By Daniel D. Joseph, Yuriko Y. Renardy
Two-fluid dynamics is a hard topic wealthy in physics and prac tical purposes. the various finest difficulties are tied to the lack of balance that's discovered in preferential positioning and shaping of the interface, in order that interfacial balance is a tremendous participant during this drama. quite often, options of equations governing the dynamics of 2 fluids usually are not uniquely made up our minds by way of the boundary information and assorted configurations of stream fit with a similar information. this can be one this is because balance experiences are vital; we have to comprehend which of the potential suggestions are good to foretell what may be saw. after we begun our reports within the early 1980's, it used to be in no way glaring that balance concept may possibly actu best friend paintings within the opposed setting of pervasive nonuniqueness. We have been pleasantly shocked, even astounded, by way of the level to which it does paintings. there are numerous basic strategies, known as simple flows, that are by no means sturdy, yet we could regularly compute progress premiums and verify the wavelength and frequency of the volatile mode which grows the quickest. This proce dure appears to be like to paintings good even in deeply nonlinear regimes the place linear idea isn't really strictly legitimate, simply as Lord Rayleigh confirmed some time past in his calculation of the scale of drops due to capillary-induced pinch-off of an inviscid jet.
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Additional info for Fundamentals of Two-Fluid Dynamics: Part I: Mathematical Theory and Applications
Example text
Suppose T(t) = d is the interface between two layers lying between r = a and r = b. The fluid with viscosity p,2 occupies the region a :::; r :::; d, and the fluid with viscosity p,1 occupies the region d :::; r :::; b. 13) 8r' 8x- ' Pr= ! _ ), r 88 r2 and for Couette flow between rotating cylinders, 36 Chapter I. lntroduction 8p {}(J = 0. 15) in d ::::;; r ::::;; b; and in a ::::;; r ::::;; d by v2 = A2r + B2jr. 17) where k = (b2 -~)/(~ -a2), is positive, we may, without losing generality, consider the case for which M and il2 - fit are positive.
Let the total perturbed velocity be denoted by U + ii where ii is the perturbation. The continuity of velocity is then expressed as U 1 + ii1 = U 2 + ii2 at y = lt + h(x), where subscipts 1 and 2 refer to the lower and upper fiuids, respectively. But now note that the basic fiows U 1 and U 2 are defined for the domains 0 :::; y :::; lt and lt :::; y :::; l, respectively, so that if the perturbed interface height lt + h(x) lies in 0 :::; y :::; lt, then there is the complication of having to make sense out ofwhat is meant by U 2 there.
41) and the conditions mentioned thereafter. ow, 42 Chapter I. Introduction Hence, the flow that minimizes Fl' is the one that rnaximizes D or W. 46) it rnaximizes f n u, the volume flux of the fluid. guration is the one which minimizes viscous energy dissipation for given volume flux, or, equivalently, rnaximizes the flow rate for a given pressure gradient. Alternatively, we rnay say that the pressure gradient is minimized for a given flow rate, or that the flow rate is maximized for a given pressure gradient.
