On the Relativistic Harmonic Oscillator

In view of interest in relativistic harmonic oscillations in media, through which the speed of light is orders of magnitude smaller than in vacuum, the solution of the equation of motion is analyzed in the extreme- and weak-relativistic limits. Using scaled variables, it is shown rigorously how the equation of motion exhibits the characteristics of a boundary-layer problem in the extreme-relativistic limit: The solution differs from a sharp asymptotic pattern only around the turning points of oscillations over a vanishingly small fraction of the period. The sharp asymptotic pattern of the solution is a saw-tooth composed of linear segments. The velocity profile tends to a periodic step function and the phase-space plot tends to a rectangle. An expansion of the solution in terms of a small parameter that measures the proximity to the limit (v/c) → 1 yields an excellent approximation for the solution throughout the whole period of oscillations. In the weak-relativistic limit the same approach yields an approximation to the solution that is significantly better than in traditional asymptotic expansion procedures.

A numerical study of hybrid nanofluids near an irregular 3D surface having slips and exothermic effects

The current article presents a comprehensive investigation of MHD viscous flow of hybrid-nanofluids (Al2O3 − Ag/ water and (Al2O3 − Cu/) over a horizontally irregular 3D plane with non-uniform thickness combined with slip effects. The foremost aim of conducting this study is to enhance thermal transportation. Based on the following novelties, the subject study holds tremendous significance: i. A comparative analysis of two hybrid nanofluids with hybrid-base fluid together with slip effects ii. An exclusive study where the Tiwari and Das nanofluid model is employed combined with Fourier's heat flux model iii. Development of finite-difference code which implements the three-stage Lobatto IIIa approach for the designed problem. We have used suitable scaling transformations to convert the three-dimensional conservation equations of mass, momentum, and energy into a dimensionless system of boundary layer equations. The numerical solution of the coupled non-linear boundary layer problem is determined using the built-in finite-difference code designed to employ the three-stage Lobatto IIIa formula. A comprehensive assessment is carried out in how the velocity components, temperature, skin friction, and heat transfer rate are affected by the physical parameters of interest. The same is presented through graphs and in tabular form to offer a pictorial overview. The fluctuating trends of skin friction coefficients (x, y-directions) and Nusselt number are investigated to explore the physical landscape of the current study. The findings of this study offer a noticeable contrast to their existing counterparts.

Gradient Damage Analysis of a Cylinder Under Torsion: Bifurcation and Size Effects

In this work, an elastic-damage evolution analysis is carried out for a cylinder under torsion made of a material obeying a gradient damage model with softening. Both semi-analytical and asymptotic approaches are developed to analyze the elastic, axisymmetric and bifurcation stages. We show the existence of a fundamental branch where the damage field is asymmetric and localized within a finite thickness from the boundary. By minimizing a generalized Rayleigh quotient, the bifurcation time and modes are obtained as a function of the length scale $\epsilon =\ell /R$ ϵ = ℓ / R involving a material internal length and the cylinder radius. We will then focus on these size effects by assuming that $\epsilon $ ϵ is a small parameter in an asymptotic setting. After justification, specific spatial and temporal rescaled variables are introduced for the boundary layer problem. It is shown that the axisymmetric damage evolution and the bifurcation are governed by two universal functions independent of the length scale. The simulation results obtained by the semi-analytical approach are formally justified by the asymptotic methods.

MHD Marangoni boundary layer problem for hybrid nanofluids with thermal radiation

Purpose The purpose of this paper is to study flow of the Marangoni boundary layer pasta surface embedded in a porous medium saturated by a hybrid nanofluid in the presence of a magnetic field and thermal radiation. Design/methodology/approach The governing model was converted into ordinary differential equations applying proper similarity transformations. Therefore, Laplace transform was used to exactly solve the resulted equations. Hence, the influence of the velocity profile and temperature distribution was investigated under impacts of the involved parameters. Findings In the case of regular fluid, i.e. the solid volume fractions are zeros, the current results are in a very good agreement with those in the literature. It was found that the velocity decreases (increases) on increasing the parameters of copper-nanoparticles volume fraction, magnetic field and suction (permeability and injection). Further, the temperature increases (decreases) with an increase of the copper-nanoparticles volume fraction, magnetic field, injection and radiation (permeability and suction). Originality/value The current results of the Marangoni boundary layer problem for hybrid nanofluids are new, original and extend the previous problems investigated by many authors for the case of regular/nano fluids.

Boundary layer analysis for a 2-D Keller-Segel model

We study the boundary layer problem of a Keller-Segel model in a domain of two space dimensions with vanishing chemical diffusion coefficient. By using the method of matched asymptotic expansions of singular perturbation theory, we construct an accurate approximate solution which incorporates the effects of boundary layers and then use the classical energy estimates to prove the structural stability of the approximate solution as the chemical diffusion coefficient tends to zero.