Metrics and undirected cuts

Suppose thatG is an undirected graph whose edges have nonnegative integer-valued lengthsl(e), and that {s ₁,t ₁},?, {s _m ,t _m } are pairs of its vertices. Can one assign nonnegative weights to the cuts ofG such that, for each edgee, the total weight of cuts containinge does not exceedl(e) and, for eachi, the total weight of cuts ‘separating’s _i andt _i is equal to the distance (with respect tol) betweens _i andt _i ? Using linear programming duality, it follows from Papernov's multicommodity flow theorem that the answer is affirmative if the graph induced by the pairs {s ₁,t ₁},?, {s _m ,t _m } is one of the following: (i) the complete graph with four vertices, (ii) the circuit with five vertices, (iii) a union of two stars. We prove that if, in addition, each circuit inG has an even length (with respect tol) then there exists a suitable weighting of the cuts with the weights integer-valued; moreover, an algorithm of complexity O(n ³) (n is the number of vertices ofG) is developed for solving such a problem. Also a class of metrics decomposable into a nonnegative linear combination of cut-metrics is described, and it is shown that the separation problem for cut cones isNP-hard.     

On a family of Stokes flows

A family of simple Stokes flows involving sliding surfaces adjacent to surfaces at rest is considered. Principally, two specific flow configurations are investigated: (i) that arising when parts of the boundary of an infinitely long circular cylinder are rotating about the axis while other parts of the boundary are at rest, and (ii) the flow produced when a cap of a sphere is held at rest while the remainder of the sphere rotates about the symmetry axis. In each case computer plots of streamlines or constant velocity lines are presented to give a general impression of the resulting flow pattern.     

Prediction of periodic boundary layers

A relatively simple, yet efficient and accurate finite difference method is developed for the solution of the unsteady boundary layer equations for both laminar and turbulent flows. The numerical procedure is subjected to rigorous validation tests in the laminar case, comparing its predictions with exact analytical solutions, asymptotic solutions, and/or experimental results. Calculations of periodic laminar boundary layers are performed from low to very high oscillation frequencies, for small and large amplitudes, for zero as well as adverse time-mean pressure gradients, and even in the presence of significant flow reversal. The numerical method is then applied to predict a relatively simple experimental periodic turbulent boundary layer, using two well-known quasi-steady closure models. The predictions are shown to be in good agreement with the measurements, thereby demonstrating the suitability of the present numerical scheme for handling periodic turbulent boundary layers. The method is thus a useful tool for the further development of turbulence models for more complex unsteady flows.     

Coating flow theory by finite element and asymptotic analysis of the navier-stokes system

Coating flows are laminar free surface flows, preferably steady and two-dimensional, by which a liquid film is deposited on a substrate. Their theory rests on mass and momentum accounting for which Galerkin's weighted residual method, finite element basis functions, isoparametric mappings, and a new free surface parametrization prove particularly well-suited, especially in coping with the highly deformed free boundaries, irregular flow domains, and the singular nature of static and dynamic contact lines where fluid interfaces intersect solid surfaces. Typically, short forming zones of rapidly rearranging two-dimensional flow merge with simpler asymptotic regimes of developing or developed flow upstream and downstream. The two-dimensional computational domain can be shrunk in size by imposing boundary conditions from asymptotic analysis of those regimes or by matching to one-dimensional finite element solutions of asymptotic equations. The theory is laid out with special attention to conditions at free surfaces, contact lines, and open inflow and outflow boundaries. Efficient computation of predictions is described with emphasis on a grand Newton iteration that converges rapidly and brings other benefits. Sample results for curtain coating and roll coating flows of Newtonian liquids illustrate the power and effectiveness of the theory.     

A quadratic network optimization model for equilibrium single commodity trade flows

When supply and demand curves for a single commodity are approximately linear in each ofN regions and interregional transportation costs are linear, then equilibrium trade flows can be computed by solving a quadratic program of special structure. An equilibrium trade flow exists in which the routes carrying positive flow form a forest, and this solution can be efficiently computed by a tree growing algorithm.     

