Regular Perturbation Analysis Research Articles

Electrohydrodynamics of diffuse porous colloids.

The present article deals with the electrohydrodynamic motion of diffuse porous particles governed by an applied DC electric field. The spatial distribution of monomers as well as the charge distribution across the particle are considered to follow sigmoidal distribution involving decay length. Such a parameter measures the degree of inhomogeneity of the monomer distribution across the particle. The diffuse porous particles resemble several colloidal entities which are often seen in the environment as well as in biological and pharmaceutical industries. Considering the impact of bulk pH and ion steric effects, we modelled the electrohydrodynamics of such porous particulates based on the modified Boltzmann distribution for the spatial distribution of electrolyte ions and the Poisson equation for electric potential as well as the conservation of mass and momentum principles. We adopt regular perturbation analysis with weak field assumption and the perturbed equations are solved numerically to calculate the electrophoretic mobility and neutralization fraction of the particle charge during its motion as well as fluid collection efficiency. We further deduced the closed form relation between the drag force experienced by the charged porous particle and the fluid collection efficiency. In addition to the numerical results, we further derived the closed form analytical results for all the intrinsic parameters indicated above derived within the Debye-Hückel electrostatic framework and homogeneous distribution of monomers within the particle for which the decay length vanishes. The deduced mathematical results as indicated above will be useful to analyze several electrostatic and hydrodynamic features of a wide class of porous particulate and environmental entities.

Mathematical Analysis on Current–Voltage Relations via Classical Poisson–Nernst–Planck Systems with Nonzero Permanent Charges under Relaxed Electroneutrality Boundary Conditions

We focus on a quasi-one-dimensional Poisson-Nernst-Planck model with small permanent charges for ionic flows of two oppositely charged ion species through an ion channel. Of particular interest is to examine the dynamics of ionic flows in terms of I-V (current-voltage) relations with boundary layers due to the relaxation of neutral conditions on boundary concentrations. This is achieved by employing the regular perturbation analysis on the solutions established through geometric singular perturbation analysis. Rich dynamics are observed, particularly, the nonlinear interplays among different physical parameters are characterized. Critical potentials are identified, which play critical roles in the study of ionic flows and can be estimated experimentally. Numerical simulations are performed to further illustrate and provide more intuitive understandings of our analytical results.

Asymmetric rectified electric and concentration fields in multicomponent electrolytes with surface reactions.

Recent experimental studies have utilized AC electric fields and electrochemical reactions in multicomponent electrolyte solutions to control colloidal assembly. However, theoretical investigations have thus far been limited to binary electrolytes and have overlooked the impact of electrochemical reactions. In this study, we address these limitations by analyzing a system with multicomponent electrolytes, while also relaxing the assumption of ideally blocking electrodes to capture the effect of surface electrochemical reactions. Through a regular perturbation analysis in the low-applied-potential regime, we solve the Poisson-Nernst-Planck equations and obtain effective equations for electrical potential and ion concentrations. By employing a combination of numerical and analytical calculations, our analysis reveals a significant finding: electrochemical reactions alone can generate asymmetric rectified electric fields (AREFs), i.e., time-averaged, long-range electric fields, even when the diffusivities of the ionic species are equal. This finding expands our understanding beyond the conventional notion that AREFs arise solely from diffusivity contrast. Furthermore, we demonstrate that AREFs induced by electrochemical reactions can be stronger than those resulting from asymmetric diffusivities. Additionally, we report the emergence of asymmetric rectified concentration fields (ARCFs), i.e., time-averaged, long-range concentration fields, which supports the electrodiffusiophoresis mechanism of colloidal assembly observed in experiments. We also derive analytical expressions for AREFs and ARCFs, emphasizing the role of imbalances in ionic strength and charge density, respectively, as the driving forces behind their formation. The results presented in this article advance the field of colloidal assembly and also have implications for improved understanding of electrolyte transport in electrochemical devices.

Qualitative properties of zero-current ionic flows via Poisson-Nernst-Planck systems with nonuniform ion sizes

<p style='text-indent:20px;'>We consider a one-dimensional Poisson-Nernst-Planck system with two oppositely charged particles and nonuniform finite ion sizes modeled through a local hard-sphere potential. The existence and local uniqueness result is established under the framework of geometric singular perturbation theory. Treating the fi- nite ion size as a small parameter, through regular perturbation analysis, we are able to derive approximations of the individual fluxes explicitly, and this allows us to further study the qualitative properties of zero-current ionic flows, a special state among the range of the value for ionic current, which is significant for physiology. Of particular interest are the effects on the zero-current ionic flows from finite ion sizes, diffusion coefficients and ion valences. Critical potentials are identified and their important roles played in the study of ionic flow properties are characterized. Those non-intuitive observations from mathematical analysis of the system provide better understandings of the mechanism of ionic flows through membrane channels, particularly the internal dynamics of ionic flows, which cannot be detected via current technology. Numerical simulations are performed to provide more intuitive illustrations of the analytical results.</p>

