Leading-order Equations Research Articles

Asymptotic analysis of compound liquid jets at low Reynolds numbers

Asymptotic methods based on the slenderness ratio are used to obtain the leading-order equations which govern the fluid dynamics of both hollow and solid, compound jets such as those employed in the manufacture of textile fibers, composite fibers and optical fibers. These fibers consist of an inner material which may be a round jet or an annular one and which, in turn, is surrounded by an annular jet in contact with ambient air. It is shown that the leading-order equations are one-dimensional and that it is possible to obtain analytical solutions for several flow regimes for steady, compound jets. These analytical solutions are presented here. The one-dimensional leading-order equations for the fluid dynamics of annular liquid jets at low Reynolds numbers are also derived here.

BPS black holes in gauged N = 4, D = 4 supergravity

We find solutions of the bosonic sector of gauged N = 4, D = 4 SU(2) × SU(2) supergravity, which represent dilaton black holes with toroidal or spherical event horizons. The axion is consistently truncated, and the gauge group is broken to U(1) × U(1). The spherical black holes carry two electric and two magnetic abelian charges, whereas the toroidal holes have vanishing magnetic charges. The space-time metrics are warped products, and the manifolds turn out to be globally hyperbolic, in contrast to standard gauged supergravity ground states. It is shown that in the toroidal case, there are solutions preserving one quarter or one half of the supersymmetries, while for spherical topologies all supersymmetries are broken. In general, the toroidal BPS states represent naked singularities, but there is also a supersymmetric black hole with vanishing Hawking temperature. The 1 2 supersymmetric case arises for vanishing charges and mass, and represents the known domain wall solution of the Freedman-Schwarz model. It provides the background in which the black holes live. Finally, we use Chamseddine's and Volkov's Kaluza-Klein interpretation of gauged N = 4, D = 4 SU(2) × SU(2) supergravity to lift our solutions to ten and eleven dimensions and to consider them as solutions to the leading order equations of motion of the string/M-theory effective action.

Annular liquid jets and other axisymmetric free-surface flows at high Reynolds numbers

A perturbation method based on a long wavelength approximation is used to obtain the leading order equations governing the fluid dynamics of laminar, annular, round and compound liquid jets and liquid films on convex and concave cylindrical surfaces. An approximate, integral balance method is also used to determine the inviscid core and the thickness of the boundary layers of annular liquid jets near the nozzle exit. The steady state equations are transformed into parabolic ones by means of the von Mises transformation and solved in an adaptive, staggered grid to determine the axial velocity distribution and the location of the free surfaces. It is shown that, for free surface flows subject to inertia, gravity and surface tension, there is a contraction near the nozzle which increases as the Reynolds and Froude numbers are decreased, and is nearly independent of the Weber number for Weber numbers larger than about one hundred. It is also shown that this contraction depends on the flow considered, and is larger for films on convex surfaces. It is also shown that, for round jets, the acceleration of the jet's free surface is larger than that of the jet's centerline, although, sufficiently far from the nozzle exit, the axial velocity is uniform across the jet.

Leading-order equivalence of two formulations for long, annular liquid membranes

In a recent paper, the author [1] studied both analytically and numerically the steady uid dynamics of annular liquid jets by means of the models developed by Boussinesq [2] and Ramos [3]. In this short note, the formulation developed by Lee and Wang [4] for annular liquid membranes, i.e., annular liquid jets of zero thickness, in the absence of gravity is generalized to include the gravitational acceleration and compared with the one developed by Ramos [3] for annular liquid membranes and annular liquid jets. Lee and Wang [4] developed a model to study the uid dynamics of liquid membranes and studied numerically the formation of hollow spherical shells of liquid that result from the capillary instabilities of such membranes in zero gravity, and derived their equations by considering di€erential elements of the membrane and establishing conservation of mass and conservation of linear momentum along and normal to the membrane in the absence of gravity. By way of contrast, Ramos [3] considered annular liquid jets and the Euler equations for axisymmetric ows, accounted for gravity, employed an integral formulation, and assumed that the axial and radial velocity components of annular liquid jets are only functions of time and the axial coordinate. The objective of this short note is threefold. First, the model of Lee and Wang [4] is generalized to account for the gravitational acceleration, and the governing equations are based on the axial and radial velocity components rather than on those along and normal to the membrane and the resulting equations are compared with those of the model of Ramos [3]. Second, a long-wave analysis of the equations is carried out for inertia and capillary, annular liquid membranes. It is shown that the models developed by Lee and Wang [4] and Ramos [3] are governed by the same leading-order equations for long, annular liquid membranes. Lee and Wang's formulation [4]. The equations derived by Lee and Wang [4] for axisymmetric, annular liquid membranes along and normal to the membrane in the absence of gravity, may be easily generalized to account for gravity and written as

