COMBINED SHEAR FLOW IN A VISCOUS HALF-SPACE WITH A TRANSLATING BOUNDARY
D.V. Georgievskii1,2
1Moscow State University, Moscow, Russia
2Moscow Center of Fundamental and Applied Mathematics, Moscow, Russia
Keywords: unsteady shear, viscous fluid, shear stress, error function, perturbation, quadratic functional, method of integral relations, sufficient stability estimate
Abstract
We investigate the superposition of two unsteady one-dimensional shear flows of a homogeneous Newtonian viscous fluid in a half-space with a boundary that translationally moves parallel to itself. Exact solutions are expressed in terms of Stieltjes integrals in a quasi-self-similar form characteristic of diffusion-vortex processes. A linearized perturbation problem is formulated in terms of velocity and pressure variations, assuming the motion of the boundary plane remains unperturbed. This problem is analyzed using the method of integral relations. The method employs quadratic functionals and yields integral estimates sufficient to ensure that perturbations do not grow over an infinite time interval. The case in which both boundary velocity components are monotonically nondecreasing functions of time is examined analytically.
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