PREFACE INTRODUCTION Free-surface flows in nature and industry Scope of the book FUNDAMENTALS OF FLUID MECHANICS Main concepts Governing equations Elements of thermodynamics Classical boundary conditions Physically meaningful solutions and paradoxes of modeling MOVING CONTACT LINES: AN OVERVIEW Essence of the problem Experimental observations Molecular dynamics simulations Review of theories The key to the moving contact-line problem BOUNDARY CONDITIONS ON FORMING INTERFACES Modeling of interfaces Conservation laws Liquid-gas and liquid-solid interfaces Liquid-liquid interfaces Summary Open questions and possible generalizations MOVING CONTACT LINES: MATHEMATICAL DESCRIPTION Flow in the immediate vicinity of a moving contact line Dynamic wetting at small capillary numbers De-wetting and re-wetting Comparison with experiments and some estimates Examples: flows in a quasi-static regime Dynamic wetting at finite capillary numbers Liquid-liquid displacement Summary and outstanding modeling issues CUSPS, CORNERS AND COALESCENCE OF DROPS Singularities of free-surface curvature in experiments Conventional modeling "Missing" physics Singularity-free solution: cusp or corner? Coalescence of drops BREAKUP OF JETS AND RUPTURE OF FILMS Background Drop formation: emerging singularity Experiments on capillary pinch-off "Missing" physics and its qualitative verification Axisymmetric capillary pinch-off: singularity-free solution Pinch-off from a molecular viewpoint Rupture of films Summary APPENDIX A: Elements of vector and tensor calculus APPENDIX B: Equations of fluid mechanics in curvilinear coordinates APPENDIX C: Complex representation of biharmonic functions APPENDIX D: Physical properties of some fluids REFERENCES INDICES

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Capillary Flows with Forming Interfaces

Joining of different materials is a situation frequently observed in mechanical engineering and in materials science. Due to the difference in the elastic properties of the constituent materials, the junction points can be the origin of stress singularities and a possible source of damage. Hence, a full appreciation of these critical situations is of fundamental importance both from the mathematical and the engineering standpoints. In this paper, an overview of interface mechanical problems leading to stress singularities is proposed to show their relevance in engineering. The mathematical methods for the asymptotic analysis of stress singularities in multimaterial junctions and wedges composed of isotropic linear-elastic materials are reviewed and compared, with special attention to in-plane and out-of-plane loadings. This analysis mathematically demonstrates in a historical retrospective the equivalence of the eigenfunction expansion method, of the complex function representation, and of the Mellin transform technique for the determination of the order of the stress singularity in such problems. The analogies between linear elasticity and the Stokes flow of dissimilar immiscible fluids, the steady-state heat transfer across different materials, and the St. Venant torsion of composite bars are also discussed. Finally, advanced issues for the stress singularities due to joining of angularly nonhomogeneous elastic wedges are presented. This review article contains 147 references. DOI: 10.1115/1.2885134

https://staff.polito.it/alberto.carpinteri/papers/CARPINTERI_2008_N.477_AMR.pdf

On the stress singularities at multimaterial interfaces and related analogies with fluid dynamics and diffusion

The resonance for the elastic plate with a free edge is studied from the point of view of complex resonance. The variations of the real part and of the imaginary part of the complex resonance frequency as a function of the Poisson ratio ν are determined numerically. The results confirm the real resonance frequency theoretically predicted in I. Roitberg et al., Q. J. Mech. Appl. Math. 51, 1–13 (1998) for a zero Poisson ratio ν1=0, and a real resonance frequency that corresponds to a Lame mode is discovered for a Poisson ratio ν2=0.2248. It is shown that both real resonance frequencies may exist, at these two particular values of ν, because of the decoupling between the propagating Lamb mode and the set of evanescent Lamb modes.

Revisiting the edge resonance for Lamb waves in a semi-infinite plate

We present a fast direct solver methodology for the Dirichlet biharmonic problem in a rectangle. The solver is applicable in the case of the second order Stephenson scheme [J. W. Stephenson, J. Comput. Phys., 55 (1984), pp. 65-80] as well as in the case of a new fourth order scheme, which is discussed in this paper and is based on the capacitance matrix method ([B. L. Buzbee and F. W. Dorr, SIAM J. Numer. Anal., 11 (1974), pp. 1136-1150], [P. Bjorstad, SIAM J. Numer. Anal., 20 (1983), pp. 59-71]). The discrete biharmonic operator is decomposed into two components. The first is a diagonal operator in the eigenfunction basis of the Laplacian, to which the FFT algorithm is applied. The second is a low-rank perturbation operator (given by the capacitance matrix), which is due to the deviation of the discrete operators from diagonal form. The Sherman-Morrison formula [G. H. Golub and C. F. Van Loan, Matrix Computations, 3rd ed., John Hopkins University Press, Baltimore, MD, 1996] is applied to obtain a fast solution of the resulting linear system of equations.

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A Fast Direct Solver for the Biharmonic Problem in a Rectangular Grid

The main goal of this paper is to construct a spatial analog to the Kolosov–Muskhelishvili formulae using the framework of the hypercomplex function theory. We prove a generalization of Goursat's representation theorem for solutions of the biharmonic equation in three dimensions. On the basis of this result, we construct explicitly hypercomplex displacement and stress formulae in terms of two monogenic functions. Copyright © 2008 John Wiley & Sons, Ltd.

VV Meleshko

Papers

Selected topics in the history of the two-dimensional biharmonic problem