Small Perturbations of the Diffusion-Vortex Flows of a Newtonian Liquid in a Half-Plane
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l Perturbations of the Diffusion-Vortex Flows of a Newtonian Liquid in a Half-Plane D. V. Georgievskiia,b,* a Moscow
b
State University, Moscow, 119991 Russia Ishlinsky Institute of Problems of Mechanics, Russian Academy of Sciences, Moscow, 119526 Russia * e-mail: [email protected] Received November 1, 2019; revised January 10, 2020; accepted January 22, 2020
Abstract—In this paper, we study plane diffusion-vortex flows in the half-plane of a viscous incompressible fluid controlled by the boundary motion. At the boundary, either the longitudinal velocity or shear stress can be specified as functions of time. Classical self-similar solutions take place if these functions coincide with the Heaviside function. A linearized problem for relatively small initial perturbations superimposed on the kinematics in the entire half-plane is formulated. It consists of one biparabolic equation with variable coefficients with respect to the complex-valued current function and four homogeneous boundary conditions. Exponential estimates, which are estimates of the attenuation for some values of the parameters while they indicate the nature of the growth of perturbations for others, are derived using the method of integral relations. Some characteristic cases of specifying the boundary velocity or tangential stress on it are analyzed. Keywords: vortex-layer diffusion, shear flow, shear stress, incompressibility, viscosity, small perturbations, quadratic functional, attenuation, exponential estimate DOI: 10.1134/S0015462820070046
Known in continuum mechanics, the self-similar and quasi-self-similar diffusion-vortex flows in the half-plane with a specified boundary motion serve as a good approximation in modeling the boundary control of processes occurring in the range of a medium’s motion inaccessible to the application of forces. To find the parameters of the boundary motion law in which the small initial perturbations superimposed on one-dimensional unsteady diffusion-vortex flows do not increase (or exponentially decay) with time by some measure throughout the half-plane is an important problem. 1. VORTEX-LAYER DIFFUSION AT A SPECIFIED VELOCITY OF THE HALF-PLANE BOUNDARY Let the half-plane Ω = {−∞ < x1 < ∞, x2 > 0} with a boundary Σ = {−∞ < x1 < ∞ , x2 = 0} be occupied by a homogeneous incompressible viscous fluid with density ρ and dynamic viscosity μ. At t < 0, the medium is at rest, and at t = 0, boundary Σ begins to move along itself at the specified speed V(t) that is piecewise continuous in time. The presence of mass force F with components F1 = 0 and F2 = F ( x2, t ) is possible. Next, we present the description in a dimensionless form, including a triple of quantities {ρ, μ,V0} in the dimensional basis, where V0 is the characteristic value of the function V(t). Thus, the classical diffusion of the discontinuity occurs if V(t) coincides with the Heaviside function h(t). Hereinafter, V(t) is the dimensionless velocity of boundary Σ in the mentioned basis. A nonstationary one-dimensional shift in a region Ω at t > 0
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