taking into account couple stresses. Despite the presence of only one non-zero component of the velocity vector, this component depends on time and two spatial coordinates. In view of the incompressibility equation, which is a special case of the mass conservation law, there is no dependence on the third spatial coordinate. The resulting redefined system of equations is considered in a non-stationary formulation. The construction of a family of exact solutions for the resulting redefined equation system begins with the analysis of the homogeneous Couette-type solution as the simplest in this class. Further, the structure of the solution gradually becomes more complicated, i.e. the profile of the only non-zero component of the velocity vector is represented as a polynomial depending on one variable (horizontal coordinate). The polynomial coefficients functionally depend on the second (vertical) coordinate and time. It is shown that, due to the strong nonlinearity and heterogeneity of the equation under study, the sum of its individual solutions is not a solution. It is also shown that, in the linearly independent basis of the power functions of the horizontal coordinate, which determine the above-mentioned polynomial, the equation in question decomposes into a chain of the simplest homogeneous and inhomogeneous parabolic partial differential equations. These equations are integrated sequentially, the order of integration being described separately. The results reported in this study extend the family of previously presented exact solutions to describing unidirectional unsteady flows.
Burmasheva et al. (Sat,) studied this question.