Thermo-diffusion impacts on the flow of an elastico-viscous fluid over an inclined heated plane with magnetized wall

The focus is in this article is to scrutinize the simultaneous significances of magnetic diffusion, thermo-diffusion and angular location on the hydromagnetic flow of an elastico-viscous fluid over an inclined heated plane with magnetized wall. The flow medium is considered to be uniformly permeable (Darcy-Brinkman porous medium) and the flow of the fluid is considerably affected due to the appearance of a strong magnetic field in the direction normal to the flow surface. The significances of Hall current, induced magnetic field and Coriolis force on flow nature is also included in the study. The leading non-dimensionalized equations are explored by regular perturbation analysis. Ultimately, the expressions for velocity field, induced magnetic field, temperature and concentration are obtained. We further derived the surface skin friction, surface current density, heat and mass fluxes. The computation of results is performed with the aid of Mathematica software and results are presented in graphical and tabular forms for distinct flow impacting parameters. Numerical simulation explores that mass diffusion factor brings growth in the fluid velocity, temperature and normal induced magnetic field while it reduces the main induced magnetic field. Magnetic diffusion develops the primary flow and primary induced magnetic field and lessens the normal flow and normal induced magnetic field. Inclination angle of the heated plane upgrades primary induced magnetic field while downgrading normal induced magnetic field.

Heat and mass transport performance of MHD elastico‐viscous fluid flow over a vertically oriented magnetized surface with magnetic and thermo diffusions

AbstractThe key objective in this study is to examine the heat and mass transport behavior of magnetohydrodynamic elastic‐viscous fluid flow over a vertically oriented magnetized surface placed in a uniform permeable regime with magnetic and thermo diffusions. The fluid is partially ionized and permeated to flow in the presence of a strong magnetic field domain. Hence the Hall current effect is considered in this investigation. The significance of rotation and induced magnetic field on the flowing nature are also scrutinized in this study. The mathematical model of the problem is converted to a similar model by introducing suitable nondimensional variables. To obtain the closed‐form solutions of the flow leading equations, the regular perturbation analysis is utilized. For the exhibition of results, figures and tables are generated with the assistance of scientific computation software MATHEMATICA. Computed results are validated with the existing result in the limiting case. Such an investigation is important in evaluating the flow characteristics of low magnetic diffusive viscoelastic fluid. A noteworthy result seen is that magnetic diffusion significantly controls the fluid flow by altering the magnetic drag force. Mass diffusion factor brings an increase in the fluid velocity. Furthermore, we observed that the surface current density along the principal flow direction is significantly reduced by magnetic diffusion and mass diffusion factor.

Equilibrium tilt of slippery elliptical rods in creeping simple shear

It is shown that shape anisotropy and intrinsic surface slip lead to equilibrium tilt of slippery particles in a creeping simple shear flow, even for nearly shape-isotropic particles with a cross-section that is close to circular provided the Navier-slip length is sufficiently large. We study a rigid particle with an elliptical cross-section, and of infinite extent in the vorticity direction, in simple shear. A Navier-slip boundary condition is imposed on its surface. When a Navier-slip length parameter $\lambda$ is infinite, an analytical solution is derived for the Stokes flow around a particle tilting in equilibrium at an angle $(1/2)\cos ^{-1}((1-k)/(1+k))$ to the flow direction where $0 \le k \le 1$ is the ratio of the semi-minor to semi-major axes of its elliptical cross-section. A regular perturbation analysis about this analytical solution is then performed for small values of $1/\lambda$ and a numerical continuation method implemented for larger values. It is found that an equilibrium continues to exist for any anisotropic particle $k < 1$ provided $\lambda \ge \lambda _{crit}(k)$ where $\lambda _{crit}(k)$ is a critical Navier-slip length parameter determined here. As the case $k \to 1$ of a circular cross-section is approached, it is found that $\lambda _{crit}(k) \to \infty$ , so the range of Navier-slip lengths allowing equilibrium tilt shrinks as shape anistropy is lost. Novel theoretical connections with equilibria for constant-pressure gas bubbles with surface tension are also pointed out.

