Brian R. Seymour

Fluid mechanics of stenosed arteries

Abstract A finite difference scheme is used to investigate flow in a tube with an occlusion. The results are interpreted in the context of blood flow in stenosed arteries. Numerical results for steady and pulsatile flows confirm, in a quantitative sense, that a high shear stress is not likely to initiate atherosclerosis lesions. The study of unsteady flow reveals several interesting new features. It appears that there is a correlation between regions of recirculation, which are a prominent feature of the unsteady flow, and the location of lesions. Experimental measurements for steady flow complement the numerical study and show qualitative agreement.

IV. – Nonlinear Geometrical Acoustics

Abstract : The development of the theory of nonlinear wave propagation in both bounded and semi-infinite dissipative media is followed from its origins in the theories of linear geometrical acoustics, simple waves, and acceleration fronts. In Part I, Sections 2 to 5, we consider examples in which only one component wave is excited and describe the effects of three types of dissipative mechanisms on unidirectional waves. In Part II, Sections 6 to 9 we consider nonlinear waves in media of finite extent. In general, more than one component wave is excited. The coupling between the interaction and distortion of the different components is described. The effect of radiation from the boundaries is included for both transient and forced, time-periodic motions.

Pulse Propagation in a Nonlinear Viscoelastic Rod of Finite Length

The problem of impact on a, nonlinear viscoelastic rod of finite length is considered. It is shown that, in the high frequency or geometrical acoustics limit, the disturbance in the rod may be represented as the superposition of two modulated simple waves traveling in opposite directions which do not interact in the body of the material. Based on this result the impact and initial boundary value problems are reduced to finding the solution of a nonlinear difference equation, which is solved exactly. We examine the competing effects of amplitude dispersion, which may produce shocks, and dissipation and give conditions under which each dominates. The corresponding results for an inhomogeneous rod are also given.

A note on the resonant interaction between a surface wave and two interfacial waves

Hill & Foda (1998) and Jamali (1998) have presented theoretical and experimental studies of the resonant interaction between a surface wave and two oblique interfacial waves. Despite many similarities between the findings there is one seemingly major difference. Hill & Fodas (1998) analysis indicated that there are only narrow bands of frequency, density ratio and direction angle within which growth is possible. On the other hand, Jamali (1998) predicted and observed wave growth over wide ranges of frequency and direction angle, and for all the density ratios that he investigated. We show that Hill & Fodas (1998) second-order representation of the dynamic interfacial boundary condition is missing a term proportional to the time derivative of the square of the velocity shear across the interface. When this missing term is included in the analysis, the resulting predictions are consistent with the laboratory experiments.

Viscoplastic fluid displacements in horizontal narrow eccentric annuli : stratification and travelling wave solutions

We consider laminar displacement flows in narrow eccentric annuli, oriented horizontally, between two fluids of Herschel–Bulkley type, (i.e. including Newtonian, power-law and Bingham models). This situation is modelled via a Hele-Shaw approach. Whereas slumping and stratification would be expected in the absence of any imposed flow rate, for a displacement flow we show that there are often steady-state travelling wave solutions in this displacement. These may exist even at large eccentricities and for large density differences between the fluids. When heavy fluids displace light fluids, annular eccentricity opposes buoyancy and steady states are more prevalent than when light fluids displace heavy fluids. For large ratios of buoyancy forces to viscous forces we derive a lubrication-style displacement model. This simplification allows us to find necessary and sufficient conditions under which a displacement can be steady, which can be expressed conveniently in terms of a consistency ratio. It is interesting that buoyancy does not appear in the critical conditions for a horizontal well. Instead a competition between fluid rheologies and eccentricity is the determining factor. Buoyancy acts only to determine the axial length of the steady-state profile.

Asymptotic analysis of a surface-interfacial wave interaction

The three-dimensional interaction of a surface wave with two oblique interfacial waves in a horizontally infinite two-layer fluid is analyzed asymptotically. The nondimensional density difference is taken as a perturbation parameter and simple expressions for the growth rates and kinematic properties of the waves are obtained. The results show that the interfacial wavelengths are an order smaller than the surface wavelength. Also, to the leading-order approximation, the interfacial waves have a frequency half that of the surface wave, and their directions differ by 180° in the horizontal plane. The interaction coefficients are found to be equal at the leading order. The asymptotic solution is compared with the exact solution, and an excellent agreement is obtained for the range of applicability of the asymptotic theory. The analysis is extended to interactions in a medium with sidewalls. A previous laboratory flume study is addressed, and the asymptotic theory is used to explain the experimental observations.

Transient soil response in a porous seabed with variable permeability

The soil permeability of many natural marine sediments decreases with depth because of consolidation under overburden pressure. This is accompanied by a decrease in porosity and void ratio that also affect the permeability. Conventional theories for wave-induced soil response have assumed a homogeneous porous seabed. This paper presents a new approach for the wave-induced response in a soil matrix, with variable permeability as a function of burial depth. The soil matrix considered is unsaturated and anisotropic, and is subject to a three-dimensional wave system. The pore pressure and effective stresses induced by such a system are obtained from a set of equations incorporating a variable permeability. Verification is available through reduction to the simple case of uniform permeability. The results indicate that the effect of variable soil permeability on pore pressure and vertical effective stress may be significant, especially in a gravelled seabed and for unsaturated sandy soils.

Exact Solutions Describing Soliton-Like Interactions in a Nondispersive Medium

When two solitons collide they interact locally but emerge unscattered from the interaction. In this paper we show that there are certain media, whose responses are governed by the nonlinear, nondispersive, wave equation, in which any two pulses travelling in opposite directions interact nonlinearly for a finite time when they collide but then part unaffected by the interaction. For these media thedependence of the wave speed on the amplitude of the disturbance, although special, is not pathological. In fact, by a suitable choice of the constants in the expression for the sound speed, the responses of many real materials can be modeled. A concise representation is obtained for the exact solution of the initial value problem in terms of the solution of a first order system of linear ordinary differential equations. An explicit solution is found for piecewise linear initial data.

Nonlinear resonant oscillations in open tubes

A gas in a tube, one end of which is open, is driven by a periodic applied velocity or pressure at or near a resonant frequency. Damping is introduced into the system by radiation of energy through the open end. It is shown that shocks are possible at an open end and that there is a critical level of damping which ensures a continuous gas response for all frequencies. At the critical level the amplitude of the response is O (e 1/3 ), where e is the amplitude of the input, and it is bounded by the amplitude predicted by linear theory. There is agreement with the qualitative experimental results available.

Nonlinear resonant oscillations in closed tubes of variable cross-section

An axisymmetric tube with a variable cross-sectional area, closed at both ends, containing a polytropic gas is oscillated parallel to its axis at or near a resonant frequency. The resonant gas oscillations in an equivalent tube of constant cross-section contain shocks. We show how cone, horn and bulb resonators produce shockless periodic outputs. The output consists of a dominant fundamental mode, where its amplitude and detuning are connected by a cubic equation - the amplitude-frequency relation. For the same gas, a cone resonator exhibits a hardening behaviour, while a bulb resonator may exhibit a hardening or softening behaviour. These theoretical results agree qualitatively with available experimental results and are the basis for resonant macrosonic synthesis (RMS).