Leading-order Equations Research Articles

The generalized Onsager model for a binary gas mixture

AbstractThe Onsager model for the secondary flow field in a high-speed rotating cylinder is extended to incorporate the difference in mass of the two species in a binary gas mixture. The base flow is an isothermal solid-body rotation in which there is a balance between the radial pressure gradient and the centrifugal force density for each species. Explicit expressions for the radial variation of the pressure, mass/mole fractions, and from these the radial variation of the viscosity, thermal conductivity and diffusion coefficient, are derived, and these are used in the computation of the secondary flow. For the secondary flow, the mass, momentum and energy equations in axisymmetric coordinates are expanded in an asymptotic series in a parameter $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\epsilon = (\Delta m/ m_{av})$, where $\Delta m$ is the difference in the molecular masses of the two species, and the average molecular mass $m_{av}$ is defined as $m_{av}= (\rho _{w1} m_1 + \rho _{w2} m_2)/\rho _w$, where $\rho _{w1}$ and $\rho _{w2}$ are the mass densities of the two species at the wall, and $\rho _w = \rho _{w1} + \rho _{w2}$. The equation for the master potential and the boundary conditions are derived correct to $O(\epsilon ^2)$. The leading-order equation for the master potential contains a self-adjoint sixth-order operator in the radial direction, which is different from the generalized Onsager model (Pradhan & Kumaran, J. Fluid Mech., vol. 686, 2011, pp. 109–159), since the species mass difference is included in the computation of the density, viscosity and thermal conductivity in the base state. This is solved, subject to boundary conditions, to obtain the leading approximation for the secondary flow, followed by a solution of the diffusion equation for the leading correction to the species mole fractions. The $O(\epsilon )$ and $O(\epsilon ^2)$ equations contain inhomogeneous terms that depend on the lower-order solutions, and these are solved in a hierarchical manner to obtain the $O(\epsilon )$ and $O(\epsilon ^2)$ corrections to the master potential. A similar hierarchical procedure is used for the Carrier–Maslen model for the end-cap secondary flow. The results of the Onsager hierarchy, up to $O(\epsilon ^2)$, are compared with the results of direct simulation Monte Carlo simulations for a binary hard-sphere gas mixture for secondary flow due to a wall temperature gradient, inflow/outflow of gas along the axis, as well as mass and momentum sources in the flow. There is excellent agreement between the solutions for the secondary flow correct to $O(\epsilon ^2)$ and the simulations, to within 15 %, even at a Reynolds number as low as 100, and length/diameter ratio as low as 2, for a low stratification parameter $\mathcal{A}$ of 0.707, and when the secondary flow velocity is as high as 0.2 times the maximum base flow velocity, and the ratio $2 \Delta m / (m_1 + m_2)$ is as high as 0.5. Here, the Reynolds number $\mathit{Re}= \rho _w \varOmega R^2 / \mu $, the stratification parameter $\mathcal{A}= \sqrt{m \varOmega ^2 R^2 / (2k_BT)}$, $R$ and $\varOmega $ are the cylinder radius and angular velocity, $m$ is the molecular mass, $\rho _w$ is the wall density, $\mu $ is the viscosity and $T$ is the temperature. The leading-order solutions do capture the qualitative trends, but are not in quantitative agreement.

Kasner solutions, climbing scalars and big-bang singularity

We elaborate on a recently discovered phenomenon where a scalar field close to big-bang is forced to climb a steep potential by its dynamics. We analyze the phenomenon in more general terms by writing the leading order equations of motion near the singularity. We formulate the conditions for climbing to exist in the case of several scalars and after inclusion of higher-derivative corrections and we apply our results to some models of moduli stabilization. We analyze an example with steep stabilizing potential and notice again a related critical behavior: for a potential steepness above a critical value, going backwards towards big-bang, the scalar undergoes wilder oscillations, with the steep potential pushing it back at every passage and not allowing the scalar to escape to infinity.Whereas it was pointed out earlier that there are possibleimplications of the climbing phase to CMB, we point out here anotherpotential application, to the issue of initial conditions ininflation.