Modeling the time-dependent growth of single-crystal fibers

A model is developed to simulate time-dependent behavior in laser-heated miniature pedestal growth (LHPG) of single-crystal fibers. This process is meniscus controlled since the diameter of the growing fiber changes in response to variations between the equilibrium and the instantaneous contact angle at the melt/solidifying-solid interface. One of the operational problems associated with the LHPG technique is controlling diameter fluctuations in the growing fiber due to process perturbations. Smooth source feed and fiber pull rates, steady laser-heat input, and suppression of mechanical vibrations are necessary to achieve a constant fiber diameter. We derive a system of time-dependent, one-dimensional equations of motion and heat transfer as the leading order equations in an asymptotic expansion based upon the melt slenderness ratio (fiber radius/melt height). This system is solved numerically to predict the thermal profiles, mean positions of the melting and solidifying fronts, melt/gas interface, melt volume, and fiber diameter. System response to perturbations in the pull rates, thermal environment, and gravitational field is investigated.

1-D models for thin filaments of liquid-crystalline polymers: Coupling of orientation and flow in the stability of simple solutions

Slender asymptotic fiber models are derived from Doi-type 3-D equations for free surface flows of liquid-crystalline polymers. Leading order equations and self-consistent corrections are presented for a variety of physical regimes. We then explore the coupling of orientation effects to slender elongational flow behavior, with particular focus on the interplay between the Rayleigh capillary instability and both stabilizing and destabilizing orientation behavior. In the simple context of constant solutions, we identify physical regimes and precise conditions under which the Rayleigh instability may be completely arrested, as well as other regimes where orientation reduces but does not cancel capillary instability. In addition, we identify sources of additional orientation-dominated instabilities that are evident in both the uniaxial and biaxial nematic liquid crystal order parameters. These models and stability analyses lay the foundation for applications to fiber spinning processes.

Long-time effects of bottom topography in shallow water

We present and discuss new shallow water equations that provide an estimate of the long-time asymptotic effects of slowly varying bottom topography and weak hydrostatic imbalance on the vertically averaged horizontal velocity of an incompressible fluid with a free surface which is moving under the force of gravity. We consider the regime where the Froude number is much smaller than the aspect ratio δ of the shallow domain. The new equations are obtained at first order in an asymptotic expansion of the solutions of the Euler equations for a shallow fluid by using the small parameter δ 2. The leading order equations in this expansion enforce hydrostatic balance while those obtained at first order retain certain nonhydrostatic effects. Both sets of equations conserve energy and circulation, convect potential vorticity and have a Hamiltonian formulation. The corresponding energy and enstrophy are quadratic integrals with which we can bound the cumulative influence of the nonhydrostatic effects.

FORCE FIELDS ON INVISCID, SLENDER, ANNULAR LIQUID JETS

Regular perturbation expansions are used to analyse the fluid dynamics of unsteady, inviscid, slender, thin, incompressible (constant density), axisymmetric, upward and downward, annular liquid jets subjected to non-homogeneous, conservative body forces when both the annular jets are very thin and the gases enclosed by and surrounding the jet are dynamically passive. Both inertia- and capillarity-dominated annular jets are considered. It is shown that, for inertia-dominated jets, closure of the leading-order equations is achieved at second order in the perturbation parameter, which is the slenderness ratio, whereas closure is achieved at first order for capillarity-dominated jets. The steady leading-order equations are solved numerically by means of both an adaptive finite difference method which maps the curvilinear geometry of the jet onto a unit square and a fourth-order-accurate Runge–Kutta technique. It is shown that the fluid dynamics of steady, annular liquid jets is very sensitive to the Froude and Weber numbers and nozzle exit angle in the presence of non-homogeneous, conservative body forces. For upward jets with inwardly or axially directed velocities at the nozzle exit the effect of the non-homogeneous, conservative body forces is to increase the leading-order axial velocity component, decrease the jet's mean radius and move the stagnation point downstream. For downward jets with radially outward velocity at the nozzle exit the axial velocity component decreases monotonically as the magnitude of the non-homogeneous, conservative body forces is increased.

