By Jean-Marc Vanden-Broeck
ISBN-10: 0511729618
ISBN-13: 9780511729614
ISBN-10: 0521811902
ISBN-13: 9780521811903
Unfastened floor difficulties happen in lots of facets of technological know-how and of way of life comparable to the waves on a seashore, bubbles emerging in a tumbler of champagne, melting ice, pouring flows from a box and sails billowing within the wind. accordingly, the impression of floor rigidity on gravity-capillary flows remains to be a fertile box of study in utilized arithmetic and engineering. focusing on functions coming up from fluid dynamics, Vanden-Broeck attracts upon his years of expertise within the box to handle the numerous demanding situations keen on trying to describe such flows mathematically. while cautious numerical suggestions are applied to unravel the elemental equations, an emphasis is put upon the reader constructing a deep realizing of the constitution of the ensuing recommendations. the writer additionally stories suitable strategies in fluid mechanics to aid readers from different medical fields who're attracted to loose boundary difficulties.
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Additional info for Gravity-Capillary Free-Surface Flows (Cambridge Monographs on Mechanics)
Sample text
32) t → 0. 12) as ∞ (γ2 −γ1 )/π γ1 /π w = (t − d) an tn . 35) n=0 There are, of course, many other possible choices for G(t). 34) can be multiplied by any function analytic in |t| ≤ 1. 7) is satisfied. 35) after N terms and finding the coefficients an , n = 0, . . , N − 1 by collocation. This is the approach we will use when solving problems where the effects of gravity or surface tension are included in the dynamic boundary condition. 36) and therefore the present problem has the exact solution w= t−d 1 − td (γ2 −γ1 )/π tγ1 /π .
18. 16 in the complex t-plane. an expansion, as follows: τ − iθ = − ln 1+t − 1−t ∞ Bn tn . 12). 16). 19) yields u − iv ∼ f 1/2 as f → 0. 53) yields τ − iθ ∼ ln(1 − t) as t → 1. 59) is not singular and can be represented in the unit circle of the t-plane by a Taylor expansion. 57). One might argue that other singularities occur at the separation points A and B. 53). 60) n=0 where G(t) = 1−t . 12). The only difference is that the series has been rewritten as the exponential of a series. 56) are satisfied by assuming that the coefficients Bn are real and that Bn = 0 when n is even.
13), we find that α + πµ = γ. 13) gives z = Af γ/π . 19), w= π −π/γ π/γ−1 A z . 20) Flows inside corners will occur in many flow configurations described in this book and we will refer often to the above local analysis. e. if the angle HGL is rotated). The only difference is that α is then different from zero. 20). 6 as a flow inside a corner when γ < π and as a flow around a corner when γ > π. 21) where φD is the value of φ at the point D. Here we have used the classical O notation to indicate an estimate of the behaviour of a function.
Gravity-Capillary Free-Surface Flows (Cambridge Monographs on Mechanics) by Jean-Marc Vanden-Broeck
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