Petrov type I condition and dual fluid dynamics

Recently Lysov and Strominger [arXiv:1104.5502] showed that imposing Petrov type I condition on a $(p+1)$-dimensional timelike hypersurface embedded in a $(p+2)$-dimensional vacuum Einstein gravity reduces the degrees of freedom in the extrinsic curvature of the hypersurface to that of a fluid on the hypersurface, and that the leading-order Einstein constraint equations in terms of the mean curvature of the embedding give the incompressible Navier-Stokes equations of the dual fluid. In this paper we show that the non-relativistic fluid dual to vacuum Einstein gravity does not satisfy the Petrov type I condition at next order, unless additional constraint such as the irrotational condition is added. In addition, we show that this procedure can be inversed to derive the non-relativistic hydrodynamics with higher order corrections through imposing the Petrov type I condition, and that some second order transport coefficients can be extracted, but the dual "Petrov type I fluid" does not match the dual fluid constructed from the geometry of vacuum Einstein gravity in the non-relativistic limit. We discuss the procedure both on the finite cutoff surface via the non-relativistic hydrodynamic expansion and on the highly accelerated surface via the near horizon expansion.

Ordering of two small parameters in the shallow water wave problem

The classical problem of irrotational long waves on the surface of a shallow layer of an ideal fluid moving under the influence of gravity as well as surface tension is considered. A systematic procedure for deriving an equation for surface elevation for a prescribed relation between the orders of the two expansion parameters, the amplitude parameter α and the long wavelength (or shallowness) parameter β, is developed. Unlike the heuristic approaches found in the literature, when modifications are made in the equation for surface elevation itself, the procedure starts from the consistently truncated asymptotic expansions for unidirectional waves, a counterpart of the Boussinesq system of equations for the surface elevation and the bottom velocity, from which the leading-order and higher order equations for the surface elevation can be obtained by iterations. The relations between the orders of the two small parameters are taken in the form β = O(αn) and α = O(βm) with n and m specified to some important particular cases. The analysis shows, in particular, that some evolution equations, proposed before as model equations in other physical contexts (such as the Gardner equation, the modified Korteweg–de Vries (KdV) equation and the so-called fifth-order KdV equation), can emerge as the leading-order equations in the asymptotic expansion for the unidirectional water waves, on equal footing with the KdV equation. The results related to the higher orders of approximation provide a set of consistent higher order model equations for unidirectional water waves which replace the KdV equation with higher order corrections in the case of non-standard ordering when the parameters α and β are not of the same order of magnitude. The shortcomings of certain models used in the literature become apparent as a result of the subsequent analysis. It is also shown that various model equations obtained by assuming a prescribed relation β = O(αn) between the orders of the two small parameters can be equivalently treated as obtained by applying transformations of variables which scale out the parameter β, in favor of α. It allows us to consider the nonlinearity-dispersion balance, epitomized by the soliton equations, as existing for any β, provided that α → 0, but leads to a prescription, in asymptotic terms, of the region of time and space where the equations are valid and so the corresponding dynamics are expected to occur.

Asymptotic solutions and new insights for cylinder and core–shell polymer gels in a solvent

A polymer gel in a solvent can swell to a large extent, and the large deformation caused by the swelling makes studies on the behaviors of gels difficult due to the high nonlinearity. In this paper, we study three homogeneous deformations of a cylinder gel in the cases of free-swelling, swellings under uniaxial constraint and under equal biaxial constraint and the inhomogeneous deformation of a spherical core–shell gel. The aim is to deduce new insights from the asymptotic solutions. In particular, the roles of the chemical potential are fully explored through appropriate parameters. For homogeneous deformations, the governing nonlinear algebraic equations are solved using perturbation methods. Analytical formulae for the stretch and stress are obtained and the key controlling parameters are found, which lead to some interesting insightful information. Also, according to some critical values of this parameter, the plane of the polymer network parameter and the solvent parameter can be divided into five domains, in which the gel has different behaviors. The inhomogeneous deformation of the core–shell gel (which can be used as a drug-carrier in applications) is governed by a complicated variable-coefficient nonlinear differential equation. It is not possible to obtain the exact solution. However, when the above-mentioned parameter is small or large, we manage to construct asymptotic solutions. In the first case, two key transformations are introduced, which make the leading-order equation twice integrable. In the second case, we use an extended method of matched asymptotics to construct the leading-order solution. For a core–shell gel, debonding can happen due to the swelling-induced stress. Our analytical solutions suggest that the best strategy for preventing this is to increase the initial stretch, since such an increase has no influence on the volume ratio but leads to the reduction of the stress. Also, how the geometry of the shell reduces the volume ratio is explicitly revealed. All the analytical results are also validated by numerical solutions.

