Andrew L. Hazel

Effects of size and shape (aspect ratio) on the hemodynamics of saccular aneurysms: A possible index for surgical treatment of intracranial aneurysms

OBJECTIVE The present study was undertaken to explore the relationship between the characteristic geometry of aneurysms prone to rupture and the blood flow patterns therein, using microsurgically produced aneurysms that simulated human middle cerebral artery aneurysms in scale and shape. METHODS We measured in vivo velocity profiles using our 20-MHz, 80-channel, Doppler ultrasound velocimeter. We produced small (< or =5 mm, 5 cases) and large (6-13 mm, 12 cases) aneurysms with round, dumbbell, or multilobular shapes. RESULTS The fundamental patterns of intra-aneurysmal flow were composed of inflow, circulating flow, and outflow. The inflow, which entered the aneurysm only during the systolic phase, was strongly influenced by the position and size of the neck and the flow ratio into the distal branches. The outflow was usually nonpulsatile and of low velocity. The circulating flow depended on the aspect ratio (depth/neck width). A single recirculation zone was observed in aneurysms with aspect ratios of less than 1.6. This circulation did not seem to extend to areas with aspect ratios greater than this value; in aneurysms with aspect ratios of more than 1.6, a much slower circulation was observed near the dome. Furthermore, in the dome of dumbbell-shaped aneurysms and daughter aneurysms, no flow was detected. Intra-aneurysmal flow was determined by the aspect ratio, rather than the aneurysm size. CONCLUSION The localized, extremely low-flow condition that was observed in the dome of aneurysms with aspect ratios of more than 1.6 is a common flow characteristic in the geometry of ruptured aneurysms, so great care should be taken for patients with unruptured intracranial aneurysms with aspect ratios of more than 1.6.

The steady propagation of a semi-infinite bubble into a tube of elliptical or rectangular cross-section

This paper investigates the propagation of an air finger into a fluid-filled, axially uniform tube of elliptical or rectangular cross-section with transverse length scale a and aspect ratio [alpha]. Gravity is assumed to act parallel to the tubes axis. The problem is studied numerically by a finite-element-based direct solution of the free-surface Stokes equations. In rectangular tubes, our results for the pressure drop across the bubble tip, [Delta]p, are in good agreement with the asymptotic predictions of Wong et al. (1995b) at low values of the capillary number, Ca (ratio of viscous to surface-tension forces). At larger Ca, Wong et al.s (1995b) predictions are found to underestimate [Delta]p. In both elliptical and rectangular tubes, the ratio [Delta]p([alpha])/[Delta]p([alpha] = 1) is approximately independent of Ca and thus equal to the ratio of the static meniscus curvatures. In non-axisymmetric tubes, the air-liquid interface develops a noticeable asymmetry near the bubble tip at all values of the capillary number. The tip asymmetry decays with increasing distance from the bubble tip, but the decay rate becomes very small as Ca increases. For example, in a rectangular tube with [alpha] = 1.5, when Ca = 10, the maximum and minimum finger radii still differ by more than 10% at a distance 100a behind the finger tip. At large Ca the air finger ultimately becomes axisymmetric with radius r[infty infinity]. In this regime, we find that r[infty infinity] in elliptical and rectangular tubes is related to r[infty infinity] in circular and square tubes, respectively, by a simple, empirical scaling law. The scaling has the physical interpretation that for rectangular and elliptical tubes of a given cross-sectional area, the propagation speed of an air finger, which is driven by the injection of air at a constant volumetric rate, is independent of the tubes aspect ratio. For smaller Ca (Ca [alpha]^ dry spots will develop on the tube walls at all values of Ca.