High-Kappa Limits of the Time-Dependent Ginzburg–Landau Model

The time-dependent Ginzburg–Landau model of superconductivity is examined in the high-$\kappa $, high magnetic field setting. This work generalizes the previous result for the steady-state model with a constant applied magnetic field. The significance of this generalization lies in the ability to incorporate the effects of both the applied magnetic field and applied current or voltage. Thus, it is possible to use the simplified setting obtained in this paper to study the motion and “pinning” of vortices in the presence of an applied current and a variable applied field. The results within are obtained via a formal asymptotic expansion of the Ginzburg–Landau equations in terms of $\kappa$, which yields a simplified system of leading-order equations. The formal asymptotic expansion is then justified by showing the solution to the full time-dependent Ginzburg-Landau equations converges to the solution of the leading-order equations as $\kappa \to \infty $. Computational results are also given that show the simplified leading-order model is indeed an accurate approximation to the solution of the full system of equations, even for moderate values of $\kappa$.

Models for thin viscous sheets

Leading-order equations governing the dynamics of a two-dimensional thin viscous sheet are derived. The inclusion of inertia effects is found to result in an ill-posed model when the sheet is compressed, and the resulting paradox is resolved by rescaling the equations over new length-and timescales which depend on the Reynolds number of the flow and the aspect ratio of the sheet. Physically this implies a dominant lengthscale for transverse displacements during viscous buckling. The theory is generalized to give new models for fully three-dimensional sheets.

PLANAR LIQUID SHEETS AT LOW REYNOLDS NUMBERS

Asymptotic methods are employed to derive the leading-order equations which govern the fluid dynamics of time-dependent, incompressible, planar liquid sheets at low Reynolds numbers using as small parameter the slenderness ratio. Analytical and numerical solutions of relevance to both steady film casting processes and plane stagnation flows are obtained with the leading-order equations. It is shown that for steady film casting processes the model which accounts for both gravity and low-Reynolds-number effects predicts thicker and slower planar liquid sheets than those which neglect a surface curvature term or assume that Reynolds number is zero, because the neglect of the curvature term and the assumption of zero Reynolds number are not justified at high take-up velocities owing to the large velocity gradients that occur at the take-up point. It is also shown that for Reynolds number/Froude number ratios larger than one, models which neglect the surface curvature or assume a zero Reynolds number predict velocity profiles which are either concave or exhibit an inflection point, whereas the model which accounts for both curvature and low-Reynolds-number effects predicts convex velocity profiles. For plane stagnation flows it is shown that models which account for both low-Reynolds-number and curvature effects predict nearly identical results to those of models which assume zero Reynolds number. These two models also predict a faster thickening of the planar liquid sheet than models which account for low- Reynolds-number effects but neglect the surface curvature. This curvature term is very large near the stagnation point and cannot be neglected there. It is also shown that the thickening of the sheet occurs closer to the stagnation point as the Reynolds number/Froude number ratio is increased, i.e. as the magnitude of the gravitational acceleration is increased. In addition it is shown that large surface tension introduces a third-order spatial derivative in the axial momentum equation at leading order.

Inviscid, slender, annular liquid jets

Regular perturbation expansions are used to analyse unsteady, inviscid, slender, incompressible (constant density), axisymmetric, annular liquid jets when the gases enclosed by and surrounding the jet are dynamically passive. Both inertia- and capillarity-dominated annular jets are considered. It is shown that, for inertia-dominated jets, closure of the leading order equations is achieved at second order in the perturbation parameter which is the slenderness ratio, whereas closure is achieved at first order for capillarity-dominated jets. The leading order equations are used to determine the fluid dynamics of steady annular jets.

General class of BPS saturated dyonic black holes as exact superstring solutions

We show that a four-parameter generating solution for a general class of four-dimensional, spherically-symmetric, static, dyonic BPS saturated solutions of leading-order effective equations of toroidally compactified heterotic string theory is an exact string solution. The corresponding ten-dimensional background defines a conformal sigma-model which is a particular case of a ‘chiral null model’ with curved ‘transverse’ part. The exact conformal invariance is a consequence of the chiral null structure of the ‘electric’ part of the model and the N = 4 world-sheet supersymmetry of its transverse ‘magnetic’ part. The sigma-model action has a remarkable covariance under both target space T-duality and the electro-magnetic S-duality transformations, and it illustrates the relationship between string-string duality in six dimensions and S-duality in four dimensions. In general, there exists a large class of exact six-dimensional superstring solutions described by chiral null models with four-dimensional transverse parts represented by N = 4 supersymmetric σ-models with metrics conformal to hyper-Kähler ones.