Internal solitary waves with a weakly stratified critical layer

Motivated by observations of solitary waves in the ocean and atmosphere, this paper considers the evolution of long weakly nonlinear internal waves in an incompressible Boussinesq fluid. The motion is restricted to the vertical plane. The basic state consists of stable horizontal shear flow and density stratification. On a long time scale, the waves evolve and reach a quasi-steady régime where weak nonlinearity and weak dispersion are in balance. In many circumstances, this régime is described by a Korteweg-de-Vries equation. However, when the linear long-wave speed equals the basic flow velocity at a certain height, the critical level, the traditional assumption of weak nonlinearity breaks down due to the appearance of a singularity in the leading-order modal equation, implying a strong modification of the flow in the so-called critical layer. Since the relevant geophysical flows have high Reynolds and Péclet numbers, we invoke nonlinear effects to resolve this singularity. Viscosity and thermal conductivity are considered small but finite. Their presence renders the nonlinear-critical-layer solution unique. Crucially, the density stratification degree is assumed small at the critical level; this has the consequence that the leading-order singularity is then identical to that in an unstratified flow. Thus the asymptotic methodology employed previously for that case can be adapted to this present study. In this critical layer, the flow is fully nonlinear but laminar and quasi-steady, with a strong rearrangement of the buoyancy and vorticity contours. This inner flow is matched at the edges of the critical layer with the outer flow. The final outcome for spatially localized solutions is an integro-differential evolution equation, whose form depends on the critical-layer shape, and especially on the wave polarity, that is, depression or elevation. For a steady travelling wave, this evolution equation when expressed in terms of the streamfunction amplitude is not a Korteweg-de Vries equation, as it contains additional nonlinear terms necessary at a certain order of the asymptotic expansion when matching with the inner flow. However, this steady evolution equation can be transformed with an appropriate change of variables into a Korteweg-de-Vries equation. An analysis of the wave mean flow interaction is given. The horizontal basic stable flow is altered at the critical level at a slow viscous time scale by the nonlinear D-wave in the quasi-steady state régime. In most cases, the mean kinetic energy is likely to decay at the same time scale. The D-wave critical-layer thickness is found inversely proportional to the amplitude of the leading-order viscous outer flow. The D-wave critical-layer symmetry vis-à-vis the critical level is broken; the departure is proportional to the local Richardson number, and so is greater than the analogous asymmetry encountered in an unstratified shear flow. The mean flow horizontal momentum and kinetic energy are unchanged by the nonlinear E-wave in the quasi-steady régime, the exchanges taking place only during the formation of the critical layer and in the transition régime when the nonlinear E-wave is unsteady.

A simplified two-dimensional model of the melt spinning of semi-crystalline hollow compound fibers

A two-dimensional model of the melt spinning of semi-crystalline hollow compound fibers is presented. The model accounts for the orientation of the polymer molecules by means of a Doi–Edwards formulation for the molecular orientation tensor, and for the crystallization of the polymer by means of the Avrami–Kolmogorov kinetics with a modification for the flow-induced crystallization, and uses a Newtonian rheology where the dynamic viscosity is a function of the temperature, molecular orientation and degree of crystallization. The model is based on the leading-order one-dimensional equations for the fiber's geometry and axial and radial velocity components determined from an asymptotic analysis of slender fibers at low Reynolds numbers, and two-dimensional equations for the temperature, molecular orientation tensor and crystallization. It is shown that almost complete molecular orientation is achieved close to the maximum swell cross-section due to the large contraction of the fiber there, whereas, for the conditions considered here, the ultimate degree of crystallization is not achieved at the take-up cross-section. It is also shown that there are non-uniformities in the temperature and crystallinity profiles at the take-up cross-section which may have an effect on the fiber's properties.

