We continue our study of the adaptation from spherical to doubly periodic slot domains of the poloidal-toroidal representation of vector fields. Building on the successful construction of an orthogonal quinquepartite decomposition of doubly periodic vector fields of arbitrary divergence with integral representations for the projections of known vector fields and equivalent scalar representations for unknown vector fields (Part 1), we now present a decomposition of vector field equations into an equivalent set of scalar field equations. The Stokes equations for slow viscous incompressible fluid flow in an arbitrary force field are treated as an example, and for them the application of the decomposition uncouples the conservation of momentum equation from the conservation of mass constraint. The resulting scalar equations are then solved by elementary methods. The extension to generalised Stokes equations resulting from the application of various time discretisation schemes to the Navier-Stokes equations is also solved.

We consider a theoretical model for the flow of Newtonian fluid through a long flexible-walled channel which is formed from four compliant and rigid compartments arranged alternately in series. We drive the flow using a fixed upstream flux and derive a spatially one-dimensional model using a flow profile assumption. The compliant compartments of the channel are assumed subject to a large external pressure, so the system admits a highly collapsed steady state. Using both a global (linear) stability eigensolver and fully nonlinear simulations, we show that these highly collapsed steady states admit a primary global oscillatory instability similar to observations in a single channel. We also show that in some regions of the parameter space the system admits a secondary mode of instability which can interact with the primary mode and lead to significant changes in the structure of the neutral stability curves. Finally, we apply the predictions of this model to the flow of blood through the central retinal vein and examine the conditions required for the onset of self-excited oscillation. We show that the neutral stability curve of the primary mode of instability discussed above agrees well with canine experimental measurements of the onset of retinal venous pulsation, although there is a large discrepancy in the oscillation frequency.

An analysis is developed for the behaviour of a cloud of cavitation bubbles during both the growth and collapse phases. The theory is based on a multipole method exploiting a modified variational principle developed by Miles [“Nonlinear surface waves in closed basins”, J. Fluid Mech.75 (1976) 418–448] for water waves. Calculations record that bubbles grow approximately spherically, but that a staggered collapse ensues, with the outermost bubbles in the cloud collapsing first of all, leading to a cascade of bubble collapses with very high pressures developed near the cloud centroid. A more complex phenomenon occurs for bubbles of variable radius with local zones of collapse, with a complex frequency spectrum associated with each individual bubble, leading to both local and global collective behaviour.

Two-dimensional free-surface flows produced by a submerged source in a fluid of infinite depth are considered. It is assumed that the point on the free surface just above the source is a stagnation point and that the fluid outside two shear layers is at rest. The free-surface profile and the shape of the shear layers are determined numerically by using a series-truncation method. It is shown that there is a solution for each value of the Froude number F > 0. When F tends to infinity, the flow also describes a thin jet impinging in a fluid at rest.

A general theory is given for autonomous perturbations of non-linear autonomous second order oscillators. It is found using a multiple scales method. A central part of it requires computation of Fourier coefficients for representation of the underlying oscillations, and these coefficients are found as convergent expansions in a suitable parameter.

The paper uses the factorisation method to discuss solutions of period three for the difference equation

which has been proposed as a simple mathematical model for the effect of frequency dependent selection in genetics. Numerical values are obtained for the critical values of a at which solutions of period three first appear. In addition, the interval in which stable solutions are possible has been determined. Exact solutions are given for the case a = 4 and these have been used to check the results.

A recent paper by Leipnik and Pearce introduced the gamma Weibull distribution. One of the main properties given is its characteristic function, which is expressed as an infinite sum. In this note, we provide a simpler representation in terms of the well-known hypergeometric functions in some special cases. We also derive expressions for moments of the distribution.

This note introduces quasi-local-balance for discrete-time Markov chains with absorbing states. From quasi-local-balance product-form quasi-stationary distributions are derived by analogy with product-form stationary distributions for Markov chains that satisfy local balance.

We give a simple and transparent proof for the square-root method of solving the continuous-time algebraic Riccati equation. We examine some benefits of combining the square-root method with other solution methods. The iterative square-root method is also discussed. Finally, paradigm numerical examples are given to compare the square-root method with the Schur method.

We would like to present a method to compute the incompatibility operator in any system of curvilinear coordinates (components). The procedure is independent of the metric in the sense that the expression can be obtained by means of the basis vectors only, which are first defined as normal or tangential to the domain boundary, and then extended to the whole domain. It is an intrinsic method, to some extent, since the chosen curvilinear system depends solely on the geometry of the domain boundary. As an application, the in-extenso expression of incompatibility in a spherical system is given.

It has been known for a long time that the equivariant $2+1$ wave map into the $2$-sphere blows up if the initial data are chosen appropriately. Here, we present numerical evidence for the stability of the blow-up phenomenon under explicit violations of equivariance.

