Nonstationary Boundary Layer of a Fluid with the Ladyzhenskaya Rheological Law
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Journal of Mathematical Sciences, Vol. 249, No. 6, September, 2020
NONSTATIONARY BOUNDARY LAYER OF A FLUID WITH THE LADYZHENSKAYA RHEOLOGICAL LAW R. R. Bulatova Lomonosov Moscow State University Moscow 119991, Russia [email protected]
V. N. Samokhin Moscow Politechnic University 2A Pryanishnikova St., Moscow 127550, Russia [email protected]
G. A. Chechkin
∗
Lomonosov Moscow State University Institute of Mathematics with Computing Centre 112, Chernyshevskogo St., Ufa 450008, Russia [email protected]
UDC 517.946
The unique solvability of the problem for the equations governing a nonstationary boundary layer of a viscous fluid subject to the Ladyzhenskaya rheological law is studied in the literature for the first time. We prove the existence and uniqueness of a solution to the problem. Bibliography: 7 titles. We study the boundary layer appearing in the case of a nonstationary flow of a viscous incompressible fluid near a symmetric body subject to the Ladyzhenskaya rheological law. Using the Crocco transformation, we can reduce the nonstationary system of boundary layer equations to one quasilinear equation in some neighborhood of critical points. We note that the stationary case was studied in [1]. The unique solvability of the problem is proved globally in time. We refer to [2] for the results on the well-posedness of boundary value and initial-boundary value problems for systems of Prandtl equations and, for example, to [3]==[5] for the generalized system of Prandtl type equations of boundary layer for a fluid with the modified rheological law due to Ladyzhenskaya.
1
The Crocco Transformation
In the case of a two-dimensional nonstationary flow, the modified system of boundary layer equations has the form ∗
To whom the correspondence should be addressed.
Translated from Problemy Matematicheskogo Analiza 103, 2020, pp. 31-42. c 2020 Springer Science+Business Media, LLC 1072-3374/20/2496-0850
850
ν(1 + 3d(uy )2 )uyy − ut − uux − vuy = −Ut − U Ux ,
(1.1)
ux + vy = 0. Here, ν and d are constants depending on the fluid properties, the fluid density ρ is assumed to be equal to 1, U (t, x) is a given function connected with the pressure p(t, x) by the relation −px = Ut + U Ux . The system of equations (1.1) is considered in the domain D = {0 < t < ∞, 0 < x < X, 0 < y < ∞} with initial and boundary conditions u(0, x, y) = u0 (x, y), v(t, x, 0) = v0 (t, x),
u(t, 0, y) = 0,
u(t, x, 0) = 0,
u(t, x, y) → U (t, x),
y → ∞,
(1.2)
where U (t, 0) = 0, Ux (t, 0) > 0, U (t, x) > 0 for x > 0 and the function v0 (t, x) is given. The conditions U (t, 0) = 0 and u(t, 0, y) = 0 define a point x = 0 where the exterior fluid flow stops and the boundary layer is symmetric with respect to this point. In addition, we assume that U (t, x) = xV (t, x),
(1.3)
V (t, x) > 0 and V , Vx , Vt , v0 are bounded for 0 < x X. We look for solutions in the class of functions admitting the asymptotics |u(t, x, y) − U (t, x)| ∼ exp {−M y 2 },
y → ∞.
Definition 1.1. A solution to the problem (1.1), (1.2) is a pair of
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