Hele-Shaw and Stokes flow with a source or sink : stability of spherical solutions

Qualitative aspects of mathematical models for the dynamics of liquids with a moving boundary are studied. These models describe for instance groundwater flow, extraction of oil, the growth of tumours and viscous sintering in the production of glass. Stability of radially symmetric solutions and decay properties of perturbations are studied for the case that in a single point fluid is injected or extracted. For the motion of the moving boundary a nonlinear non-local evolution equation is derived. The domain is rescaled in such a way that the spherical solution is represented by a stationary solution. Because of this rescaling, the evolution operator is time dependent. The nonlinear stability results are based on linearisation, energy estimates and the principle of linearised stability. The Hele-Shaw model is studied for several boundary conditions, describing various physical situations. In the case of zero pressure on the boundary, it is proved for the injection problem that balls around the injection point are asymptotically stable with respect to small star-shaped perturbations. If surface tension regularisation is included, then balls are stable even for the case of suction under additional assumptions on the initial geometry, suction speed and dimension. Moreover, perturbations turn out to decay algebraically fast. For two dimensional suction, the influence of surface tension dominates the influence of the sink for large time. As a consequence, no condition on the suction speed is necessary. In contrast to the two dimensional problem there is a bound on the suction speed for the 3D problem. In dimensions higher or equal to four the influence of the sink dominates the influence of surface tension. This leads to linear instability for the spherical solution for any suction speed. Making use of the autonomous character of the evolution equation, existence of nontrivial self-similarly vanishing solutions to the three dimensional suction problem with surface tension is proved. These solutions are found as bifurcation solutions from the trivial spherical solution. The suction speed plays the role of bifurcation parameter. Moreover, one branch of bifurcation solutions turns out to be stable with respect to a certain class of perturbations. For the closely related Stokes flow stability of the spherical solution in the case of injection has been proved for dimensions two and three. For the suction problem for these dimensions the spherical solution is linearly unstable

Stability of self-similar extinction solutions for a 3D Hele-Shaw suction problem

We present a stability result for a class of non-trivial self-similarly vanishing solutions to a 3D Hele-Shaw moving boundary problem with surface tension and single-point suction. These solutions are domains that bifurcate from the trivial spherical solution. The moving domains have a geometric centre located at the suction point and they are axially symmetric. We show stability with respect to perturbations that preserve these properties

Perturbation theorems for Hele-Shaw flows and their applications

In this work, we give a perturbation theorem for strong polynomial solutions to the zero surface tension Hele-Shaw equation driven by injection or suction, so called the Polubarinova-Galin equation. This theorem enables us to explore properties of solutions with initial functions close to but are not polynomial. Applications of this theorem are given in the suction or injection case. In the former case, we show that if the initial domain is close to a disk, most of fluid will be sucked before the strong solution blows up. In the later case, we obtain precise large-time rescaling behaviors for large data to Hele-Shaw flows in terms of invariant Richardson complex moments. This rescaling behavior result generalizes a recent result regarding large-time rescaling behavior for small data in terms of moments. As a byproduct of a theorem in this paper, a short proof of existence and uniqueness of strong solutions to the Polubarinova-Galin equation is given.Comment: 25 page

Some studies on the deformation of the membrane in an RF MEMS switch

Radio Frequency (RF) switches of Micro Electro Mechanical Systems (MEMS) are appealing to the mobile industry because of their energy efficiency and ability to accommodate more frequency bands. However, the electromechanical coupling of the electrical circuit to the mechanical components in RF MEMS switches is not fully understood. In this paper, we consider the problem of mechanical deformation of electrodes in RF MEMS switch due to the electrostatic forces caused by the difference in voltage between the electrodes. It is known from previous studies of this problem, that the solution exhibits multiple deformation states for a given electrostatic force. Subsequently, the capacity of the switch that depends on the deformation of electrodes displays a hysteresis behaviour against the voltage in the switch. We investigate the present problem along two lines of attack. First, we solve for the deformation states of electrodes using numerical methods such as finite difference and shooting methods. Subsequently, a relationship between capacity and voltage of the RF MEMS switch is constructed. The solutions obtained are exemplified using the continuation and bifurcation package AUTO. Second, we focus on the analytical methods for a simplified version of the problem and on the stability analysis for the solutions of deformation states. The stability analysis shows that there exists a continuous path of equilibrium deformation states between the open and closed state

Determinants of postnatal spleen tissue regeneration and organogenesis

Abstract The spleen is an organ that filters the blood and is responsible for generating blood-borne immune responses. It is also an organ with a remarkable capacity to regenerate. Techniques for splenic auto-transplantation have emerged to take advantage of this characteristic and rebuild spleen tissue in individuals undergoing splenectomy. While this procedure has been performed for decades, the underlying mechanisms controlling spleen regeneration have remained elusive. Insights into secondary lymphoid organogenesis and the roles of stromal organiser cells and lymphotoxin signalling in lymph node development have helped reveal similar requirements for spleen regeneration. These factors are now considered in the regulation of embryonic and postnatal spleen formation, and in the establishment of mature white pulp and marginal zone compartments which are essential for spleen-mediated immunity. A greater understanding of the cellular and molecular mechanisms which control spleen development will assist in the design of more precise and efficient tissue grafting methods for spleen regeneration on demand. Regeneration of organs which harbour functional white pulp tissue will also offer novel opportunities for effective immunotherapy against cancer as well as infectious diseases

Biochemical and structural characterization of mycobacterial aspartyl-tRNA synthetase AspS, a promising TB drug target.

The human pathogen Mycobacterium tuberculosis is the causative agent of pulmonary tuberculosis (TB), a disease with high worldwide mortality rates. Current treatment programs are under significant threat from multi-drug and extensively-drug resistant strains of M. tuberculosis, and it is essential to identify new inhibitors and their targets. We generated spontaneous resistant mutants in Mycobacterium bovis BCG in the presence of 10× the minimum inhibitory concentration (MIC) of compound 1, a previously identified potent inhibitor of mycobacterial growth in culture. Whole genome sequencing of two resistant mutants revealed in one case a single nucleotide polymorphism in the gene aspS at 535GAC>535AAC (D179N), while in the second mutant a single nucleotide polymorphism was identified upstream of the aspS promoter region. We probed whole cell target engagement by overexpressing either M. bovis BCG aspS or Mycobacterium smegmatis aspS, which resulted in a ten-fold and greater than ten-fold increase, respectively, of the MIC against compound 1. To analyse the impact of inhibitor 1 on M. tuberculosis AspS (Mt-AspS) activity we over-expressed, purified and characterised the kinetics of this enzyme using a robust tRNA-independent assay adapted to a high-throughput screening format. Finally, to aid hit-to-lead optimization, the crystal structure of apo M. smegmatis AspS was determined to a resolution of 2.4 Å