1,721,055 research outputs found
Mixed systems of conservation laws in industrial mathematical modelling motion
No full textMany mathematical models of evolutionary industrial processes may be written as N x N systems of conservation laws in terms of N independent variables comprising time and N-1 space variables. If such systems posses real and distinct eigenvalues, they are said to be strictly hyperbolic. For "mixed systems", however, the eigenvalues may be equal at points in phase space or even fail to be real, so that the problem has both hyperbolic and elliptic characteristics. In this case the system is ill-posed and requires the specification of boundary conditions that can violate causality. Mathematical models of physical processes that lead to mixed equations are discussed and reviewed, and some of the properties of mixed systems are compared to those of hyperbolic systems. The significance of prototype systems that have been proposed specifically to analyse such properties is considered, and attention is then turned to the archetypal mixed system; the two-phase flow equations. Possible resolutions of the two-phase flow dilemma are compared, and a manner in which the modelling may be approached via a more general rational asymptotic scheme is indicated
Determining the viscosity of a carbon paste used in smelting
No full textIndustrial Mathematics is growing enormously in popularity around the world. This book deals with real industrial problems from real industries. Presented as a series of case studies by some of the world's most active and successful industrial mathematicians, this volume shows clearly how the process of mathematical collaboration with industry can not only work successfully for the industrial partner, but also lead to interesting and important mathematics. The book begins with a brief introduction, where the equations that most of the studies are based upon are summarized. Thirteen different problems are then considered, ranging from the cooking of cereal to the analysis of epidemic waves in animal populations. Throughout the work the emphasis is on telling industry what they really want to know. This book is suitable for all final year undergraduates, master's students, and Ph.D. students who are working on practical mathematical modeling
A high-Reynolds-number cross-flow with injection and suction
No full textThe effect upon a high-Reynolds-number cross-flow of an upstream injection slot and a downstream suction slot of given geometries and strengths is examined. It is shown that the problem may be reduced to a single nonlinear singular integrodifferential equation. It transpires that, in the resulting flow, a total of five different regimes may be identified. For critical suction, the suction strength is just sufficient to reingest all of the previously injected flow. For weaker suctions, the flow is either subcritical, in which case the injected flow that cannot be reingested forms a layer downstream of the suction slot, or subsubcritical, in which case the lowpressure region produced by the injection is of sufficient strength that the ‘suction’ slot exhausts, rather than ingests, fluid. For suctions stronger than the critical value, the flow is supercritical, and the suction slot ingests some of the cross-flow as well as the previously injected flow, leading to an order-of-magnitude increase in the mass flow into the slot. Finally, for supersupercritical flow, when the suction strength is an order of magnitude larger than in cases previously considered, the injection slot is effectively absent and the mass flow into the slot once again jumps by an order of magnitude. In each case the equation governing the flow is solved asymptotically and numerically. Some limiting cases are also identified in which closed-form solutions may be determined
Film cooling effectiveness for subsonic slot injection into a cross flow
No full textA model is developed that allows the prediction of the film cooling effectiveness produced by slot injection into a uniform cross flow. The model relies on the fact that when the slot pressure exceeds the cross flow pressure by a small amount only so that injection is weak, the resulting small parameter may be exploited to solve the flow problem. The energy equation for the flow may then be solved to determine how much protection cold gas injection gives to the wall downstream of a slot. Although the leading order energy equation must be solved numerically, a simple asymptotic expression may also be derived to allow predictions of heat transfer at large distances from the injection slot
De-icing by slot injection
No full textWe consider the removal of ice from a plate in a cold cross flow by injection of hot fluid through a slot in the plate. De-icing of this sort is required in a number of diverse industrial scenarios, and is particularly relevant to the aviation industry, where the presence of ice on aircraft wings is a major safety hazard. Thin aerofoil theory is used to determine the flow above the injected fluid layer, and this is coupled to flow and energy equations in the injected layer and the ice. The key non-dimensional parameters and ratios in the problem are identified. The result is a nonlinear singular integro-differential equation which is coupled to a convection/diffusion equation and a Stefan condition. Some special cases are discussed and some asymptotic limits are identified. The problem is then solved numerically, and results for a number of different cases are presented
The unsteady motion of two-dimensional flags with bending stiffness
No full textThe motion of a two-dimensional flag at a time-dependent angle of incidence to an irrotational flow of an inviscid, incompressible fluid is examined. The flag is modelled as a thin, flexible, impermeable membrane of finite mass with bending stiffness. The flag is fixed at the leading edge where it is assumed to be either freely hinged or clamped with zero gradient. The angle of incidence to the outer flow is assumed to be small and thin aerofoil theory and simple beam theory are employed to obtain a partial singular integro-differential equation for the flag shape. Steady solutions to the problem are calculated analytically for various limiting cases and numerically for order one values of a non-dimensional parameter that measures the relative importance of outer flow momentum flux and flexural rigidity. For the unsteady problem, the stability of steady solutions depends only upon two non-dimensional parameters. Stability analysis is performed in order to identify the regions of instability. The resulting quadratic eigenvalue problem is solved numerically and the marginal stability curves for both the hinged and the clamped flags are constructed. These curves show that both stable and unstable solutions may exist for various values of the mass and flexural rigidity of the membrane and for both methods of attachment at the leading edge. In order to confirm the results of the linear stability analysis, the full unsteady flag equation is solved numerically using an explicit method. The numerical solutions agree with the predictions of the linear stability analysis and also identify the shapes that the flag adopts according to the magnitude of the flexural rigidity and mass
Modelling drainage of the precorneal tear film after a blink
No full textWe study the drainage of the precorneal tear film in humans. A fluid dynamic model for the drainage of the aqueous layer is developed that includes the effects of evaporation and gravity. The model may be reduced to a single nonlinear partial differential equation for the thickness of the aqueous layer. The equation is solved numerically and accurate times for film rupture are obtained for physically realistic parameters. The results indicate that although gravity and evaporation are not the most dominant effects in some parts of the film, they can nevertheless materially affect the film drainage process and should therefore be included in models for tear film drainage. <br/
Simultaneous measurements of conductivity and thickness for polymer electrolyte films: a simulation study
No full textPlanar cells provide an attractive alternative to traditional cylindrical cells for polymer electrolyte conductivity measurement as they not only allow sample quantity minimisation but also equilibration with vapour. They are also a key component in the parallel screening of combinatorially synthesised arrays of samples. A finite element simulation technique is used to calculate the complex cell constants of planar cells for a wide range of cell geometries. The real part and the phase of the cell impedance at the high frequency limit are identified as the most sensitive indicators of the polymer thickness, and thereby the cell constant, through which the measured impedance value yields the conductivity. Cell geometries are identified for which the conductivity may be determined in an optimally accurate fashion
On the unsteady motion of two-dimensional sails
No full textAn equation is derived to describe the motion of a two-dimensional inextensible sail at a small, time-dependent, angle of incidence to a uniform two-dimensional flow. The equation derived is a singular partial integro-differential equation, which in the steady case reduces to the sail equation of Voelz. A number of limiting versions of the equation are derived and analysed for cases where the relative mass of the sail is large or small. For general unsteady sail motions the governing equation must be solved numerically. A scheme is proposed that employs Chebyshev polynomials to approximate the position of the sail; ordinary differential equations are derived to determine the relevant Chebyshev coefficients and a number of examples are illustrated and discussed. It is found that in some cases where the angle of attack changes sign the tension may become large; in these instances the underlying physical assumptions of the model may be violated
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