oomph-lib – An Object-Oriented Multi-Physics Finite-Element Library

This paper discusses certain aspects of the design and implementation of oomph-lib, an object-oriented multi-physics finite-element library, available as open-source software at http://www.oomph-lib.org. The main aim of the library is to provide an environment that facilitates the robust, adaptive solution of multi-physics problems by monolithic discretisations, while maximising the potential for code re-use. This is achieved by the extensive use of object-oriented programming techniques, including multiple inheritance, function overloading and template (generic) programming, which allow existing objects to be (re-)used in many different ways without having to change their original implementation. These ideas are illustrated by considering some specific issues that arise when implementing monolithic finite-element discretisations of large-displacement fluidstructure- interaction problems within an Arbitrary Lagrangian Eulerian (ALE) framework. We also discuss the development of wrapper classes that permit the generic and efficient evaluation of the so-called “shape derivatives”, the derivatives of the discretised fluid equations with respect to those solid mechanics degrees of freedom that affect the nodal positions in the fluid mesh. Finally, we apply the methodology in several examples.

Three-dimensional airway reopening: The steady propagation of a semi-infinite bubble into a buckled elastic tube

We consider the steady propagation of an air finger into a buckled elastic tube initially filled with viscous fluid. This study is motivated by the physiological problem of pulmonary airway reopening. The system is modelled using geometrically nonlinear Kirchhoff–Love shell theory coupled to the free-surface Stokes equations. The resulting three-dimensional fluid–structure-interaction problem is solved numerically by a fully coupled finite element method. The system is governed by three dimensionless parameters: (i) the capillary number, Ca=[mu]U/[sigma]*, represents the ratio of viscous to surface-tension forces, where [mu] is the fluid viscosity, U is the fingers propagation speed and [sigma]* is the surface tension at the air–liquid interface; (ii) [sigma]=[sigma]*/(RK) represents the ratio of surface tension to elastic forces, where R is the undeformed radius of the tube and K its bending modulus; and (iii) A[infty infinity]=A*[infty infinity]/(4R2), characterizes the initial degree of tube collapse, where A*[infty infinity] is the cross-sectional area of the tube far ahead of the bubble. The generic behaviour of the system is found to be very similar to that observed in previous two-dimensional models (Gaver et al. 1996; Heil 2000). In particular, we find a two-branch behaviour in the relationship between dimensionless propagation speed, Ca, and dimensionless bubble pressure, p*b/([sigma]*/R). At low Ca, a decrease in p*b is required to increase the propagation speed. We present a simple model that explains this behaviour and why it occurs in both two and three dimensions. At high Ca, p*b increases monotonically with propagation speed and p*b/([sigma]*/R) [is proportional to] Ca for sufficiently large values of [sigma] and Ca. In a frame of reference moving with the finger velocity, an open vortex develops ahead of the bubble tip at low Ca, but as Ca increases, the flow topology changes and the vortex disappears. An increase in dimensional surface tension, [sigma]*, causes an increase in the bubble pressure required to drive the air finger at a given speed; p*b also increases with A*[infty infinity] and higher bubble pressures are required to open less strongly buckled tubes. This unexpected finding could have important physiological ramifications. If [sigma]* is sufficiently small, steady airway reopening can occur when the bubble pressure is lower than the external (pleural) pressure, in which case the airway remains buckled (non-axisymmetric) after the passage of the air finger. Furthermore, we find that the maximum wall shear stresses exerted on the airways during reopening may be large enough to damage the lung tissue.

Steady finite-Reynolds-number flows in three-dimensional collapsible tubes

A fully coupled finite-element method is used to investigate the steady flow of a viscous fluid through a thin-walled elastic tube mounted between two rigid tubes. The steady three-dimensional Navier–Stokes equations are solved simultaneously with the equations of geometrically nonlinear Kirchhoff–Love shell theory. If the transmural (internal minus external) pressure acting on the tube is sufficiently negative then the tube buckles non-axisymmetrically and the subsequent large deformations lead to a strong interaction between the fluid and solid mechanics. The main effect of fluid inertia on the macroscopic behaviour of the system is due to the Bernoulli effect, which induces an additional local pressure drop when the tube buckles and its cross-sectional area is reduced. Thus, the tube collapses more strongly than it would in the absence of fluid inertia. Typical tube shapes and flow fields are presented. In strongly collapsed tubes, at finite values of the Reynolds number, two ’jets‘ develop downstream of the region of strongest collapse and persist for considerable axial distances. For sufficiently high values of the Reynolds number, these jets impact upon the sidewalls and spread azimuthally. The consequent azimuthal transport of momentum dramatically changes the axial velocity profiles, which become approximately

Surface-tension-induced buckling of liquid-lined elastic tubes: a model for pulmonary airway closure

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The Steady Propagation of an Air Finger into a Rectangular Tube

-shaped when the flow enters the rigid downstream pipe. Further convection of momentum causes the development of a ring-shaped velocity profile before the ultimate return to a parabolic profile far downstream.

