Numerical study of the collapse of a bubble subjected to a lithotripter pulse

The collapse of a bubble subjected to a lithotripter pulse is studied numerically. The goal is to record the pressure exerted along the stone, as a measure of potential stone damage. It is found that the pressure due to buble collapse is much larger than that of the lithotripter pulse. Furthermore, the pressure greatly depends on the geometry of the problem (initial stand-off distance and bubble size) and on the properties of the pulse (amplitude and width)

Shock-induced collapse of a gas bubble in shockwave lithotripsy

The shock-induced collapse of a pre-existing nucleus near a solid surface in the focal region of a lithotripter is investigated. The entire flow field of the collapse of a single gas bubble subjected to a lithotripter pulse is simulated using a high-order accurate shock- and interface-capturing scheme, and the wall pressure is considered as an indication of potential damage. Results from the computations show the same qualitative behavior as that observed in experiments: a re-entrant jet forms in the direction of propagation of the pulse and penetrates the bubble during collapse, ultimately hitting the distal side and generating a water-hammer shock. As a result of the propagation of this wave, wall pressures on the order of 1 GPa may be achieved for bubbles collapsing close to the wall. The wall pressure decreases with initial stand-off distance and pulse width and increases with pulse amplitude. For the stand-off distances considered in the present work, the wall pressure due to bubble collapse is larger than that due to the incoming shockwave; the region over which this holds may extend to ten initial radii. The present results indicate that shock-induced collapse is a mechanism with high potential for damage in shockwave lithotripsy

Numerical simulations of non-spherical bubble collapse

A high-order accurate shock- and interface-capturing scheme is used to simulate the collapse of a gas bubble in water. In order to better understand the damage caused by collapsing bubbles, the dynamics of the shock-induced and Rayleigh collapse of a bubble near a planar rigid surface and in a free field are analysed. Collapse times, bubble displacements, interfacial velocities and surface pressures are quantified as a function of the pressure ratio driving the collapse and of the initial bubble stand-off distance from the wall; these quantities are compared to the available theory and experiments and show good agreement with the data for both the bubble dynamics and the propagation of the shock emitted upon the collapse. Non-spherical collapse involves the formation of a re-entrant jet directed towards the wall or in the direction of propagation of the incoming shock. In shock-induced collapse, very high jet velocities can be achieved, and the finite time for shock propagation through the bubble may be non-negligible compared to the collapse time for the pressure ratios of interest. Several types of shock waves are generated during the collapse, including precursor and water-hammer shocks that arise from the re-entrant jet formation and its impact upon the distal side of the bubble, respectively. The water-hammer shock can generate very high pressures on the wall, far exceeding those from the incident shock. The potential damage to the neighbouring surface is quantified by measuring the wall pressure. The range of stand-off distances and the surface area for which amplification of the incident shock due to bubble collapse occurs is determined

Non-spherical collapse of an air bubble subjected to a lithotripter pulse

In order to better understand the contribution of bubble collapse to stone comminution in shockwave lithotripsy, the shockinduced and Rayleigh collapse of a spherical air bubble is investigated using numerical simulations, and the free-field collapse of a cavitation bubble is studied experimentally. In shock-induced collapse near a wall, it is found that the presence of the bubble greatly amplifies the pressure recorded at the stone surface; the functional dependence of the wall pressure on the initial standoff distance and the amplitude are presented. In Rayleigh collapse near a solid surface, the proximity of the wall retards the flow and leads to a more prominent jet. Experiments show that re-entrant jets form in the collapse of cavitation bubbles excited by lithotripter shockwaves in a fashion comparable to previous studies of collapse near a solid surface

Turbulence diffusion effects at material interfaces, with application to the Rayleigh-Taylor instability

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/106473/1/AIAA2013-3121.pd

A Simple Method to Improve the Accuracy of Advection in Discontinuous Galerkin Methods for Navier-Stokes Simulations

Peer Reviewedhttps://deepblue.lib.umich.edu/bitstream/2027.42/140426/1/6.2014-1276.pd

A New Family of Discontinuous Galerkin Schemes for Diffusion Problems

Peer Reviewedhttps://deepblue.lib.umich.edu/bitstream/2027.42/143057/1/6.2017-3444.pd

Corporate Governance, Executive Compensation and Securities Litigation

It is generally accepted that good corporate governance, executive compensation and the threat of litigation are all important mechanisms for incentivizing managers of public corporations. While there are significant and robust literatures analyzing each of these policy instruments in isolation, their mutual relationship and interaction has received somewhat less attention. Such neglect is mildly surprising in light of a strong intuition that the three devices are structurally related to one another (either as complements or substitutes). In this paper, we construct an agency cost model of the firm in which corporate governance protections, executive compensation levels, and litigation incentives are all endogenously determined. We then test the predictions of the model using a firm-level data set including governance, executive compensation, and securities litigation variables. Consistent with our predictions, we find governance and compensation to be structural substitutes with one another, so that more protective governance structures tend to coincide with lower-powered incentives in executive contracts. Also consistent with our predictions, we find executive compensation and shareholder litigation appear to be structural complements to one another, so that higher powered incentives tend to catalyze more frequent litigation. In fact, we estimate that each 1% increase in the incentive component of a CEO\u27s contract predicts 0.3% increase in the likelihood of a securities class action and a $3.4 million dollar increase in expected settlement costs. In addition, the complementarity of executive compensation and litigation allows us to formulate new ways to test for the effects of legal reform, such as the Private Securities Litigation Reform Act of 1995. The results of our preliminary tests appear inconsistent with the claims of the statute\u27s proponents that the PSLRA systematically discouraged non-meritorious litigation without burdening meritorious claims, particularly for firms with relatively low volatility

Damage potential of the shock-induced collapse of a gas bubble

Numerical simulations are used to evaluate the damage potential of the shock-induced collapse of a pre-existing gas bubble near a rigid surface. In the context of shock wave lithotripsy, a medical procedure where focused shock waves are used to pulverize kidney stones, shock-induced bubble collapse represents a potential mechanism by which the shock energy directed at the stone may be amplified and concentrated. First the bubble dynamics of shock-induced collapse are discussed. As an indication of the damage potential, the wall pressure is considered. It is found that, for bubbles initially close to the wall, local pressures greater than 1 GPa are achieved. For larger stand-off distances, the wall pressure is inversely proportional to the location of bubble collapse. From this relationship, it is found that bubbles within a certain initial stand-off distance from the wall amplify the pressure of the incoming shock. Furthermore, the extent along the wall over which the pressure due to bubble collapse is higher than that of the pulse is estimated. In addition, the present computational fluid dynamics simulations are used as input into an elastic waves propagation code, in order to investigate the stresses generated within kidney stone in the context of shock wave lithotripsy. The present work shows that the shock-induced collapse of a gas bubble has potential not only for erosion along the stone surface, but also for structural damage within the stone due to internal wave reflection and interference.http://deepblue.lib.umich.edu/bitstream/2027.42/84221/1/CAV2009-final177.pd