Showing papers in "Journal of Engineering Mathematics in 1996"
TL;DR: In this article, methods of computing periodic Green's functions of Stokes flow representing the flow due to triply-, doubly-, and singly-periodic arrays of three-dimensional or two-dimensional point forces are reviewed, developed, and discussed with emphasis on efficient numerical computation.
Abstract: Methods of computing periodic Green’s functions of Stokes flow representing the flow due to triply-, doubly-, and singly-periodic arrays of three-dimensional or two-dimensional point forces are reviewed, developed, and discussed with emphasis on efficient numerical computation. The standard representation in terms of Fourier series requires a prohibitive computational effort for use with singularity and boundary-integral-equation methods; alternative representations based on variations of Ewald’s summation method involving various types of splitting between physical and Fourier space with partial sums that decay in a Gaussian or exponential manner, allow for efficient numerical computation. The physical changes undergone by the flow in deriving singly- and doubly- periodic Green’s functions from their triply-periodic counterparts are considered.
69 citations
TL;DR: Previous biomechanical studies of modes of locomotion are extended to include analyses of three-dimensional flow fields and, in some cases, a rotlet field (curl of a stokeslet) needs to be incorporated in the models.
Abstract: Helical distributions of stokeslets can valuably model microbial locomotion through a fluid, and also the flow field generated, wherever a flagellum actively executes helical undulations (as in many single-celled algae and protozoa) or where (as in many bacteria) the action of rotary motors causes a passive structure of helical shape (which may be a flagellum or else the cell body itself) to rotate. Here, previous biomechanical studies of such modes of locomotion are extended to include analyses of three-dimensional flow fields. In some cases, a rotlet field (curl of a stokeslet) needs to be incorporated in the models. For example, spirochete swimming is modelled by combined helical distributions of stokeslets and rotlets; the computed flow field being confined to within distances of less than twice the radius of the cell body’s helical shape from its axis, while including a powerful jet-like interior flow through the coils of the swimming spirochete.
66 citations
TL;DR: In this article, the fundamental singularity of Stokes flow (the stokeslet) in the context of applications to locomotion and feeding currents in micro-organisms is discussed. But the authors focus on the image system of the Stokeslet in a rigid plane boundary.
Abstract: The paper discusses the fundamental singularity of Stokes flow (the stokeslet) in the context of applications to locomotion and feeding currents in micro-organisms. The image system for a stokeslet in a rigid plane boundary may be derived from Lorentz’s mirror image technique [1] or by an appropriate limit of Oseen’s solution for a sphere near a plane boundary [2]. An alternative derivation using Fourier transform methods [3] leads to an immediate physical interpretation of the image system in terms of a stokeslet and its multipole derivatives. The schematic illustration of a stokeslet and its image system in a plane boundary are exploited to explain the fluid dynamical principles of ciliary propulsion. For a point force oriented normal to the plane boundary, the resulting axisymmetric motion leads to a Stokes stream function representation which illustrates the toroidal eddy structure of the flow field. A similar eddy structure is also obtained for the two-dimensional system, although in this case, the toroidal structure is replaced by two eddies. This closed streamline model is developed to model chaotic filtration through the concept of a ‘blinking stokeslet’, a stokeslet alternating its vertical position according to a specific protocol. The resulting behaviour is illustrated via Poincare sections, particle dispersion and length of particle path tracings. Sessile micro-organisms may exploit a similar process so they can filter as large a volume of liquid as possible in search of food and nutrients.
41 citations
TL;DR: In this article, the authors compare the representation of Stokes flows at very low Reynolds number by a surface distribution of stokeslets and dipoles along the flagellar centreline, with the velocity of each crosssection expressed as a centreline value of the combined fields of singularities beyond a certain cutoff distance.