The problem of scattering of surface water waves by a horizontal circular cylinder totally submerged in deep water is well studied in the literature within the framework of linearised theory with the remarkable conclusion that a normally incident wave train experiences no reflection. However, if the cross-section of the cylinder is not circular then it experiences reflection in general. The present paper studies the case when the cylinder is not quite circular and derives expressions for reflection and transmission coefficients correct to order ∈, where ∈ is a measure of small departure of the cylinder cross-section from circularity. A simplified perturbation analysis is employed to derive two independent boundary value problems (BVP) up to first order in ∈. The first BVP corresponds to the problem of water wave scattering by a submerged circular cylinder. The reflection coefficient up to first order and the first order correction to the transmission coefficient arise in the second BVP in a natural way and are obtained by a suitable use of Green' integral theorem without solving the second BVP. Assuming a Fourier expansion of the shape function, these are evaluated approximately. It is noticed that for some particular shapes of the cylinder, these vanish. Also, the numerical results for the transmission coefficients up to first order for a nearly circular cylinder for which the reflection coefficients up to first order vanish, are given in tabular form. It is observed that for many other smooth cylinders, the result for a circular cylinder that the reflection coefficient vanishes, is also approximately valid.

Methods for integral equations are used to derive upper and lower pointwise bounds for the solution of a nonlinear boundary value problem arising in the steady-state finite cable model of cell membranes. Test calculations are performed to illustrate the results and the accuracy achieved is significantly better than that obtained previously by other methods.

The homotopy analysis method (HAM) is applied to a nonlinear ordinary differential equation (ODE) emerging from a closure model of the von Kármán–Howarth equation which models the decay of isotropic turbulence. In the infinite Reynolds number limit, the von Kármán–Howarth equation admits a symmetry reduction leading to the aforementioned one-parameter ODE. Though the latter equation is not fully integrable, it can be integrated once for two particular parameter values and, for one of these values, the relevant boundary conditions can also be satisfied. The key result of this paper is that for the generic case, HAM is employed such that solutions for arbitrary parameter values are derived. We obtain explicit analytical solutions by recursive formulas with constant coefficients, using some transformations of variables in order to express the solutions in polynomial form. We also prove that the Loitsyansky invariant is a conservation law for the asymptotic form of the original equation.

In a glass furnace solid batches of material are fed into a chamber and radiation heating applied. An individual batch is melted over the course of several minutes to form molten glass. A travelling front within the batch designates the progress of the melting, a process characterized by multiple radiation reflections. This results in an effective conductivity within the melting zone that is significantly larger than that in the unmelted batch. Approximations based on these disparate conductivities enable accurate explicit expressions for the almost constant melting front speed and the associated temperature profile to be derived. Our results compare favourably with existing numerical simulations of the process, with the advantage of being both analytic and relatively simple. These predictions may be useful in suggesting how a furnace might be most effectively controlled under varying batch conditions, as well as ensuring the quality of the glass sheets produced.

Stokes’ axisymmetrical translational motion of a slip sphere, located anywhere on the diameter of a virtual spherical fluid ‘cell’, is investigated. The fluid is micropolar and flows are parallel to the line connecting the two centres. An infinite-series solution is presented for the stream function, pressure field, vorticity, microrotation component, shear stress and couple stress of the flow. Basset-type slip boundary conditions on the sphere surface are used for velocity and microrotation. The Happel and Kuwabara boundary conditions are used on the fictitious surface of the cell model. Numerical results for the normalized drag force acting on the sphere are obtained with excellent convergence for various values of the volume fraction, the relative distance between the centre of the sphere and the virtual envelope, the vortex viscosity parameter and the slip coefficients of the sphere. In the special case when the spherical particle is in the concentric position with the cell surface, the numerical values of the normalized drag force agree with the available values in the literature.

In this paper we derive extremality and comparison results for explicit and implicit initial and boundary value problems of first-order differential equations. Both the differential equations and the boundary conditions may involve discontinuities.

It is observed (among other things) that a theorem on bilinear and bilateral generating functions, which was given recently in the predecessor of this Journal, does not hold true as stated and proved earlier. Several possible remedies and generalizations, which indeed are relevant to the present investigation of various other results on bilinear and bilateral generating functions, are also considered.

In the paper King [8], a new class of source solutions was derived for the nonlinear diffusion equation for diffusivities of the form D(c) = D0cm/(l - vc)m+2. Here we extend this method for the nonlinear diffusion and convection equation

to obtain mass-conserving source solutions for a nonlinear conductivity function K(c) = K0cm+2/(l - vc)m+1. In particular we consider the cases m = -1,0, and 1, where fully analytical solutions are available. Furthermore we provide source solutions for the exponential forms of the diffusivity and conductivity as given by D(c) = D0c−2e−n/c and K(c) = K0ce−n/c.

The essential aspects of the Boundary Integral Equation Method for the numerical solution of elliptic type boundary value problems are presented. A numerical example for a stress concentration problem in classical elasticity in three dimensions is given along with several examples for a class of scalar problems in elastic torsion of non-cylindrical bars. Some discussion and criticism of the method itself and in comparison with widely used field methods is also presented.