Head-on collision of the solitary waves in fluid-filled elastic tubes

In the present work, by employing the field equations given in [15] and the extended PLK method derived in [9], we have studied the head-on collision of solitary waves in arteries. Introducing a set of stretched coordinates which include some unknown functions characterizing the higher order dispersive effects and the trajectory functions to be determined from the removal of possible secularities that might occur in the solution. Expanding these unknown functions and the field variables into power series of the smallness parameter epsilon and introducing the resulting expansions into the field equations we obtained the sets of partial differential equations. By solving these differential equations and imposing the requirements for the removal of possible secularities we obtained the speed correction terms and the trajectory functions. The results of our calculation show that both the evolution equations and the phase shifts resulting from the head-on collision of solitary waves are quite different from those of Xue [15], who employed the incorrect formulation of Su and Mirie [4]. As opposed to the result of previous works on the same subject, in the present work the phase shifts depend on the amplitudes of both colliding waves.Publisher's Versio

A note on the interactions of nonlinear waves governed by the generalized boussinesq equation

In this work, based on a one dimensional model, the interaction of two solitary waves propagating in opposite directions in a fluid whose equations are governed by the generalized Boussinesq equation, by use of the Poincare-Lighthill-Kuo (PLK) method. It is shown that bidirectional solitary waves are propagated, and the head-on collision of these two solitons occur. The phase shifts and the trajectories of these two solitons after the collisions are obtained.Author Post PrintPublisher's Versio

Interactions of Nonlinear Waves in Fluid-Filled Elastic Tubes

In this work, treating an artery as a prestressed thin-walled elastic tube and the blood as an inviscid fluid, the interactions of two nonlinear waves propagating in opposite directions are studied in the longwave approximation by use of the extended PLK (Poincaré-Lighthill-Kuo) perturbation method. The results show that up to O(k 3 ), where k is the wave number, the head-on collision of two solitary waves is elastic and the solitary waves preserve their original properties after the interaction. The leading-order analytical phase shifts and the trajectories of two solitons after the collision are derived explicitly

Re-visiting the head-on collision problem between two solitary waves in shallow water

Upon discovering the wrongness of the statement "although this term does not cause any secularity for this order it will cause secularity at higher order expansion, therefore, that term must vanish" by Su and Mirie [4], in the present work, we studied the head-on collision of two solitary waves propagating in shallow water by introducing a set of stretched coordinates in which the trajectory functions are of order of epsilon(2), where epsilon is the smallness parameter measuring non-linearity. Expanding the field variables and trajectory functions into power series in epsilon, we obtained a set of differential equations governing various terms in the perturbation expansion. By solving them under non-secularity condition we obtained the evolution equations and also the expressions for phase functions. By seeking a progressive wave solution to these evolution equations we have determined the speed correction terms and the phase shifts. As opposed to the result of Su and Mine [4] and similar works, our calculations show that the phase shifts depend on both amplitudes of the colliding waves.Publisher's VersionAuthor Post Prin

On head-on collision between two solitary waves in shallow water: the use of the extended PLK method

In the present work, we examined the head-on collision of solitary waves in shallow water theory, through the use of extended Poincare–Lighthill–Kuo (PLK) method based on the combination of reductive perturbation method with strained coordinates. Motivated with the result obtained by Ozden and Demiray (Int J Nonlinear Mech 69:66–70, 2015), we introduced a set of stretched coordinates that include some unknown functions which are to be determined so as to remove secularities that might occur in the solution. By expanding these unknown functions and the field variables into power series in the smallness parameter ϵ, introducing them into the field equations and imposing the conditions to remove the secularities, we obtained some evolution equations. By seeking a progressive wave solution to these evolution equations, we determined the speed correction terms and the phase-shift functions. The result obtained here is exactly the same with found by Ozden and Demiray (Int J Nonlinear Mech 69:66–70, 2015), wherein the analysis employed by Su and Mirie (J Fluid Mech 98:509–525, 1980) is utilized.Publisher's VersionAuthor Post Prin

Interactions of nonlinear electron-acoustic solitary waves with vortex electron distribution

In the present work, based on a one dimensional model, we consider the head-on-collision of nonlinear electron-acoustic waves in a plasma composed of a cold electron fluid, hot electrons obeying a trapped/vortex-like distribution, and stationary ions. The analysis is based on the use of extended Poincare, Lighthill-Kuo method [C. H. Su and R. M. Mirie, J. Fluid Mech. 98, 509 (1980); R. M. Mirie and C. H. Su, J. Fluid Mech. 115, 475 (1982)]. It is shown that, for the first order approximation, the waves propagating in opposite directions are characterized by modified Korteweg-de Vries equations. In contrary to the results of previous investigations on this subject, we showed that the phase shifts are functions of both amplitudes of the colliding waves. The numerical results indicate that the waves with larger amplitude experience smaller phase shifts. Such a result seems to be plausible from physical considerations.Publisher's Versio

Non-linear solitary sound waves in lipid membranes and their possible role in biological signaling.

