Mark A. Kelmanson

Computational linear rheology of general branch-on-branch polymers

We present a general algorithm for predicting the linear rheology of branched polymers. While the method draws heavily on existing theoretical understanding of the relaxation processes in entangled polymer melts, a number of new concepts are developed to handle diverse polymer architectures including branch-on-branch structures. We validate the algorithm with experimental examples from model polymer architectures to fix the parameters of the model. We use experimentally determined parameters to generate a numerical ensemble of branched metallocene-catalyzed polyethylene resins. Application of our algorithm shows the importance of branch-on-branch chains in the system and predicts the linear rheology with good quantitative agreement over a wide range of branching density and molecular weight.

Boundary Integral Equation Analyses of Singular, Potential, and Biharmonic Problems

Content.- 1 - General Introduction.- 2 - An Integral Equation Method for the Solution of Singular Slow Flow Problems.- 3 - Modified Integral Equation Solution of Viscous Flows Near Sharp Corners.- 4 - Solution of Nonlinear Elliptic Equations with Boundary Singularities by an Integral Equation Method.- 5 - Boundary Integral Equation Solution of Viscous Flows with Free Surfaces.- 6 - A Boundary Integral Equation Method for the Study of Slow Flow in Bearings with Arbitrary Geometrics.- 7 - General Conclusions.

An integral equation method for the solution of singular slow flow problems

A biharmonic boundary integral equation (BBIE) method is used to solve a two dimensional contained viscous flow problem. In order to achieve a greater accuracy than is usually possible in this type of method analytic expressions are used for the piecewise integration of all the kernel functions rather than the more time-consuming method of Gaussian quadrature.

On the decay and drift of free-surface perturbations in viscous thin-film flow exterior to a rotating cylinder

Free–surface viscous flow exterior to a circular cylinder rotating about its horizontal axis in a vertical gravitational field is considered. When the mean film thickness is small compared with the cylinder radius a, numerical simulations of the full Stokes equations reveal a surface–amplitude decay rate so slow that computational expense precludes investigation of large–time dynamics. However, numerical integrations of the simpler lubrication approximation are achievable to large times, and these reveal not only a slow decay to steady state, but also a gravity–induced phase lag, relative to the cylinder, in the wave modes in the free–surface elevation. A naive–expansion analysis reveals a complicated evolution in time with four different time–scales. Firstly, there is the fast process of rotating with the cylinder on a time–scale 1/ω, where ω is the angular velocity of the cylinder. Secondly, surface tension squeezes the free surface to a cylindrical shape on the time–scale μa4/s3, where μ is the dynamic viscosity of the fluid and s its surface tension. After this time, disturbances to the steady state take the form of an eccentricity of the cylindrical shape of the free surface that drifts in phase on the third time–scale of μ2a2/ω2g24, where ω is the density of the fluid and g the gravitational acceleration, and decays exponentially on the fourth and slowest time–scale of μ2μ3a6/μ2g2s7. The naive–expansion analysis thus suggests the drift rate and exponential decay rate in the fundamental mode to be proportional to 4 and 7 respectively. A rescaling is suggested wherein is the length–scale, whence a four–term, two–time–scale expansion for the film thickness yields explicit formulae for both the decay and drift rates, the former of which can be used in practical experiments to enforce the fastest possible decay to the steady state. An unusual delay in the resolution of secularity in the two–time–scale expansion is explained, as is the ability of the two–time–scale expansion to capture the four–time–scale physics. Results are presented for a selection of (non–dimensionalized) surface tension and gravity parameters, and excellent agreement is demonstrated between our asymptotics and the results of numerical simulations over varying time–scales. The convergence to the independently derived steady state is demonstrated, and a detailed explanation is presented of the influential physical mechanisms inherent in the multiple–scale expansion.

Steady, viscous, free-surface flow on a rotating cylinder

The commonly observed phenomenon of steady, viscous, free-surface flow on the outer surface of a rotating cylinder is investigated by means of an iterative, integral-equation formulation applied to the Stokes approximation of the Navier-Stokes equations. The method of solution places no restriction on the thickness of the fluid layer residing on the cylinder surface; indeed, results are presented for cases where the layer thickness is of the same order of magnitude as the cylinder radius. Free-surface profiles and free-surface velocity distributions are presented for a range of flow parameters. Where appropriate, comparisons are made with the results of thinfilm theory; excellent agreement is observed. For all film thicknesses and surface tensions, results show a high degree of symmetry about a horizontal axis even though the gravity field is vertical. A proof is presented that, for vanishing surface tension, this is a consequence of the Stokes approximation.