Effects on I–V relations from small permanent charge and channel geometry via classical Poisson–Nernst–Planck equations with multiple cations

We analyse a quasi-one-dimensional Poisson–Nernst–Planck system for ionic flow through a membrane channel with nonzero but small permanent charge. The system includes three ion species, two cations with the same valences and one anion. The cross-sectional area of the channel is included in the system, which provides certain information of the geometry of the three-dimensional channel. This is critical for our analysis. We treat the model as a boundary value problem of a singularly perturbed system of differential equations. Under the framework of geometric singular perturbation theory, together with specific structures of the model, the existence and local uniqueness of solutions to the boundary value problem for small permanent charges is established. Furthermore, treating the permanent charge as a small parameter, via regular perturbation analysis, we are able to derive an approximation of the I–V (current–voltage) relations explicitly, from which the permanent charge and channel geometry effects on ionic flows are analysed in detail. Critical potentials are identified and their roles in characterising the ionic flow properties of interest are studied. Some critical potentials can be experimentally estimated, and this provides an efficient way to adjust the boundary pot to observe distinct dynamics of ionic flows. Mathematical analysis indicates that a short and narrow filter within which the permanent charge is confined is expected in order to optimise the effect of permanent charges. This is consistent with the typical structure of an ion channel.

Small Permanent Charge Effects on Individual Fluxes via Poisson–Nernst–Planck Models with Multiple Cations

A quasi-one-dimensional Poisson–Nernst–Planck system for ionic flow through a membrane channel is studied. Nonzero but small permanent charge, the major structural quantity of an ion channel, is included in the model. The system includes three ion species, two cations with the same valences and one anion, which provides more correlations/interactions between ions compared to the case included only two oppositely charged particles. The cross-section area of the channel is included in the system, which provides certain information of the geometry of the three-dimensional channel. This is crucial for our analysis. Under the framework of geometric singular perturbation theory, more importantly, the specific structure of the model, the existence and local uniqueness of solutions to the system for small permanent charges is established. Furthermore, treating the permanent charge as a small parameter, through regular perturbation analysis, we are able to derive approximations of the individual fluxes explicitly, and this allows us to examine the small permanent charge effects on ionic flows in detail. Of particular interest is the competition between two cations, which is related to the selectivity phenomena of ion channels. Critical potentials are identified and their roles in characterizing ionic flow properties are studied. Some critical potentials can be estimated experimentally, and this provides an efficient way to adjust/control boundary conditions (electric potential and concentrations) to observe distinct qualitative properties of ionic flows. Mathematical analysis further indicates that to optimize the effect of permanent charges, a short and narrow filter, within which the permanent charge is confined, is expected, which is consistent with the typical structure of an ion channel.

Dynamics of Traveling Waves for the Perturbed Generalized KdV Equation

This paper is devoted to the existence of traveling, solitary and periodic waves for the perturbed generalized KdV by applying geometric singular perturbation, differential manifold theory and the regular perturbation analysis of Hamiltonian systems. Under the assumptions that the distributed delay kernel is the strong general one and the average delay is sufficiently small, traveling, solitary and periodic waves are shown to exist in the perturbed system. It is further proved that the wave speed is decreasing by analyzing the ratio of Abelian integrals, and we analyze these functions by using the theory of analytic functions and algebraic geometry. Moreover, the upper and lower bounds of the limit wave speed are presented. The relationship between wavelength and wave speed of traveling waves is also established.

Competition between Cations via Classical Poisson-Nernst-Planck Models with Nonzero but Small Permanent Charges.

We study a one-dimensional Poisson–Nernst–Planck system for ionic flow through a membrane channel. Nonzero but small permanent charge, the major structural quantity of an ion channel, is included in the model. Two cations with the same valences and one anion are included in the model, which provides more rich and complicated correlations/interactions between ions. The cross-section area of the channel is included in the system, and it provides certain information of the geometry of the three-dimensional channel, which is critical for our analysis. Geometric singular perturbation analysis is employed to establish the existence and local uniqueness of solutions to the system for small permanent charges. Treating the permanent charge as a small parameter, through regular perturbation analysis, we are able to derive approximations of the individual fluxes explicitly, and this allows us to study the competition between two cations, which is related to the selectivity phenomena of ion channels. Numerical simulations are performed to provide a more intuitive illustration of our analytical results, and they are consistent.

