Strong confinement of active microalgae leads to inversion of vortex flow and enhanced mixing

Microorganisms swimming through viscous fluids imprint their propulsion mechanisms in the flow fields they generate. Extreme confinement of these swimmers between rigid boundaries often arises in natural and technological contexts, yet measurements of their mechanics in this regime are absent. Here, we show that strongly confining the microalga Chlamydomonas between two parallel plates not only inhibits its motility through contact friction with the walls but also leads, for purely mechanical reasons, to inversion of the surrounding vortex flows. Insights from the experiment lead to a simplified theoretical description of flow fields based on a quasi-2D Brinkman approximation to the Stokes equation rather than the usual method of images. We argue that this vortex flow inversion provides the advantage of enhanced fluid mixing despite higher friction. Overall, our results offer a comprehensive framework for analyzing the collective flows of strongly confined swimmers.

Using Experimentally Calibrated Regularized Stokeslets to Assess Bacterial Flagellar Motility Near a Surface

The presence of a nearby boundary is likely to be important in the life cycle and evolution of motile flagellate bacteria. This has led many authors to employ numerical simulations to model near-surface bacterial motion and compute hydrodynamic boundary effects. A common choice has been the method of images for regularized Stokeslets (MIRS); however, the method requires discretization sizes and regularization parameters that are not specified by any theory. To determine appropriate regularization parameters for given discretization choices in MIRS, we conducted dynamically similar macroscopic experiments and fit the simulations to the data. In the experiments, we measured the torque on cylinders and helices of different wavelengths as they rotated in a viscous fluid at various distances to a boundary. We found that differences between experiments and optimized simulations were less than 5% when using surface discretizations for cylinders and centerline discretizations for helices. Having determined optimal regularization parameters, we used MIRS to simulate an idealized free-swimming bacterium constructed of a cylindrical cell body and a helical flagellum moving near a boundary. We assessed the swimming performance of many bacterial morphologies by computing swimming speed, motor rotation rate, Purcell’s propulsive efficiency, energy cost per swimming distance, and a new metabolic energy cost defined to be the energy cost per body mass per swimming distance. All five measures predicted that the optimal flagellar wavelength is eight times the helical radius independently of body size and surface proximity. Although the measures disagreed on the optimal body size, they all predicted that body size is an important factor in the energy cost of bacterial motility near and far from a surface.

The method of images in the pricing of barrier derivatives in three dimensions

The thesis describes the joint distributions of minima, maxima and endpoint values for a three dimensional Wiener process. In particular, the results provide the point cumulative distributions for the maxima and/or minima of the components of the process. The densities are obtained explicitly for special type of correlations by the method of images; the analysis requires a detailed study of partitions of the sphere by means of spherical triangles. The joint densities obtained can be used to obtain explicit expressions for price of options in financial mathematics. We provide closed-form expressions for the price of several barrier type derivatives with a three dimensional geometric Wiener process as underlying. These solutions are found for special correlation matrices and are given by linear combinations of three dimensional Gaussian cumulative distributions. In order to extend the results to a wider set of correlation matrices the method of random correlations is outlined.

The method of images in the pricing of barrier derivatives in three dimensions

The thesis describes the joint distributions of minima, maxima and endpoint values for a three dimensional Wiener process. In particular, the results provide the point cumulative distributions for the maxima and/or minima of the components of the process. The densities are obtained explicitly for special type of correlations by the method of images; the analysis requires a detailed study of partitions of the sphere by means of spherical triangles. The joint densities obtained can be used to obtain explicit expressions for price of options in financial mathematics. We provide closed-form expressions for the price of several barrier type derivatives with a three dimensional geometric Wiener process as underlying. These solutions are found for special correlation matrices and are given by linear combinations of three dimensional Gaussian cumulative distributions. In order to extend the results to a wider set of correlation matrices the method of random correlations is outlined.

Numerical determination of aerodynamic characteristics of an airfoil in a ground effect

В работе изучается движение профиля над экраном на различных относительных высотах. Рассмотрены следующие методы его моделирования: условие неподвижного экрана и метод зеркального отображения для моделирования обращённого движения и условие экрана, движущегося со скоростью профиля, что моделирует прямое движение. Целью работы является выбор метода моделирования экрана, при котором обтекание профиля соответствует действительности и оценка разницы между рассмотренными методами. Задача решена в открытом пакете OpenFOAM методом контрольного объёма, где совместно решены уравнения Навье-Стокса и неразрывности, осреднённые по Рейнольдсу. Произведена верификация и валидация математической модели и найдено сеточно-независимое решение. Выбраны два профиля в плане: сегментный и симметричный. Рассмотрены несколько относительных высот. В работе построены эпюры скоростей под профилем, представлены картины обтекания профилей, исследованы их основные эксплуатационные характеристики: коэффициент подъёмной силы и коэффициент сопротивления в зависимости от относительной высоты. Построено распределение коэффициента давления по поверхности рассматриваемых профилей в зависимости от граничных условий и относительных высот. В результате анализа показано различие происходящих физических процессов при обтекании профилей в прямом и обращённом движении. Данная работа позволяет сделать вывод о том, каким образом проводить физический эксперимент для различных профилей, показывает преимущество использования метода зеркальных отображений или подвижного экрана при проведении эксперимента. In this article the movement of the profile above the screen at different relative heights is reviewed. The following methods of its modeling are considered: the condition of a stationary screen and the method of images for simulating reverse motion and the condition of a screen moving with the profile speed that simulate forward motion are considered. The aim of the work is to select a screen simulation method for a physical experiment. An open-source packet OpenFOAM based on finite-volume method is used to solve the Navier-Stokes and continuity equations averaged by Reynolds method. The mathematical model is verified and validated, and a grid-independent solution is found. Two profiles are selected: segmental and symmetrical. Several relative heights are considered. The velocitiy profiles under the airfoil are constructed, the patterns of the flow around the airfoils are presented. The dependences of coefficients on the studied parameters and the distribution of the pressure coefficient over the profile are studied and analyzed. As a result of the analysis, the difference between the physical processes when flowing around the airfoils is in forward and reverse motion is shown. This work allows us to make a conclusion about how to conduct a model experiment for various profiles, shows the advantage of using the method of images or a movable screen in the experiment.