Abstract: Lorentz [1] pioneered the representation of flows at very low Reynolds number by a surface distribution of stokeslets — whose strengths, nowadays, are computed by surface-velocity collocations. That method is here compared with a representation widely used in flagellar hydrodynamics, by a curvilinear distribution of stokeslets and dipoles along the flagellar centreline; with the velocity of each cross-section expressed as a centreline value of the combined fields of singularities beyond a certain cutoff distance. The latter is also a good representation, and offers moreover some computational advantages. This paper establishes the equivalence of the two representations, and identifies those properties of Stokes flows which make both the dipoles and the cutoff essential to that equivalence.
32 citations
TL;DR: In this paper, a mathematical formulation is presented which is used to calculate the flow field of a two-dimensional Stokes fluid that is represented by a lattice of unit cells with pores inside.
Abstract: In this paper a mathematical formulation is presented which is used to calculate the flow field of a two-dimensional Stokes fluid that is represented by a lattice of unit cells with pores inside. The formulation is described in terms of an integral equation based on Lorentz’s formulation, whereby the fundamental solution is used that represents the flow due to a periodic lattice of point forces. The derived integral equation is applied to model the viscous sintering phenomenon, viz. the process that occurs (for example) during the densification of a porous glass heated to such a high temperature that it becomes a viscous fluid. The numerical simulation is carried out by solving the governing Stokes flow equations for a fixed domain through a Boundary Element Method (BEM). The resulting velocity field then determines an approximate geometry at a next time point which is obtained by an implicit integration method. From this formulation quite a few theoretical insights can be obtained of the viscous sintering process with respect to both pore size and pore distribution of the porous glass. In particular, this model is able to examine the consequences of microstructure on the evolution of pore-size distribution, as will be demonstrated for several example problems.
28 citations
TL;DR: In this article, a heat equation with nonlinear boundary conditions is used to model the evolution of a chemical reaction within a small cell, and an integro-differential equation reformulation is derived, a perturbation solution is developed and a modified product integration method is discussed.
Abstract: This paper is concerned with the modelling of the evolution of a chemical reaction within a small cell. Mathematically the problem consists of a heat equation with nonlinear boundary conditions. Through an integro-differential equation reformulation, an asymptotic result is derived, a perturbation solution is developed, and a modified product integration method is discussed. Finally, an alternative integral formulation is presented which acts as a check on the previous results and permits high accuracy numerical solutions.
27 citations
TL;DR: In this paper, a boundary integral method for the simulation of the time-dependent deformation of axisymmetric Newtonian or non-Newtonian drops suspended in a Newtonian fluid subjected to an axismmetric flow field is developed.
Abstract: A boundary integral method for the simulation of the time-dependent deformation of axisymmetric Newtonian or non-Newtonian drops suspended in a Newtonian fluid subjected to an axisymmetric flow field is developed. The boundary integral formulation for Stokes flow is used and the non-Newtonian stress is treated as a source term which yields an extra integral over the domain of the drop. By transforming the integral representation for the velocity to cylindrical coordinates we can reduce the dimension of the computational problem. The integral equation for the velocity remains of the same form as in Cartesian coordinates, and the Green's functions are transformed explicitly to cylindrical coordinates. Besides a numerical validation of the method we present simulation results for a Newtonian drop and a drop consisting of an Oldroyd-B fluid. The results for the Newtonian drop are consistent with results from the literature. The deformation process of the non-Newtonian drop for small capillary numbers appears to be governed by two relaxation times.
27 citations
TL;DR: The complementary finite-element programming method and algorithm for solving finite deformation problems in nonsmooth mechanics are presented in this paper, which provides a dual approach for the numerical solutions of the mixed boundary-value problem governed by nonsnooth physical laws.
Abstract: The complementary finite-element programming method and algorithm for solving finite deformation problems in nonsmooth mechanics are presented. This method provides a dual approach for the numerical solutions of the mixed boundary-value problem governed by nonsmooth physical laws. Application to non-smooth plastic flow is illustrated.
26 citations
TL;DR: In this paper, a nonlinear integral-equation model is derived in terms of hodograph variables for free-surface flow past an arbitrary bottom obstruction, and results are presented for a semi-circular obstruction and are compared with those of Forbes and Schwartz.