Asymptotic Analysis of Flow Processes at Drawing of Single Optical Microfibres

Microstructured optical fibres (i.e. fibres that contain holes) have assumed a high profile in recent years, and given rise to many novel optical devices. The problem of manufacturing such fibres by heating and then drawing a preform is considered for both the cases of annular microfibres and annular capillaries. A fluid-mechanics model suggested in literature that uses asymptotic analysis based on the small aspect ratio of capillaries is analysed and revised. The leading-order equations are examined in some asymptotic limits, many of which give valuable practical information about the control parameters that influence the drawing process. Additionally, the solution obtained for a single capillary provides a suitable basis for describing more complicated fibre structures.

Asymptotic Analysis of Flow Processes at Drawing of Single Optical Microfibres

Microstructured optical fibres (i.e. fibres that contain holes) have assumed a high profile in recent years, and given rise to many novel optical devices. The problem of manufacturing such fibres by heating and then drawing a preform is considered for both the cases of annular microfibres and annular capillaries. A fluid-mechanics model suggested in literature that uses asymptotic analysis based on the small aspect ratio of capillaries is analysed and revised. The leading-order equations are examined in some asymptotic limits, many of which give valuable practical information about the control parameters that influence the drawing process. Additionally, the solution obtained for a single capillary provides a suitable basis for describing more complicated fibre structures.

Thin film dynamics on a prolate spheroid with application to the cornea

The tear film on the front of the eye is critical to proper eyesight; in many mathematical models of the tear film, the tear film is assumed to be on a flat substrate. We re-examine this assumption by studying the effect of a substrate which is representative of the human cornea. We study the flow of a thin fluid film on a prolate spheroid which is a good approximation to the shape of the human cornea. Two lubrication models for the dynamics of the film are studied in prolate spheroidal coordinates which are appropriate for this situation. One is a self-consistent leading-order hyperbolic partial differential equation (PDE) valid for relatively large substrate curvature; the other retains the next higher-order terms resulting in a fourth-order parabolic PDE for the film dynamics. The former is studied for both Newtonian and Ellis (shear thinning) fluids; for typical tear film parameter values, the shear thinning is too small to be significant in this model. For larger shear thinning, we find a significant effect on finite-time singularities. The second model is studied for a Newtonian fluid and allows for a meniscus at one end of the domain. We do not find a strong effect on the thinning rate at the center of the cornea. We conclude that the corneal shape does not have a significant effect on the thinning rate of the tear film for typical conditions.

Gravitational radiation and angular momentum flux from a slowly rotating dynamical black hole

A four-dimensional asymptotic expansion scheme is used to study the next-order effects of the nonlinearity near a spinning dynamical black hole. The angular-momentum flux and energy flux formula are then obtained by constructing the reference frame in terms of the compatible constant spinors and the compatibility of the coupling leading-order Newman-Penrose equations. By using the slow rotation and small-tide approximation for a spinning black hole, the horizon cross-section we chose is spherical symmetric. It turns out the flux formula is rather simple and can be compared with the known results. Directly from the energy flux formula of the slow-rotating dynamical horizon, we find that the physically reasonable condition on requiring the positivity of the gravitational energy flux yields that the shear will monotonically decrease with time. Thus a slow-rotating dynamical horizon will asymptotically approach an isolated horizon during late time.

A model for nonlinear interactions of internal gravity waves with saturated regions

A model for interactions between non-hydrostatic gravity waves and deep convective narrow hot towers is presented. The starting point of the derivation are the conservation laws for mass, momentum and energy for compressible flows combined with a bulk micro-physic model. Using multiscale asymptotics, a set of leading order equations is extracted, valid for the specific scales of the investigated regime. These are a timescale of 100 s, a horizontal and vertical lengthscale of 10 km for the wave dynamics plus a second horizontal lengthscale of 1 km for the narrow hot towers. Because of the comparatively short horizontal scales, Coriolis effects are negligible in this regime. The leading order equations are then closed by applying conditional averages over the hot tower lengthscale, leading to a closed model for the wave-scale that retains the net effects of the smaller scale dynamics. By assuming a systematically small saturation deficit in the ansatz, the small vertical displacements arising in this regime suffice to induce leading order changes of the saturated area fraction. The latter is the essential parameter in the model arising from the micro-physics.