Wave propagation effects possible in solid composite materials by use of stabilized negative-stiffness components

Effects possible on wave propagation in solid composite materials by use of a stabilized negative stiffness phase are explored. One composite treated is an infinite periodic laminate comprised of two different homogeneous, isotropic linear elastic phases, for which uniaxial strain perpendicular to the laminae applies. Longitudinal plane wave propagation in this direction is analyzed, showing that the composite is stable provided both phases have merely strongly elliptic moduli. By tuning the negative bulk modulus thus permitted in one phase, the no-pass zones between dispersion curves can be substantially enlarged, the initiation frequency of the lowest no-pass zone can be made to approach zero, and the bands of permitted propagating frequencies can be substantially diminished, as can the phase and group velocities of long-wavelength waves. When material damping is added via linear viscoelasticity, a tuned negative bulk modulus dramatically enhances wave amplitude attenuation. The second composite considered is a matrix containing a random distribution of spherical inclusions, both materials being homogeneous, isotropic and linear elastic. Applying our recent demonstration for dilute distributions that such inclusions can have a negative bulk modulus while the overall composite remains stable, J. R. Willis’ variational approach is employed to analyze mean longitudinal plane waves. For low frequency/long wavelength waves, a dilute distribution of such inclusions can significantly reduce the wave speed. The next order correction in a regular perturbation analysis in frequency captures wave amplitude attenuation. It shows that at a mere 3.6% volume fraction of inclusions having a negative bulk modulus well within the stable range, the attenuation term (appearing in an exponential) has over twice the value attainable by a random distribution of rigid spherical particles of any volume fraction. These results suggest an effective and efficient means of attenuating waves in elastic solids via elastic wave scattering, even without use of material damping.

Regular factorizations and perturbation analysis for the core inverse of linear operators in Hilbert spaces

ABSTRACTThis paper concerns the regular factorization and expression of the core inverse in a Hilbert space. Utilizing the regular factorization, we first give some characterizations for the existence and the expression of the group inverse and core inverse. Based on these, we prove that the core inverse of the perturbed operator has the simplest possible expression if and only if the perturbation is range-preserving, and derive an explicit expression under the rank-preserving perturbation. Thus we can conclude that both the range-preserving perturbation and the rank-preserving perturbation are all continuous perturbations. The obtained results extend and improve many recent ones in matrix theory and operator theory.

Evolutionary pulsational mode dynamics in nonthermal turbulent viscous astrofluids

The pulsational mode of gravitational collapse in a partially ionized self-gravitating inhomogeneous viscous nonthermal nonextensive astrofluid in the presence of turbulence pressure is illustratively analyzed. The constitutive thermal species, lighter electrons and ions, are thermostatistically treated with the nonthermal $\kappa$ -distribution laws. The inertial species, such as identical heavier neutral and charged dust microspheres, are modelled in the turbulent fluid framework. All the possible linear processes responsible for dust-dust collisions are accounted. The Larson logatropic equations of state relating the dust thermal (linear) and turbulence (nonlinear) pressures with dust densities are included. A regular linear normal perturbation analysis (local) over the complex astrocloud ensues in a generalized quartic dispersion relation with unique nature of plasma-dependent multi-parametric coefficients. A numerical standpoint is provided to showcase the basic mode features in a judicious astronomical paradigm. It is shown that both the kinematic viscosity of the dust fluids and nonthermality parameter (kappa, the power-law tail index) of the thermal species act as stabilizing (damping) agent against the gravity; and so forth. The underlying evolutionary microphysics is explored. The significance of redistributing astrofluid material via waveinduced accretion in dynamic nonhomologic structureless cloud collapse leading to hierarchical astrostructure formation is actualized.

Stability of gravito-coupled complex gyratory astrofluids

We analyze the gravitational instability of complex rotating astrofluids in the presence of dynamic role of dark matter in a homogeneous hydrostatic equilibrium framework. The effects of the lowest-order fluid viscoelasticity, Coriolis force, fluid turbulence and inter-layer frictional coupling dynamics are concurrently considered in spatially-flat geometry. The Coriolis rotation is relative to the center of the entire fluid mass distribution, contributed by both the gyratory bright (visible) and dark (invisible) sectors, conjugated via the mutual gravitational interaction. The turbulence effects are included via the modified Larson equation of state. We use a regular Fourier-based linear perturbation analysis over the rotating fluid field equations to obtain a unique form of quartic dispersion relation with variable coefficients. We numerically carry out the dispersion analysis in two extreme limits: hydrodynamic (low-frequency) and kinetic (high-frequency) regimes. It is demonstrated that, in the former regime, the gas as well as dark matter rotations have stabilizing effects on the Jeans instability of the bi-fluidic admixture. In contrast, in the latter, the rotations play destabilizing roles on the instability. An interesting feature noted here is that the magnitude of the group velocity of the fluctuations throughout increases with both the gas and dark matter rotation frequencies, and vice-versa. We, finally, hope that the obtained results could be helpful in understanding the top-down kinetic mechanisms of bounded structure formation via gravitational collapse dynamics.