Modified Integral Equation Solution of Viscous Flows Near Sharp Corners

Solutions of the biharmonic equation governing steady two dimensional viscous flow of an incompressible newtonian fluid are obtained by employing a direct biharmonic boundary integral equation (BBIE) method in which Green’s Theorem is used to reformulate the differential equation as a pair of coupled integral equations.

On the stability of viscous free–surface flow supported by a rotating cylinder

Using an adaptive finite–element (FE) scheme developed recently by the authors, we shed new light on the long–standing fundamental problem of the unsteady free–surface Stokes flow exterior to a circular cylinder rotating about its horizontal axis in a vertical gravitational field. For supportable loads, we observe that the steadystate is more readily attained for near–maximal fluid loads on the cylinder than for significantly sub–maximal loads. For the latter, we investigate large–time dynamics by means of a finite–difference approximation to the thin–film equations, which is also used to validate the adaptive FE simulations (applied to the full Stokes equations) for these significantly sub–maximal loads. Conversely, by comparing results of the two methods, we assess the validity of the thin–film approximation as either the load is increased or the rotation rate of the cylinder is decreased. Results are presented on the independent effects of gravity, surface tension and initial film thickness on the decay to steady–state. Finally, new numerical simulations of load shedding are presented.

Finite Element Simulation of Three-Dimensional Free-Surface Flow Problems

An adaptive finite element algorithm is described for the stable solution of three-dimensional free-surface-flow problems based primarily on the use of node movement. The algorithm also includes a discrete remeshing procedure which enhances its accuracy and robustness. The spatial discretisation allows an isoparametric piecewise-quadratic approximation of the domain geometry for accurate resolution of the curved free surface. The technique is illustrated through an implementation for surface-tension-dominated viscous flows modelled in terms of the Stokes equations with suitable boundary conditions on the deforming free surface. Two three-dimensional test problems are used to demonstrate the performance of the method: a liquid bridge problem and the formation of a fluid droplet.

On inertial effects in the Moffatt–Pukhnachov coating-flow problem

The effects are investigated of including inertial terms, in both small- and large-surface-tension limits, in a remodelling of the influential and fundamental problem first formulated by Moffatt and Pukhnachov in 1977: that of viscous thin-film free-surface Stokes flow exterior to a circular cylinder rotating about its horizontal axis in a vertical gravitational field. An analysis of the non-dimensionalizations of previous related literature is made and the precise manner in which different rescalings lead to the asymptotic promotion or demotion of pure-inertial flux terms over gravitational-inertial terms is highlighted. An asymptotic mass-conserving evolution equation for a perturbed-film thickness is derived and solved using two-timescale asymptotics with a strained fast timescale. By using an algebraic manipulator to automate the asymptotics to high orders in the small expansion parameter of the ratio of the film thickness to the cylinder radius, consistent a posteriori truncations are obtained. Via two-timescale and numerical solutions of the evolution equation, new light is shed on diverse effects of inertia in both small- and large-surface-tension limits, in each of which a critical Reynolds number is discovered above which the thin-film evolution equation has no steady-state solution due to the strength of the destabilizing inertial centrifugal force. Extensions of the theory to the treatment of thicker films are discussed.

An integral equation justification of the boundary conditions of the driven-cavity problem

Abstract The driven-cavity problem, a renowned bench-mark problem of computational, incompressible fluid dynamics, is physically unrealistic insofar as the inherent boundary singularities (where the moving lid meets the stationary walls) imply the necessity of an infinite force to drive the flow: this follows from G.I. Taylors analysis of the so-called scaper problem. Using a boundary integral equation (BIE) formulation employing a suitable Greens function, we investigate herein, in the Strokes approximation, the effect of introducing small “leaks” to replace the singularities, thus rendering the problem physically realizable, The BIE approach used here incorporates functional forms of both the asymptotic far-field and singular near-field solution behaviours, in order to improve the accuracy of the numerical solution. Surprisingly, we find that the introduction of the leaks effects notably the global flow field a distance of the order of 100 leak widths away from the leaks. However, we observe that, as the leak width tends to zero, there is exellent agreement between our results and Taylors thus justifying the use of the seemingly unrealizble boundary conditions in the driven-cavity problem. We also discover that the far-field, asymptotic, closed-form solution mentioned above is a remarkably accurate representation of the flow even in the near-field. Several streamline plots, over a range of spatial scales, are presented.