An asymptotic expansion for the vortex-induced vibrations of a circular cylinder

This paper investigates the vortex-induced vibrations (VIV) of a spring-mounted circular cylinder. We compute analytically the leading-order equations describing the nonlinear interaction of the fluid and structure modes by carrying out an asymptotic analysis of the Navier–Stokes equations close to the threshold of instability of the fluid-only system. We show that vortex-shedding can occur at subcritical Reynolds numbers as a result of the coupled system being linearly unstable to the structure mode. We also show that resonance occurs when the frequency of the nonlinear limit cycle matches the natural frequency of the cylinder, the displacement being then in phase with the flow-induced lift fluctuations. Using an extension of this model meant to encompass the effect of the low-order added-mass and damping forces induced by the displaced fluid, we show that the amount of energy that can be extracted from the flow can be optimized by an appropriate choice of the structural parameters. Finally, we suggest a possible connection between the present ‘exact’ model and the empirical wake oscillator model used to study VIV at high Reynolds numbers. We show that for the low Reynolds numbers considered here, the effect of the structure on the fluid can be represented by a first coupling term proportional to the cylinder acceleration in the fluid equation, and by a second term of lower magnitude, which can stem either from an integral term or from a term proportional to the third derivative of the cylinder position.

Leading-Order Dynamics of Gravity-Driven Flows in Free-Standing Soap Films

We derive the leading-order equations that govern the dynamics of the flow in a falling, free-standing soap film. Starting with the incompressible Navier–Stokes equations, we carry out an asymptotic analysis using parameters that correspond to a common experimental setup. We account for the effects of inertia, surface elasticity, pressure, viscous stresses, gravity, and air drag. We find that the dynamics of the flow is dominated by the effects of inertia, surface elasticity, gravity, and air drag. We solve the leading-order equations to compute the steady-state profiles of velocity, thickness, and pressure in an experiment in which the film is in the Marangoni elasticity regime. The computational results, which include a Marangoni shock, are in good accord with the experimental measurements.

Resummation of large logarithms in the heavy-quark effects on the parton distributions inside the virtual photon

We discuss the resummation of the large logarithmic terms appearing in the heavy quark effects on parton distribution functions inside the virtual photon. We incorporate heavy quark mass effects by changing the initial condition of the leading-order DGLAP evolution equation. In a certain kinematical limit, we recover the logarithmic terms of the next-to-leading order heavy quark effects obtained in the previous work. This method enables us to resum the large logarithmic terms due to heavy quark mass effects on the parton distributions in the virtual photon. We numerically calculate parton distributions using the formulae derived in this work, and discuss the property of the resummed heavy quark effects.

A systematic derivation of a consistent set of “Boussinesq equations”

The often used “Boussinesq equations” for the determination of the coupled flow and temperature field in natural convection are systematically deduced by an asymptotic approach. With the nondimensional temperature difference that drives the flow, ɛ, as a perturbation parameter the leading order equations are identified as the appropriate equations, named “asymptotic Boussinesq equations”. These equations appear as the distinguished limit \(\varepsilon\rightarrow0\) and \({Ec}\rightarrow 0\) with \({Ec}/\varepsilon =const.\) The equations are compared to “Boussinesq equations” of other studies and used to calculate Nusselt numbers in laminar and turbulent flows in infinite vertical channels as an example and for the justification of the asymptotic approach.