Abstract: A new nonlinear integral-equation model is derived in terms of hodograph variables for free-surface flow past an arbitrary bottom obstruction. A numerical method, carefully chosen to solve the resulting nonlinear algebraic equations and a simple, yet effective radiation condition have led to some very encouraging results. In this paper, results are presented for a semi-circular obstruction and are compared with those of Forbes and Schwartz [1]. It is shown that the wave resistance calculated from our nonlinear model exhibits a good agreement with that predicted by the linear model for a large range of Froude numbers for a small disturbance. The small-Froude-number non-uniformity associated with the linear model is also discussed.
25 citations
TL;DR: In this paper, the steady states of a combustion model were shown to have critical points (turning points in the bifurcation diagram) for certain ranges of parameter values and the time evolution for the solution for values of λ just above its critical value 403-1 was discussed.
Abstract: The steady states of a combustion model, derived in a previous paper, were shown to have critical points (turning points in the bifurcation diagram) for certain ranges of parameter values. Here attention is fixed on the heat release parameter λ and the time evolution for the solution for values of λ just above its critical value 403-1 is discussed. It is shown that the solution develops a three-stage structure, with the solution both approaching and leaving the critical point on a relatively short time scale. However, the majority of the time is spent in moving slowly past the critical point, on an 403-2 time scale. The solution finally attains its values on the upper solution branch, except in the special case of the exponential approximation and when reactant consumption is neglected. Here the temperature develops a singularity at a finite time tB, of O(log(tB−t)), though the fluid velocity remains finite at tB.
25 citations
TL;DR: In this paper, the authors studied the problem of slow incompressible flow past a circular cylinder using a hybrid method which combines numerical computation and asymptotic analysis, and they showed how to go beyond all those terms to find the sum of all terms containing e times a power of 1/log e.
Abstract: The solution for slow incompressible flow past a circular cylinder involves terms in powers of 1/log e, e times powers of 1/ log e, etc., where e is the Reynolds number. Previously we showed how to determine the sum of all terms in powers of 1/ log e. Now we show how to go beyond all those terms to find the sum of all terms containing e times a power of 1/log e. The first sum gives the drag coefficient and represents a symmetric flow in the Stokes region near the cylinder. The second term reveals the asymmetry of the flow near the body. This problem is studied using a hybrid method which combines numerical computation and asymptotic analysis.
TL;DR: In this article, the image system for a velocity field of the Oseen tensor in a fluid region bounded by a rigid spherical container is derived, and the reflection formula for a plane wall is derived as a limiting case of the new solution.
Abstract: The image system for a velocity field of the Oseen tensor in a fluid region bounded by a rigid spherical container is derived. The Green’s function and image system due to a nearby boundary constitute two themes explored in the pioneering (1896) paper by Lorentz. The special structure of our image system facilitates its incorporation as kernels for integral representations of velocity fields (another theme in the Lorentz paper) for a domain bounded by a spherical wall. The reflection formula for a plane wall is derived as a limiting case of the new solution.
TL;DR: In this paper, a simple formula is derived for the velocity potential in the vicinity of a cylinder, and in particular on the cylinder surfaces, which allows hydrodynamic quantities such as forces to be easily evaluated.
Abstract: The full linear problem of the scattering of water waves by an array of N bottom-mounted vertical circular cylinders situated in a channel of constant depth and width is solved using the method of multipoles. A simple formula is derived for the velocity potential in the vicinity of a cylinder, and in particular on the cylinder surfaces, which allows hydrodynamic quantities such as forces to be easily evaluated. The simplicity of the solution makes the evaluation of quantities of interest straightforward and extensive results are given. An approximate solution for the forces on the cylinders, based on the assumption that the wavelength of the incident wave is long compared with the cylinder radii, is also given, and this is compared with results from the ‘exact’ linear solution.
TL;DR: In this paper, a train of small-amplitude surface waves is obliquely incident on a fixed, thin, vertical plate submerged in deep water, and an appropriate one-term Galerkin approximation is employed to calculate very accurate upper and lower bounds for the reflection and transmission coefficients for any angle of incidence and any wave number.
