Jaqalanka Closure Report

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Wave Solutions

In classical continuum physics, a wave is a mechanical disturbance. Whether the disturbance is stationary or traveling and whether it is caused by the motion of atoms and molecules or the vibration of a lattice structure, a wave can be understood as a specific type of solution of an appropriate mathematical equation modeling the underlying physics. Typical models consist of partial differential equations that exhibit certain general properties, e.g., hyperbolicity. This, in turn, leads to the possibility of wave solutions. Various analytical techniques (integral transforms, complex variables, reduction to ordinary differential equations, etc.) are available to find wave solutions of linear partial differential equations. Furthermore, linear hyperbolic equations with higher-order derivatives provide the mathematical underpinning of the phenomenon of dispersion, i.e., the dependence of a wave's phase speed on its wavenumber. For systems of nonlinear first-order hyperbolic equations, there also exists a general theory for finding wave solutions. In addition, nonlinear parabolic partial differential equations are sometimes said to posses wave solutions, though they lack hyperbolicity, because it may be possible to find solutions that translate in space with time. Unfortunately, an all-encompassing methodology for solution of partial differential equations with any possible combination of nonlinearities does not exist. Thus, nonlinear wave solutions must be sought on a case-by-case basis depending on the governing equation.Comment: 22 pages, 3 figures; to appear in the Mathematical Preliminaries and Methods section of the Encyclopedia of Thermal Stresses, ed. R.B. Hetnarski, Springer (2014), to appea

Solutions to the complex Korteweg-de Vries equation: Blow-up solutions and non-singular solutions

In the paper two kinds of solutions are derived for the complex Korteweg-de Vries equation, including blow-up solutions and non-singular solutions. We derive blow-up solutions from known 1-soliton solution and a double-pole solution. There is a complex Miura transformation between the complex Korteweg-de Vries equation and a modified Korteweg-de Vries equation. Using the transformation, solitons, breathers and rational solutions to the complex Korteweg-de Vries equation are obtained from those of the modified Korteweg-de Vries equation. Dynamics of the obtained solutions are illustrated.Comment: 12 figure

Exact solutions for hydrodynamic interactions of two squirming spheres

We provide exact solutions of the Stokes equations for a squirming sphere close to a no-slip surface, both planar and spherical, and for the interactions between two squirmers, in three dimensions. These allow the hydrodynamic interactions of swimming microscopic organisms with confining boundaries, or each other, to be determined for arbitrary separation and, in particular, in the close proximity regime where approximate methods based on point singularity descriptions cease to be valid. We give a detailed description of the circular motion of an arbitrary squirmer moving parallel to a no-slip spherical boundary or flat free surface at close separation, finding that the circling generically has opposite sense at free surfaces and at solid boundaries. While the asymptotic interaction is symmetric under head-tail reversal of the swimmer, in the near field microscopic structure can result in significant asymmetry. We also find the translational velocity towards the surface for a simple model with only the lowest two squirming modes. By comparing these to asymptotic approximations of the interaction we find that the transition from near- to far-field behaviour occurs at a separation of about two swimmer diameters. These solutions are for the rotational velocity about the wall normal, or common diameter of two spheres, and the translational speed along that same direction, and are obtained using the Lorentz reciprocal theorem for Stokes flows in conjunction with known solutions for the conjugate Stokes drag problems, the derivations of which are demonstrated here for completeness. The analogous motions in the perpendicular directions, i.e. parallel to the wall, currently cannot be calculated exactly since the relevant Stokes drag solutions needed for the reciprocal theorem are not available.Comment: 27 pages, 7 figure

Answers and Solutions

Answers and solutions to the puzzles contained in this issue

Answers and Solutions

Answers and solutions for the puzzles in this issue

Answers and Solutions

Answers and solutions to the puzzles contained in this issue