Alteration of Mean Wall Shear Stress Near an Oscillating Stagnation Point

We use a fully coupled, three-dimensional, finite-element method to study the evolution of the surface-tension-driven instabilities of a liquid layer that lines an elastic tube, a simple model for pulmonary airway closure. The equations of large-displacement shell theory are used to describe the deformations of the tube and are coupled to the Navier–Stokes equations, describing the motion of the liquid. The liquid layer is susceptible to a capillary instability, whereby an initially uniform layer can develop a series of axisymmetric peaks and troughs, analogous to the classical instability that causes liquid jets to break up into droplets. For sufficiently high values of the liquids surface tension, relative to the bending stiffness of the tube, the additional compressive load induced by the development of the axisymmetric instability can induce non-axisymmetric buckling of the tube wall. Once the tube has buckled, a strong destabilizing feedback between the fluid and solid mechanics leads to an extremely rapid further collapse and occlusion of the gas-conveying core of the tube by the liquid. We find that such occlusion is possible even when the volume of the liquid is too small to form an occluding liquid bridge in the axisymmetric tube.

Tube geometry can force switchlike transitions in the behavior of propagating bubbles

The steady propagation of an air finger into a fluid-filled tube of uniform rectangular cross-section is investigated. This paper is primarily focused on the influence of the aspect ratio, α, on the flow properties, but the effects of a transverse gravitational field are also considered. The three-dimensional interfacial problem is solved numerically using the object-oriented multi-physics finite-element library oomph-lib and the results agree with our previous experimental results (de Lo´ zar et al. Phys. Rev. Lett. vol. 99, 2007, article 234501) to within the ±1% experimental error. At a fixed capillary number Ca (ratio of viscous to surface-tension forces) the pressure drops across the finger tip and relative finger widths decrease with increasing α. The dependence of the wet fraction m (the relative quantity of liquid that remains on the tube walls after the propagation of the finger) is more complicated: m decreases with increasing α for low Ca but it increases with α at high Ca. Our results also indicate that the system is approximately quasi-two-dimensional for α 8, when we obtain quantitative agreement with McLean & Saffman’s two-dimensional model for the relative finger width as a function of the governing parameter 1/B =12α2Ca. The action of gravity causes an increase in the pressure drops, finger widths and wet fractions at fixed capillary number. In particular, when the Bond number (ratio of gravitational to surface-tension forces) is greater than one the finger lifts off the bottom wall of the tube leading to dramatic increases in the finger width and wet fraction at a given Ca. For α 3 a previously unobserved flow regime has been identified in which a small recirculation flow is situated in front of the finger tip, shielding it from any contaminants in the flow. In addition, for α 2 the capillary number, Cac, above which global recirculation flows disappear has been observed to follow the simple empirical law: Ca2/3 c α =1.21.

Oscillatory bubbles induced by geometrical constraint

The site opposite an end-to-side anastomosis, resulting from femoral bypass surgery, and the carotid sinus are two regions well known to be prone to fibrous intimal hyperplasia or atherogenesis, respectively. The blood flow at these two sites features a stagnation point, which oscillates in strength and position. Mathematical models are used to determine some of the features of such a flow; in particular, the mean wall shear stress is calculated. The positional oscillations cause a significant change in the distribution and magnitude of the mean wall shear stress from that of the well-studied case of a stagnation point that oscillates only in strength. It is therefore proposed that the recorded effect of time dependence in the flow upon atherogenesis could still be a result of the distribution of the mean and not the time-varying components of the wall shear stress.