Pressure-driven flow of a thin viscous sheet

Systematic asymptotic expansions are used to find the leading-order equations for the pressure-driven flow of a thin sheet of viscous fluid. Assuming the fluid geometry to be slender with non-negligible curvatures, the Navier–Stokes equations with appropriate free-surface conditions are simplified to give a ‘shell-theory’ model. The fluid geometry is not known in advance and a time-dependent coordinate frame has to be employed. The effects of surface tension, gravity and inertia can also be incorporated in the model.

A perturbation solution for the flexible rotating disk: non-linear equilibrium and stability under transverse loading

A singular perturbation solution is developed for the transversely loaded, flexible rotating disk using the von Kármán large deflection theory of thin plates. No solution exists for these equations in the limit of vanishing dimensionless bending stiffness, ϵ. This fact, along with the desire to have the terms describing non-linear membrane stretching appear in the leading order equations, motivates a perturbation expansion for displacement of the form ω ≈ ϵ −1/2 ω 0 + ω 1 + …. The dimensionless transverse loading term is retained in the leading order equations only if it is rescaled by ϵ 1/2 . This rescaling of transverse load magnitude constitutes a similitude prediction for unifying the equilibrium and stability measurements of different flexible, rotating disks. Experimental measurements of the stability of flexible rotating disks under transverse point loading support this prediction. In addition, the numerically computed stability of the perturbation solution agrees quantitatively with the experimental measurements.

Continuous monitoring of ash content in fuel coals

In this paper, we use a comprehensive perturbation theory to develop an Eulerian model for nonlinear longitudinal waves in tapered elastic rods. The equations of motion, when cast in their Eulerian form, are ideal for fixed-grid numerical techniques such as finite-difference methods, finite-volume methods, and modern high-resolution shock-capturing schemes. The leading-order equations in our perturbation formalism are verified using a control-volume analysis, then linearized to recover a classical model for longitudinal waves in ultrasonic horns. We assess the mathematical structure of our leading-order model, deduce the eigenvalues and eigenvectors of the Jacobian matrix, and verify its hyperbolicity. Solutions are obtained using the space-time conservation element and solution element (CESE) method, a novel numerical technique for first-order hyperbolic systems. In our simulations, the CESE method is shown to effectively capture the evolution of traveling waves such as shocks and rarefactions, as well as salient features of nonlinearity.

Spin-induced orbital precession and its modulation of the gravitational waveforms from merging binaries.

Merging compact binaries are currently regarded as the most promising source of gravitational waves for the planned Earth-based LIGO/VIRGO laser-interferometer detector system, and will be an important source also for similar, lower-frequency detectors that might be flown in space (e.g., the proposed LISA mission). During the orbital inspiral, if one or both bodies are rapidly rotating, the general relativistic spin-orbit and spin-spin coupling (i.e., the dragging of inertial frames by the bodies' spins) cause the binary's orbital plane to precess. In this paper we analyze the resulting modulation of the inspiral gravitational waveform, using post2-Newtonian equations to describe the precession of the orbital plane, but only the leading-order (Newtonian, quadrupole-moment approximation) equations to describe the orbit, the radiation reaction, the inspiral, and the wave generation. We derive all the formulas one needs to readily compute the spin-modulated gravitational waveform (within the post-Newtonian approximation and the approximation that the precession frequency is much smaller than the orbital frequency). We also develop intuition into what the modulated signals look like, by a variety of means. We provide approximate, analytical solutions for the precessional motion and the modulated waveforms for two important special cases: the case where the bodies have nearly equal masses and the case where one of the bodies has negligible spin. For these cases, for almost all choices of binary parameters, the motion is a simple precession of the orbital angular momentum around the nearly fixed direction of the total angular momentum, with a few tens of precession periods as the waves sweep through the LIGO/VIRGO observational band. However, when the spin and orbital angular momenta are approximately anti-aligned, there is a transitional-precession epoch during which their near cancellation causes the binary to lose its gyroscopic bearings and tumble in space, with a corresponding peculiar sweep of the waveform modulation. We also explore numerically the precessional behaviors that occur for general masses and spins; these typically appear quite similar to our special-case, simple-precession, and transitional-precession solutions. An Appendix develops several diagrammatic aids for understanding intuitively the relation between the precessing orbit and the modulated waveform.

