Optimal homotopy analysis and control of error for implicitly defined fully nonlinear differential equations

Implicitly defined fully nonlinear differential equations can admit solutions which have only finitely many derivatives, making their solution via analytical or numerical techniques challenging. We apply the optimal homotopy analysis method (OHAM) to the solution of implicitly defined ordinary differential equations, obtaining solutions with low error after few iterations or even one iteration of the method. This is particularly true in cases where an auxiliary nonlinear operator was employed (in contrast to the commonly used choice of an auxiliary linear operator), highlighting the need for further study on using auxiliary nonlinear operators in the HAM. Through various examples, we demonstrate that the approach is efficient for an appropriate selection of auxiliary operator and convergence control parameter

Dynamics of a planar vortex filament under the quantum local induction approximation

The Hasimoto planar vortex filament is one of the rare exact solutions to the classical local induction approximation (LIA). This solution persists in the absence of friction or other disturbances, and it maintains its form over time. As such, the dynamics of such a filament have not been extended to more complicated physical situations. We consider the planar vortex filament under the quantum LIA, which accounts for mutual friction and the velocity of a normal fluid impinging on the filament. We show that, for most interesting situations, a filament which is planar in the absence of mutual friction at zero temperature will gradually deform owing to friction effects and the normal fluid flow corresponding to warmer temperatures. The influence of friction is to induce torsion, so the filaments bend as they rotate. Furthermore, the flow of a normal fluid along the vortex filament length will result in a growth in space of the initial planar perturbations of a line filament. For warmer temperatures, these effects increase in magnitude, since the growth in space scales with the mutual friction coefficient. A number of nice qualitative results are analytical in nature, and these results are verified numerically for physically interesting cases

Quantum Hasimoto transformation and nonlinear waves on a superfluid vortex filament under the quantum local induction approximation

The Hasimoto transformation between the classical LIA (local induction approximation, a model approximating the motion of a thin vortex filament) and the nonlinear Schrödinger equation (NLS) has proven very useful in the past, since it allows one to construct new solutions to the LIA once a solution to the NLS is known. In the present paper, the quantum form of the LIA (which includes mutual friction effects) is put into correspondence with a type of complex nonlinear dispersive partial differential equation (PDE) with cubic nonlinearity (similar in form to a Ginsburg-Landau equation, with additional nonlinear terms). Transforming the quantum LIA in such a way enables one to obtain quantum vortex filament solutions once solutions to this dispersive PDE are known. From our quantum Hasimoto transformation, we determine the form and behavior of Stokes waves, a standing one-soliton, traveling waves, and similarity solutions under normal and binormal friction effects. The quantum Hasimoto transformation is useful when normal fluid velocity is relatively weak, so for the case where the normal fluid velocity is dominant we resort to other approaches. We exhibit a number of solutions that exist only in the presence of the normal fluid velocity and mutual friction terms (which would therefore not exist in the limit taken to obtain the classical LIA, decaying into line filaments under such a limit), examples of which include normal fluid driven helices, stationary and propagating topological solitons, and a vortex ring whose radius varies inversely with the normal fluid magnitude. We show that, while chaos may not be impossible under the quantum LIA, it should not be expected to arise from traveling waves along quantum vortex filaments under the quantum LIA formulation

The Biot-Savart description of Kelvin waves on a quantum vortex filament in the presence of mutual friction and a driving fluid

We study the dynamics of Kelvin waves along a quantum vortex filament in the presence of mutual friction and a driving fluid while taking into account non-local effects due to Biot-Savart integrals. The Schwarz model reduces to a nonlinear and non-local dynamical system of dimension three, the solutions of which determine the translational and rotational motion of the Kelvin waves, as well as the amplification or decay of such waves. We determine the possible qualitative behaviours of the resulting Kelvin waves. It is well known from experimental and theoretical studies that the Donnelly-Glaberson instability plays a role on the amplification or decay of Kelvin waves in the presence of a driving normal fluid velocity, and we obtain the relevant stability criterion for the non-local model. While the stability criterion is the same for local and non-local models when the wavenumber is sufficiently small, we show that large differences emerge for the large wavenumber case (tightly coiled helices). The results demonstrate that non-local effects have a stabilizing effect on the Kelvin waves, and hence larger normal fluid velocities are required for amplification of large wavenumber Kelvin waves. Additional qualitative differences between the local and non-local models are explored

Dynamics of the Rayleigh-Plesset equation modelling a gas-filled bubble immersed in an incompressible fluid

Temporal dynamics of gas-filled spherical bubbles are often described using the Rayleigh-Plesset equation, a special case of the Navier-Stokes equations that describes the oscillations of a spherical cavity in an infinite incompressible fluid. While analytical approximations and numerical simulations have previously been given in some parameter regimes, we are able to completely classify all possible dynamics exactly, in terms of only the model parameters. We present an analytical study of the solutions to the Rayleigh-Plesset equation in any number of spatial dimensions, and we demonstrate that the possible behaviors of solutions include bubbles of constant radius, bubbles with temporally oscillating radius, and bubbles with finite time collapse. Each of these behaviors can be predicted solely in terms of the spatial dimension, pressures acting on the bubble, and initial strain. In the case of oscillating bubbles, we give the amplitude and period of these oscillations in terms of an integral which is a function of the aforementioned parameters, while when the bubble collapses, we can similarly give the time of collapse in terms of these parameters. We give a systematic study of all possible behaviors, and capture special case solutions presented numerically or asymptotically in the literature. We also discuss the influence of both surface tension and viscosity when these terms are included in the Rayleigh-Plesset dynamics

Exact closed-form solution to jet flow of a polymer arising in the electrospun nanofiber elaboration process

In the present paper, we obtain the exact solution to the jet flow of a polymer arising in the electrospun nanofiber elaboration process. This allows us to improve on the approximate solutions of Colantoni and Boubaker (2014) and He, Kong, Chen, Hu, and Chen (2014). Our corrected and far simpler solution is more useful for extracting the salient physics inherent in the electrospun nanofiber elaboration process. We then consider the more robust formulation of the problem, in which the velocity of the jet and the radius of the jet depend strongly on one another, obtaining the solution in this case as well. All of the results obtained are rather simple, and give us closed-form exact solutions (rather than complicated approximations), and hence these results both maintain accuracy and have the possibility of being useful for future studies into the physics of such mechanical processes. The results also agree nicely with experimental and numerical results present in the literature