Abstract: A train of small-amplitude surface waves is obliquely incident on a fixed, thin, vertical plate submerged in deep water. The plate is infinitely long in the horizontal direction. An appropriate one-term Galerkin approximation is employed to calculate very accurate upper and lower bounds for the reflection and transmission coefficients for any angle of incidence and any wave number thereby producing very accurate numerical results.
TL;DR: The equations of motion for an incompressible fluid with friction can be written as follows as mentioned in this paper, where the equations are expressed in terms of a set of equations for the incompressibility of the fluid.
Abstract: The equations of motion for an incompressible fluid with friction can be written as follows
TL;DR: One hundred years ago, the Journal of Engineering Mathematics celebrated the appearance of a paper by the Dutch physicist H.A. Lorentz in which he put forward some seminal ideas on slow viscous flow as mentioned in this paper.
Abstract: With this special issue of the Journal of Engineering Mathematics we commemorate and celebrate the appearance, one hundred years ago (Fig. 1), of a paper [1] by the Dutch physicist H.A. Lorentz in which he put forward some seminal ideas on slow viscous flow (see also [2–4]). Lorentz (to be pronounced as Lawrence with emphasis on the first syllable) is not known, per se, for his contributions to fluid mechanics. Indeed, he was a physicist whose fame rested first and foremost on his contributions to the theory of electromagnetism, electrodynamics, the theory of electrons and the dawn of relativity. His place among his contemporaries was, perhaps, described best by Albert Einstein who wrote ([5] and [6, pp73–76]) in 1953:
“At the turn of the century the theoretical physicists of all nations considered H.A. Lorentz as the leading mind among them, and rightly so.” But then, Einstein continues as follows:
“The physicists of our time are mostly not fully aware of the decisive part which H.A. Lorentz played in shaping the fundamental ideas in theoretical physics. The reason for this strange fact is that Lorentz’s basic ideas have become so much a part of them that they are hardly able to realize quite how daring these ideas have been and to what extent they have simplified the foundations of physics.”
TL;DR: In this article, infinite periodic arrays of stokeslets in three dimensions are summed up by obtaining various rapidly converging infinite series, where the same array is parallel to and in between two plates.
Abstract: Infinite periodic arrays of stokeslets in three dimensions are summed up by obtaining various rapidly converging infinite series. The three cases treated here are: 1. Identical stokeslets distributed at constant intervals on a line parallel to a plate, 2. An array of identical stokeslets distributed on a two-dimensional periodic lattice on a plane parallel to a plate, 3. The same array, but parallel to and in between two plates. Computational results are shown and comparisons with previously averaged expressions are made.
TL;DR: In this article, a generalization of the Lorentz reciprocal theorem is developed for the creeping flow of micropolar fluids in which the continuum equations involve both the velocity and the internal spin vector fields.
Abstract: A generalization of the Lorentz reciprocal theorem is developed for the creeping flow of micropolar fluids in which the continuum equations involve both the velocity and the internal spin vector fields. In this case, the stress tensor is generally not symmetric and conservation laws for both linear and angular momentum are needed in order to describe the dynamics of the fluid continuum. This necessitates the introduction of constitutive equations for the antisymmetric part of the stress tensor and the so-called couple-stress in the medium as well. The reciprocal theorem, derived herein in the limit of negligible inertia and without external body forces and couples, provides a general integral relationship between the velocity, spin, stress and couple-stress fields of two otherwise unrelated micropolar flow fields occurring in the same fluid domain.
TL;DR: In this paper, the authors investigated both the forced and natural convection of a sphere immersed in a fluid-saturated porous medium and obtained asymptotic solutions in terms of the Rayleigh number by means of a regular perturbation method.
Abstract: Unsteady forced and natural convection around a sphere immersed in a fluid-saturated porous medium is investigated. The sphere is suddenly heated and, subsequently, maintains a constant temperature over the surface. For the forced convection problem, the method of matched asymptotic expansions is used to obtain an asymptotic solution of the energy equation in terms of the Peclet number. For the natural convection problem, asymptotic solutions in terms of the Rayleigh number are obtained by means of a regular perturbation method.
