## Optimization of PDE System

#### Optimization in the distributed parameter system.

The lumped parameter system where the system is assumed to be concentrated at a single spatial point is modelled by ordinary differential equations. On the other hand, the distributed parameter system where the system is to occupy a certain spatial and/or time domain is modelled by partial differential equations. For the past three decades, the distributed parameter systems have occupied an important place in control and system theories. Our interests are in solving the optimization problem in the distributed parameter system, especially related to the microscopic phenomena in the single crystal growth system.

## Grid Generation

#### Orthogonal grid generation

A new numerical scheme is proposed for generating an orthogonal grid in a simply-connected 2D domain. The scheme is based on the idea of decomposition of a global orthogonal transform into consecutive mappings of a conformal mapping and an auxiliary orthogonal mapping, which was suggested by Kang and Leal(J.Comput. Phys. vol.102, 78 (1992)). The method is non-iterative and flexible in adjustment of grid spacing. The grid spacing can be controlled mainly by specification of the boundary correspondence up to on three sides of the boundary. The method is also equipped with a control function that provides further degree of freedom in grid spacing adjustment. From a mathematical viewpoint, the proposed scheme can also be regarded as a numerical implementation of the constructive proof for the existence of solution of the orthogonal mapping problem in an arbitrary simply-connected domain under the condition that the boundary correspondence is specified at three sides.

## Chaotic Mixing of Micro Fluidics

#### Chaotic Mixing and Mass Transfer Enhancement

We investigate the dynamical flow patterns and the heat transfer for pulsatile flow in an axisymmetric wavy channel. There are two main objectives in this work. One is to investigate the effect of the Strouhal number and the geometric wave number on the heat transfer enhancement. The other is to analyze the dynamic flow pattern in the Eulerian and the Lagrangian points of view and to explain why the heat transfer is enhanced by the pulsatile laminar flow. The sectral representation of unsteady velocity signals is introduced for the Lagrangian flow field analysis. The method has an advantage of reducing memory and computation time drastically. As a result, the heat transfer enhancement is due to the chaotic mixing which is caused by the combination effect of the pulsatile flow and the wavy geometry.

#### Analytical studies on circulatory flows inside a drop under non-uniform electric fileds

The circulating flows formed inside a spherical drop under time-periodic non-uniform electric fields are considered and an analytical solution of the stream function distribution inside and outside the drop is obtained. The result reveals a new physics that dielectrophoretic migration is possible in a time-periodic electric field even in the situation where the dielectrophoresis is impossible if a static electric field is applied. By using the analytical solution of the stream function, fluid mixing inside a drop is also analyzed and it is found that there exists an optimal frequency of time-periodic electric field which induces the most efficient fluid mixing inside the drop.

#### Dynamics of a non-spherical bubble in a viscous fluid under a time-periodic electric field

We investigate dynamics of a non-spherical bubble filled with a permanent gas and vapor, immersed in an unbounded viscous fluid under an external time-periodic electric field. The our main concern is the instability related with large-amplitude shape oscillation when the volume oscillation loses its stability which can be applied to enhance the heat transfer rate or the reaction rate by 'microstreaming' effects, or to prevent damage in biological system due to high shear stress by shape oscillation. We assume weak electric field and use electric field on a spherical bubble surface which makes possible to decouple the effects of the electric field on the changing shape of the bubble surface to calculate electric stress. And we account for viscous effects using the potential flow solutions suggested by Kang(I.S. Kang and L.G. Leal,

*Phys. Fluids A*

**31**, 233-7 (1988)). We solve dynamical equations numerically by using Runge-Kutta-Gill method and interpret solutions by the methods of chaos physics. While the shape oscillation is damped out only after few oscillations with a pulsation of isotropic pressure such as sound field, large-amplitude of the shape oscillations are possible even with a constant electric field. And also the period of shape oscillation can be controlled by choosing the frequency of the electric field.