Download e-book for kindle: Nontraditional Methods in Mathematical Hydrodynamics by O. V. Troshkin

By O. V. Troshkin

ISBN-10: 0821802852

ISBN-13: 9780821802854

This booklet discusses a few qualitative beneficial properties of mathematical versions of incompressible fluids. 3 easy structures of hydrodynamical equations are thought of: the process of desk bound Euler equations for flows of a terrific (nonviscous) fluid, desk bound Navier-Stokes equations for flows of a viscous fluid, and Reynolds equations for the suggest pace box, strain, and pair one-point pace correlations of turbulent flows. The research issues algebraic or geometric houses of vector fields generated by means of those equations, equivalent to the overall association of streamlines, the nature and distribution of singular issues, stipulations for his or her absence or visual appeal, etc. Troshkin incorporates out a scientific software of the research to enquire stipulations for specific solvability of a couple of difficulties for those quasilinear platforms. Containing many examples of specific phenomena illustrating the final principles lined, this booklet might be of curiosity to researchers and graduate scholars operating in mathematical physics and hydrodynamics.

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Extra info for Nontraditional Methods in Mathematical Hydrodynamics

Example text

The arc A. with the endpoints is a common part of the cycle J and the boundary Q); when ¢c coincides with J, the role of Q will be performed by a curvilinear annulus bounded by J and a cycle lying in V (Figure 6). A continuum ¢c is limit (isolated) ifthere exists a sequence of points of the level C from the domain V accumulating at ¢c (conversely, there is no such sequence). THEOREM 2. For any continuum ¢c of the level C lying on the boundary cycle J of the domain V, there exists a relative neighborhood Q and a number co > 0 such that (i) if ¢c is a limit continuum, then the set ofpoints of the level C in the domain Q is the union ofa finite number n = 1, 2, ...

U+(Y) = 0. The function If/ has a unique minimum at the point(-/, yo) and a unique maximum on the boundary arc of the rectangle V complementing the interval x = -I, 0 < y < h. The flow is a semivortex (shown in §6, Figure 16a) with the opposite orientation of the streamlines. 3. U+ (y) > 0, 0 < y < h. In this case, the trace of If/ on the boundary of V attains a relative minimum at the point (-/,yo) and a positive maximum on the arc y = h, lxl ~ I. Due to the maximum principle, the numbers m = m = 1.

Taking into account (13), we obtain ~ ('l'(-l,y) - c(y)) = yk ( U_(y) - h:+i) > 0, < 0 < y < J. Since (-l, 0) - 'l'c (0) = 0, this inequality implies 0 < y < J. Using ( 13), by means of analogous considerations we find that 0 < y < J. Since, simultaneously, we come to the required inequality 'I' ;;::: c whfoh is valid on the boundary of the strip Vi. , 'I' - c is the stream function of some potential flow. Using the above inequality and the maximum principle, we come to the §4. PORTRAITS OF POTENTIAL FLOWS 4S conclusion that 1f1 - 0 ;;::: 0 on all the intervals x = const of the strip V0 • Hence, the corresponding derivative with respect to the variable yk+I is~ 0, or u(x, 0) - h:+I = lim y->+0 8 (If/ ~ y 8y 0 ) ~ 0, lxl::;; I, which proves the required statement.

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Nontraditional Methods in Mathematical Hydrodynamics by O. V. Troshkin

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