Thesis (Ph. D.)--Boston UniversityBiological macromolecules self-assemble under entropic forces to form a dynamic 20 interfacial medium where the elastic properties arise from the curvature of the entropic potential of the interface. Elastic interfaces should be capable of propagating localized perturbations analogous to sound waves. However, (1) the existence and (2) the possible role of such waves in affecting biological functions remain unexplored. Both these aspects of "sound' as a signaling mechanism in biology are explored experimentally on mixed monolayers of lipids-fluorophores-proteins at the air/water interface as a model biological interface. This study shows - for the first time - that the nonlinear susceptibility near a thermodynamic transition in a lipid monolayer results in nonlinear solitary sound waves that are of 'all or none ' nature. The state dependence of the nonlinear propagation is characterized by studying the velocity-amplitude relationship and results on distance dependence, effect of geometry and collision of solitary waves are presented. Given that the lipid bilayers and real biological membranes have such nonlinearities in their susceptibility diagrams, similar solitary phenomenon should be expected in biological membranes. In fact the observed characteristics of solitary sound waves such as, their all or none nature, a biphasic pulse shape with a long tail and optp-mechano-electro-thermal coupling etc. are strikingly similar to the phenomenon of nerve pulse propagation as observed in single nerve fibers. Finally given the strong correlation between the activity of membrane bound enzymes and the susceptibility and the fact that the later varies within a single solitary pulse, a new thermodynamic basis for biological signaling is proposed. The state of the interface controls both the nature of sound propagation and its impact on incorporated enzymes and proteins. The proof of concept is demonstrated for acetylcholine esterase embedded in a lipid monolayer, where the enzyme is spatiotemporally "knocked out" by a propagating sound wave

Dynamics of Nonlinear Gravity-Capillary Waves in Deep Water Near Resonance

The minimum phase speed of linear gravity-capillary waves in deep water ($c_\mathrm{min}$) is known to be the bifurcation point of three-dimensional solitary waves (``lumps"). In the present thesis, various aspects of unsteady gravity-capillary lumps are investigated in the context of three sets of experiments. In the first set, cinematic shadowgraph and refraction-based techniques are utilized to measure the temporal evolution of the free surface deformation pattern downstream of a surface pressure source as it moves along a towing tank, while numerical simulations using a model equation are used to extend the experimental results. The focus of this study is on exploring the characteristics of the observed periodic shedding of lump-like depressions for towing speeds close to $c_\mathrm{min}$. From the experiments, it is found that the speed-amplitude characteristics and the shape of the depressions are nearly the same as those of the freely propagating gravity-capillary lumps of inviscid potential theory. The periodic behavior is found to be analogous to the periodic generation of two-dimensional solitary waves in shallow water by a source moving at trans-critical speeds of pure gravity waves. In the second set of experiments, the effect of viscous dissipation on freely propagating lumps is examined. A steady forced lump is first generated by applying appropriate forcing and towing speed. The forcing is then removed suddenly and the change in shape and speed of the lump is measured as it propagates freely under the action of viscosity. It is found that the localized structure of the lump is maintained during the decay and the first measurement of the decay rate of gravity-capillary lumps is reported. In the third set of experiments, the interactions of state III lumps generated by two pressure sources moving in parallel straight lines are investigated. The sources are adjusted to produce nearly identical periodic responses. The first lump generated by each source, collides with the lump from the other source in the center-plane of the two sources. It was observed that a steep depression is formed during the collision but breaks up soon after and radiates energy away in the form of small-amplitude radial waves. After the collision, a quasi-steady pattern is formed with several rows of localized depressions that are similar to lumps but exhibit periodic oscillations in depth

Design of nematic liquid crystals to control microscale dynamics

Dynamics of small particles, both living such as swimming bacteria and inanimate, such as colloidal spheres, has fascinated scientists for centuries. If one could learn how to control and streamline their chaotic motion, that would open technological opportunities in areas such as the transformation of stored or environmental energy into systematic motion, micro-robotics, and transport of matter at the microscale. This overview presents an approach to command microscale dynamics by replacing an isotropic medium such as water with an anisotropic fluid, a nematic liquid crystal. Orientational order leads to new dynamic effects, such as propagation of particle-like solitary waves. Many of these effects are still awaiting their detailed mathematical description. By using plasmonic metamask photoalignment, the nematic director can be patterned into predesigned structures that control dynamics of inanimate particles through the liquid crystal enabled nonlinear electrokinetics. Moreover, plasmonic patterning of liquid crystals allows one to command the dynamics of swimming bacteria, guiding their trajectories, polarity of swimming, and concentration in space. The patterned director design can also be extended to liquid crystal elastomers, in which case the director gradients define the dynamic profile of elastomer coatings. Some of these systems form an experimental playground for the exploration of out-of-equilibrium active matter, in which the levels of activity, degree of orientational order and patterns of alignment can all be controlled independently of each other.Comment: 35 pages, 9 figures, a review based on a lectur