On predicting unsteady non-Newtonian blood flow

The influence of the non-Newtonian stress–strain relation of blood on the oscillatory shear index (OSI) and mean wall shear stress (WSS) are described. A mathematical non-dimensional model based on the momentum equation for a modified Casson’s fluid is formulated in terms of the dimensionless yield shear stress . An original direct numerical procedure is presented to predict the flow patterns. Results obtained by using a finite difference approach show a difference in OSI when blood is assumed to be a Newtonian fluid instead of a modified Casson’s fluid. The calculation of the OSI in human normal conditions under the Newtonian approach differs in 5% from the result obtained from using the Casson model.     

Simulation of polymeric flows in the injection moulding process

Recent progress in the simulation of polymeric flows of two key problems in the injection moulding process, carried out by a team at Cornell University, is briefly described. For the filling of cooled thin cavities, the fluid is characterized by a power-law viscosity with exponential temperature dependence, and interaction between the transient thermal boundary-layer and the core flow in a domain with moving boundary is essential. The earlier procedure of Hieber and Shen is modified in two aspects: a boundary-integral formulation replaces the finite-element treatment of the pressure, and an ‘energy integral’ approach is used for the transient temperature. The second problem is the steady visco-elastic flow in the juncture region where sudden changes of the geometry and large strain rates occur. The constitutive equation is postulated according to the Leonov model. The main features in the numerical implementation are: integration along a streamline to determine the elastic deformation tensors for a given velocity field, and finite-element treatment (in time-dependent form) of the pressure and fields for given stresses. In an example where the contraction ratio is 7:1, results for nominal Deborah number exceeding 100 show no numerical instability. (However, for this problem, the true Weissenberg number, i.e. the ratio of local first-normal-stress difference to shear stress turns out to be generally O(10).) The predictions also correlate very well with experimental birefringence measurements.     

Topological regular variation: I. Slow variation

Motivated by the Category Embedding Theorem, as applied to convergent automorphisms (Bingham and Ostaszewski (in press) [11]), we unify and extend the multivariate regular variation literature by a reformulation in the language of topological dynamics. Here the natural setting are metric groups, seen as normed groups (mimicking normed vector spaces). We briefly study their properties as a preliminary to establishing that the Uniform Convergence Theorem (UCT) for Baire, group-valued slowly-varying functions has two natural metric generalizations linked by the natural duality between a homogenous space and its group of homeomorphisms. Each is derivable from the other by duality. One of these explicitly extends the (topological) group version of UCT due to Bajšanski and Karamata (1969) [4] from groups to flows on a group. A multiplicative representation of the flow derived in Ostaszewski (2010) [45] demonstrates equivalence of the flow with the earlier group formulation. In companion papers we extend the theory to regularly varying functions: we establish the calculus of regular variation in Bingham and Ostaszewski (2010) [13] and we extend to locally compact, σ-compact groups the fundamental theorems on characterization and representation (Bingham and Ostaszewski (2010) [14]). In Bingham and Ostaszewski (2009) [15], working with topological flows on homogeneous spaces, we identify an index of regular variation, which in a normed-vector space context may be specified using the Riesz representation theorem, and in a locally compact group setting may be connected with Haar measure.     

Well Posedness in any Dimension for Hamiltonian Flows with Non BV Force Terms

We study existence and uniqueness for the classical dynamics of a particle in a force field in the phase space. Through an explicit control on the regularity of the trajectories, we show that this is well posed if the force belongs to the Sobolev space H ^3/4.     

Travelling waves in dilatant non-Newtonian thin films

We prove the existence of a travelling wave solution for a gravity-driven thin film of a viscous and incompressible dilatant fluid coated with an insoluble surfactant. The governing system of second order partial differential equations for the film's height h and the surfactant's concentration γ are derived by means of lubrication theory applied to the non-Newtonian Navier–Stokes equations.