Pattern solutions of the Klausmeier Model for banded vegetation in semi-arid environments I

In many semi-arid environments, vegetation cover is sparse, and is self-organized into large-scale spatial patterns. In particular, banded vegetation is typical on hillsides. Mathematical modelling is widely used to study these banded patterns, and many models are effectively extensions of a coupled reaction–diffusion–advection system proposed by Klausmeier (1999 Science 284 1826–8). However, there is currently very little mathematical theory on pattern solutions of these equations. This paper is the first in a series whose aim is a comprehensive understanding of these solutions, which can act as a springboard both for future simulation-based studies of the Klausmeier model, and for analysis of model extensions. The author focusses on a particular part of parameter space, and derives expressions for the boundaries of the parameter region in which patterns occur. The calculations are valid to leading order at large values of the ‘slope parameter’, which reflects a comparison of the rate of water flow downhill with the rate of vegetation dispersal. The form of the corresponding patterns is also studied, and the author shows that the leading order equations change close to one boundary of the parameter region in which there are patterns, leading to a homoclinic solution. Conclusions are drawn on the way in which changes in mean annual rainfall affect pattern properties, including overall biomass productivity.

Modulation of Internal Gravity Waves in a Multiscale Model for Deep Convection on Mesoscales

Abstract Starting from the conservation laws for mass, momentum, and energy together with a three-species bulk microphysics model, a model for the interaction of internal gravity waves and deep convective hot towers is derived using multiscale asymptotic techniques. From the leading-order equations, a closed model for the large-scale flow is obtained analytically by applying horizontal averages conditioned on the small-scale hot towers. No closure approximations are required besides adopting the asymptotic limit regime on which the analysis is based. The resulting model is an extension of the anelastic equations linearized about a constant background flow. Moist processes enter through the area fraction of saturated regions and through two additional dynamic equations describing the coupled evolution of the conditionally averaged small-scale vertical velocity and buoyancy. A two-way coupling between the large-scale dynamics and these small-scale quantities is obtained: moisture reduces the effective stability for the large-scale flow, and microscale up- and downdrafts define a large-scale averaged potential temperature source term. In turn, large-scale vertical velocities induce small-scale potential temperature fluctuations due to the discrepancy in effective stability between saturated and nonsaturated regions. The dispersion relation and group velocity of the system are analyzed and moisture is found to have several effects: (i) it reduces vertical energy transport by waves, (ii) it increases vertical wavenumbers but decreases the slope at which wave packets travel, (iii) it introduces a new lower horizontal cutoff wavenumber in addition to the well-known high wavenumber cutoff, and (iv) moisture can cause critical layers. Numerical examples reveal the effects of moisture on steady-state and time-dependent mountain waves in the present hot-tower regime.

On the application of concentrated solution theory to the forced convective flow of excess supporting electrolyte

A recent model for electrolysis with forced convection in a channel and an excess of dilute supporting electrolyte is extended using concentrated solution theory. Mean Spherical Approximation theory is used to calculate the necessary Onsager coefficients and ionic activities. Using asymptotic methods, we demonstrate the surprising similarities between the leading order equations for the dilute and concentrated theories. In addition, in deriving the extension, certain omissions in the original work are highlighted: an oversight in the Butler–Volmer law used in the original work is corrected, and the systematic use of similarity-like variables in electrochemical problems that do not have a similarity solution along the entire length of the diffusion layer adjacent to an electrode is demonstrated. This helps to explain the manner in which the solutions for the anode and cathode boundary layers and the bulk region decouple from each other at limiting current, enabling a much simpler solution strategy.

Influence of Surface Tension and Inner Pressure on the Process of Fibre Drawing

The present contribution deals with thermofluidynamical features occurring during the drawing of capillaries for microstructured optical fibres. Here, the process stability depends strongly on flow and thermal processes taking place as a preform is heated and drawn in the furnace. This is the case particularly for hollow fibres for which the existence of the inner hole directly depends on material parameters such as the surface tension and the rheological properties and on process parameter such as hole internal pressure and the process temperature. A fluid-mechanics model suggested in the literature that makes use of asymptotic analysis based on small aspect ratio of the micro capillaries, has been revisited and improved recently and the leading-order equations have been then examined in some asymptotic limits by Luzi et al. Starting from the novel class of solutions of the simplified equations of motion the present paper focuses on the effect of both surface tension and internal hole pressure since those are of essential importance during drawing. Thus, comparisons with experimental data are performed, in order to validate the analytical model developed in, which will be briefly presented here. The theoretical model gives very accurate predictions both when the internal hole is pressurized or when no pressure is applied, as long as the temperature does not reach too high values.