A unified perturbation analysis framework for countable Markov chains

In this paper, we are devoted to singular perturbation analysis for discrete-time or continuous-time Markov chains. We modify and extend the drift condition method, well known for regular perturbation, to develop a new framework for singular perturbation analysis. Our results extend and improve the corresponding ones in [2] for singularly perturbed Markov chains by allowing a general perturbation form, less restrictive conditions, and more computable bounds. Our analysis covers the regular perturbation analysis, and hence unifies singular and regular perturbation analysis. Furthermore, our results are illustrated by two two-dimensional Markov chains, including a discrete-time queue and a continuous-time level dependent quasi-birth-death process.

Heat transfer and cross-diffusion due to a sphere of constant thermal energy embedded in a porous medium

Heat transfer and cross-diffusion due to a sphere of constant thermal energy and concentration embedded in unbounded homogeneous porous medium in a regime where the temperature gradient produces mass flux is analytically studied using Darcy flow model. Analytical solution is obtained with regular perturbation analysis in the limit of small Rayleigh number. Due to cross-diffusion, solute front initially shows stronger convection than thermal front, but ultimately reaches steady-state at approximately the same time as that of thermal front. Quantity of heat necessary to maintain the steady-state is found to be least near the rear stagnation point and the mean Nusselt number is found to be unaffected by cross-diffusion. Nusselt number variation for different cone angles and Soret number is studied and it is found that higher improvement is achieved when cone angle is changed from 80 to 100°.

Zonal flow evolution and overstability in accretion discs

This work presents a linear analytical calculation on the stability and evolution of a compressible, viscous self-gravitating (SG) Keplerian disc with both horizontal thermal diffusion and a constant cooling time-scale when an axisymmetric structure is present and freely evolving. The calculation makes use of the shearing sheet model and is carried out for a range of cooling times. Although the solutions to the inviscid problem with no cooling or diffusion are well known, it is non-trivial to predict the effect caused by the introduction of cooling and of small diffusivities; this work focuses on perturbations of intermediate wavelengths, therefore representing an extension to the classical stability analysis on thermal and viscous instabilities. For density wave modes, the analysis can be simplified by means of a regular perturbation analysis; considering both shear and thermal diffusivities, the system is found to be overstable for intermediate and long wavelengths for values of the Toomre parameter Q ≲ 2; a non-SG instability is also detected for wavelengths ≳18H, where H is the disc scale-height, as long as γ ≲ 1.305. The regular perturbation analysis does not, however, hold for the entropy and potential vorticity slow modes as their ideal growth rates are degenerate. To understand their evolution, equations for the axisymmetric structure's amplitudes in these two quantities are analytically derived and their instability regions obtained. The instability appears boosted by increasing the value of the adiabatic index and of the Prandtl number, while it is quenched by efficient cooling.

Oblique wave scattering by an undulating porous bottom in a two-layer ice-covered fluid

The present study analyzes the reflection and transmission phenomenon of water-waves in a two-layer ice-covered system. The upper layer is covered by an ice-sheet, whereas the bottom of the lower layer is undulated and permeable. By using regular perturbation analysis and Fourier transform technique, the problem is solved and the first order reflection and transmission coefficients are determined. It is found that these coefficients depend on the shape as well as the permeability of the undulating bottom. Therefore, from the practical viewpoint, an undulating bottom topography is considered to determine all the aforesaid coefficients. The role of various system parameters, such as porosity, angle of incidence and ice parameters, are discussed to analyze the transformation of incident water wave energy from one layer to another layer. The outcomes are demonstrated in graphical forms.

Three-layer fluid flow over a small obstruction on the bottom of a channel

Many boundary value problems occur in a natural way while studying fluid flow problems in a channel. The solutions of two such boundary value problems are obtained and analysed in the context of flow problems involving three layers of fluids of different constant densities in a channel, associated with an impermeable bottom that has a small undulation. The top surface of the channel is either bounded by a rigid lid or free to the atmosphere. The fluid in each layer is assumed to be inviscid and incompressible, and the flow is irrotational and two-dimensional. Only waves that are stationary with respect to the bottom profile are considered in this paper. The effect of surface tension is neglected. In the process of obtaining solutions for both the problems, regular perturbation analysis along with a Fourier transform technique is employed to derive the first-order corrections of some important physical quantities. Two types of bottom topography, such as concave and convex, are considered to derive the profiles of the interfaces. We observe that the profiles are oscillatory in nature, representing waves of variable amplitude with distinct wave numbers propagating downstream and with no wave upstream. The observations are presented in tabular and graphical forms. doi:10.1017/S1446181114000480