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Mathematical modelling is used to examine the unsteady problem of heating and pulling an axisymmetric cylindrical glass tube with an over-pressure applied within the tube to form tapers with a near uniform bore and small wall thickness at the tip. To allow for the dependence of viscosity on temperature, a prescribed axially varying viscosity is assumed. Our motivation is the manufacture of emitter tips for mass spectrometry which provide a continuous fluid flow and do not become blocked. We demonstrate, for the first time, the feasibility of producing such emitters by this process and examine the influence of the process parameters, in particular the pulling force and over-pressure, on the geometry. For small values of these parameters a more uniform bore may be obtained but the geometry is highly sensitive to small fluctuations. Higher values of the parameters result in more variation in the bore size but less sensitivity to fluctuations. The best parameters depend on the accuracy of the puller used to manufacture the tapers and the permissible tolerances on the geometry. The model has wider application to the manufacture of other devices., Comment: 13 pages, 9 figures

We develop a model of the forces on a spherical particle suspended in flow through a curved duct under the assumption that the particle Reynolds number is small. This extends an asymptotic model of inertial lift force previously developed to study inertial migration in straight ducts. Of particular interest is the existence and location of stable equilibria within the cross-sectional plane towards which particles migrates. The Navier-Stokes equations determine the hydrodynamic forces acting on a particle. A leading order model of the forces within the cross-sectional plane is obtained through the use of a rotating coordinate system and a perturbation expansion in the particle Reynolds number of the disturbance flow. We predict the behaviour of neutrally buoyant particles at low flow rates and examine the variation in focusing position with respect to particle size and bend radius, independent of the flow rate. In this regime, the lateral focusing position of particles approximately collapses with respect to a dimensionless parameter dependent on three length scales, specifically the particle radius, duct height, and duct bend radius. Additionally, a trapezoidal shaped cross-section is considered in order to demonstrate how changes in the cross-section design influence the dynamics of particles., Comment: Submitted for review in the Journal of Fluid Mechanics

Particle-laden flows in helical channels are of interest for their applications in spiral particle separators used in the mining and mineral processing industries. In this paper, we extend the previous work of Lee, Stokes, Bertozzi (2013) by studying thin-film flows of mono-disperse particle-laden fluid in helically-wound channels of arbitrary centreline curvature and torsion, and arbitrary cross-sectional shape. In the case where the particles are uniformly distributed through the depth of the film, significant analytic progress can be made: the governing equations reduce to a single nonlinear ordinary differential equation, which is readily integrated numerically to obtain the solution subject to appropriate boundary conditions. Motivated by possible application to the design of spiral separators, we consider the effects of changing the channel centreline geometry, the cross-sectional shape and the particle density on the resulting flows and the radial distribution of particles. Our results support the findings in Arnold, Stokes, Green (2016) regarding the effect of channel centreline geometry and cross-sectional shape on flows in particle-free regions. In particle-rich regions, similar effects are seen, although the primary velocity is lower due to increased effective mixture viscosity. \ys{Of key interest, is the effect of channel geometry on the focusing of the particles, for given fluxes of fluid and particles. We find that introducing a trench into the channel cross-section, a feature often used in commercial spiral particle separators, leads to smaller radial width of the particle-rich region, i.e. sharper focusing of the particles, which is consistent with experiments showing that channel geometry influences particle separation in a spiral separator., Comment: The following article has been submitted to Physics of Fluids