Solitons, solitary waves, and voidage disturbances in gas-fluidized beds

In this paper, we consider the evolution of an initially small voidage disturbance in a gas-fluidized bed. Using a one-dimensional model proposed by Needham & Merkin (1983), Crighton (1991) has shown that weakly nonlinear waves of voidage propagate according to the Korteweg–de Vries equation with perturbation terms which can be either amplifying or dissipative, depending on the sign of a coefficient. Here, we investigate the unstable side of the threshold and examine the growth of a single KdV voidage soliton, following its development through several different regimes. As the size of the soliton increases, KdV remains the leading-order equation for some time, but the perturbation terms change, thereby altering the dependence of the amplitude on time. Eventually the disturbance attains a finite amplitude and corresponds to a fully nonlinear solitary wave solution. This matches back directly onto the KdV soliton and tends exponentially to a limiting size. We interpret the series of large-amplitude localized pulses of voidage formed in this way from initial disturbances as corresponding to the ‘voidage slugs’ observed in gas fluidization in narrow tubes.

A two-dimensional model for the dynamics of sea ice

This paper develops a systematic analysis of a sea ice pack viewed as a thin layer of coherent ice floes and open water regions at the ocean surface. The pack is driven by wind stress and Coriolis force, with responsive water drag on the base of the floes. Integration of the mass and momentum balances through the layer thickness result in a two-dimensional theory for the interface between ocean and atmosphere. The theory is presented for a plane horizontal interface, but the construction is readily extended to a non-planar interface. An interacting continua framework is adopted to describe the layer mixture of ice and water, which introduces the layer thickness h and ice area fraction A as smoothly varying functions of the plane coordinate and time, on a pack length scale and weather system timescale. It is shown how an evolution equation for A which ignores ridging can lead to the area fraction exceeding unity in maintained converging flow, which is physically invalid. This is a feature and weakness of current models, and is eliminated by artificial cut-off in numerical treatments. Here we formulate a description of the ridging process which redistributes smoothly the excess horizontal ice flux into increasing thickness of a ridging zone of area fraction A r , and a simple postulate for the vertical ridging flux yields an evolution equation for A which shows how A can approach unity asymptotically, but not exceed unity, in a maintained converging flow. This is a significant feature of the new model, and eliminates a serious physical and numerical flaw in existing models. The horizontal momentum “balance involves the gradients of the extra stress integrated through the layer thickness, extra to the integrated water pressure over the depth of a local floe edge below sea level. These extra stresses are zero in diverging flow and arise as a result of interactions between floes during converging flow. It is shown precisely how a mean stress in a floe is determined by such edge tractions, and in turn provides an interpretation of the local extra stress in the pack. The interpretation introduces the further model function f (A) which defines the fraction of ice-ice contact length over the boundary of a floe, describing an increase of the contact fraction as A increases. Model interaction mechanisms then suggest a qualitative law for the pack stress in terms of relative motions of the floes which define the pack-scale strain rates. A simple viscous law is presented for illustration, but it is shown that even this simple model can reflect a conventional motion of a failure criterion on the stresses in a ridging zone where the convergence greatly exceeds a threshold value. We have therefore defined precisely the two-dimensional ice pack stress arising in the momentum balance, and determined its relation to the contact forces between adjacent floes. The foregoing analyses hinge on the introduction of dimensionless variables and coordinate scalings which reflect the orders of magnitude of the many physical variables and their gradients in both individual floe and ice pack motions. A variety of small dimensionless parameters arise, which allows the derivation of leading-order equations defining a reduced model which describes the major balances in the motion. The distinct equations for diverging and converging flow regions indicates the existence of moving boundaries (in the two-dimensional pack domain) in the flow, satisfying appropriate matching conditions to be determined as part of the complete evolution. This feature appears to have been ignored in previous treatments. Here we illustrate the evolution of a moving boundary by constructing an exact solution to a one-dimensional pack motion which describes onshore drift due to increasing, then decreasing, wind stress. During the second phase a region of diverging flow expands from the free edge. The solution demonstrates the influence of various parameters, but, importantly, will provide a test solution for numerical algorithms which must be constructed to determine more complex one and two-dimensional motions.

Long waves at the interface between two viscous fluids

Using a perturbation method up to the second order, the equation for long waves at the interface between two viscous fluids is derived for plane Couette–Poiseuille flow, and for moderate surface tension. The leading order equation is the Burgers equation. Higher order terms take into account linear dispersive effects and stabilizing effect of surface tension, and involve two more nonlinear terms. The exactness of the coefficients of this equation has been checked by using symmetry properties. For zero gravity, Poiseuille flow is shown to be stable if and only if the velocity profile is convex. Stable Couette flow can become unstable when the Reynolds number of one fluid is decreased, while keeping the other dimensionless parameters unchanged. Introducing characteristic length scales, the interface equation is put into ‘‘canonical’’ form.