TL;DR: In this article, the fluid flow induced when free-surface travelling waves pass over a submerged circular cylinder is considered and perturbation methods are used to formulate a sequence of potential problems that are solved using a Boundary Element method.
Abstract: We consider the fluid flow induced when free-surface travelling waves pass over a submerged circular cylinder. Perturbation methods are used to formulate a sequence of potential problems that are solved using a Boundary Element method. Favourable comparison is made, where possible, with earlier work. Attention is focused, primarily, upon the time-averaged flow about the cylinder.
TL;DR: In this paper, the authors considered the torsional deformation of a non-homogeneous infinite elastic cylinder slackened by an external circular cut and established the effect of the nonhomogeneity on the stress intensity factor at the tip of the cut.
Abstract: We consider the torsional deformation of a non-homogeneous infinite elastic cylinder slackened by an external circular cut. The shear modulus of the material of the cylinder is assumed to vary with the radial coordinate by a power law. It is assumed that the lateral surface of the cylinder as well as the surface of the cut are free of stress. The main object of this study is to establish the effect of the non-homogeneity on the stress intensity factor at the tip of the cut. The problem leads to a pair of dual series relations, the solution of which is governed by a Fredholm integral equation of the second kind with a symmetric kernel. This equation is solved numerically by reducing it to an algebraic system. It is concluded that for any degree of non-homogeneity and for D, the relative depth of the cut, greater than 0.6, the cylinder may be replaced by a half-space. However, as the non-homogeneity increases, D decreases.
TL;DR: In this paper, a new class of compressible elastic solids, which includes the Blatz-Ko material as a special case, is proposed, and a closed-form solution is constructed and studied for a bifurcation problem modeling void formation in this class.
Abstract: A new class of compressible elastic solids, which includes the Blatz-Ko material as a special case, is proposed. A closed-form solution is constructed and studied for a bifurcation problem modeling void formation in this class of compressible elastic solids. The relation between the void-formation condition and the material parameters is obtained analytically. An energy comparison of the void-formation deformation and the homogeneous expansion deformation is carried out.
TL;DR: In this article, a two-dimensional free surface flow generated by a moving distribution of pressure is modeled as a viscous fluid and the bottom is covered by a thin layer of mud.
Abstract: Two-dimensional free surface flows generated by a moving distribution of pressure are considered. The bottom is assumed to be covered by a thin layer of mud. The mud is modelled as a viscous fluid. The problem is solved numerically by a boundary integral equation method. It is shown that the layer of mud produces a damping of the waves in the far field. Profiles of the free surface and of the surface of the mud are presented.
TL;DR: In this paper, the authors obtained the description of axisymmetric equilibrium holes in thin fluid layers lying on a horizontal substrate under the influence of surface tension and gravity effects in the asymptotic limit when the radius of the hole is small.
Abstract: In this paper we obtain the description of axisymmetric equilibrium holes in thin fluid layers lying on a horizontal substrate under the influence of surface tension and gravity effects in the asymptotic limit when the radius of the hole is small. For values of the contact angle between the fluid and the substrate not equal to π we demonstrate that James' (J. Fluid Mech.63, 657–664 (1974)) solution for the meniscus surrounding a narrow cylindrical rod dipped into a bath of fluid also provides the correct asymptotic solution to the present problem. In the case when the contact angle is equal to π we obtain the asymptotic solution for the first time. In both cases we obtain asymptotic expressions for the radius of the hole at the substrate and the thickness of the layer far from the hole. The correctness of these expressions is confirmed by comparison with numerical solutions to the full problem. In the light of the present study we are able to highlight shortcomings in previous studies and, in particular, show that their predictions for the thickness of the layer are correct only at leading order in the limit of small holes.
TL;DR: In this article, the free-surface flow due to a submerged source in water of finite depth is considered and the fluid is assumed to be inviscid and incompressible.
Abstract: The free-surface flow due to a submerged source in water of finite depth is considered. The fluid is assumed to be inviscid and incompressible. The problem is solved numerically by using a boundary integral equation formulation due to Hocking and Forbes [6]. The numerical results show that there is a train of waves on the free surface in accordance with the results of Mekias and Vanden-Broeck [5]. For small values of the Froude number, the amplitude of the waves is so small that the free surface is essentially flat in the far field. These waveless profiles agree with the calculations of Hocking and Forbes [6].
TL;DR: In this paper, a conformal mapping technique is used for predicting the width of the separation zone at a diverging open channel flow, and the Schwarz-Christoffel transformation is used to transform the physical boundaries of the flow into a complex plane.
Abstract: A conformal mapping technique is used for predicting the width of the separation zone at a diverging open channel flow. The Schwarz-Christoffel transformation is used to transform the physical boundaries of the flow into a complex plane and the flow field is solved using modified boundary conditions utilizing a complex velocity potential in the resulting hodograph plane. The final solution gives the width of the separation zone in nondimensionalized form and provides an inviscid solution for comparative study.
TL;DR: In this article, a simple model for burning on the circular face of a substrate is analyzed, in which the temperature develops hot and cold regions arranged in concentric circular rings, and a linearized study shows the parameter values for which small amplitude patterns are stable.
Abstract: A simple model for burning on the circular face of a substrate is analyzed. It is shown that spatial patterns can form, in which the temperature develops hot and cold regions arranged in concentric circular rings. A linearized study shows the parameter values for which small amplitude patterns are stable. The fully non-linear equations are then solved using an efficient shooting method in the spatial variable, and an extremely complicated bifurcation diagram is obtained, from which it follows that multiple solutions occur at the same values of the defining parameters. The effect of heat leakage at the edges of the circular region is considered, and complicated non-linear behaviour occurs in this case also. Seven different temperature patterns, all co-existing at the same parameter values, are presented in a particular instance.
TL;DR: In this paper, the low Reynolds number motion and deformation of a neutrally buoyant drop due to its interaction with a translating solid particle (immersed in the same fluid) is studied by means of a system of second-kind Fredholm integral equations.
Abstract: The low Reynolds number motion and deformation of a neutrally buoyant drop (immersed in a different viscous fluid) due to its interaction with a translating solid particle (immersed in the same fluid) is studied. This is achieved by means of a system of second-kind Fredholm integral equations. It is shown that the resulting system of integral equations possesses a unique continuous solution, and thus the proposed form of solution is assured to provide the unique regular solution of the present interaction problem.
TL;DR: In this article, the effect of surface deformation upon the surface velocity of the fluid particles is examined in some detail, and the influence of surface roughness is explicitely taken into account and, among other results, cases of absence of slip caused by the attachment of the liquid film to the solid surface and of slippage solely induced by surface Roughness are found.
Abstract: The mechanism of surface diffusion is taken at the basis of the phenomenon of slippage of the contact line of a liquid film. With the aid of the condition of continuity of the traction vectors at the solid-liquid interface, we obtained an evolution equation for the velocity of the fluid particles at the wall which shows a marked resemblance with Millikan's equation for the slippage coefficient of gases and reduces, in the limit of small surface diffusivity, to the classical Stokes-Einstein model. The influence of surface roughness is explicitely taken into account and, among other results, cases of absence of slip caused by the attachment of the liquid film to the solid surface and of slippage solely induced by surface roughness are found. Finally, the effect of the surface deformation upon the surface velocity of the fluid particles is examined in some detail.
TL;DR: In this article, a simple derivation of the Lorentz theorem is presented which gives the perturbation pressure and velocity due to the presence of a plane wall introduced into an unlimited viscous fluid of given pressure and velocities obeying the Stokes equation.
Abstract: A simple derivation of the Lorentz theorem is presented which gives the perturbation pressure and velocity due to the presence of a plane wall introduced into an unlimited viscous fluid of given pressure and velocity obeying the Stokes equation.