Mild solutions of stochastic Navier‐Stokes equation with jump noise in ‐spaces

In this paper, we study solvability of the local mild solution of stochastic Navier‐Stokes equation with jump noise in ‐spaces. This research work has mainly carried out by exploiting some interesting works of Tosio Kato.     

Optimal Control Problems with State Constraints in Fluid Mechanics and Combustion

Using nonlinear programming theory in Banach spaces we derive a version of Pontryagin's maximum principle that can be applied to distributed parameter systems under control and state constrains. The results are applied to fluid mechanics and combustion problems. Accepted 3 December 1996     

Stochastic 2-D Navier—Stokes Equation

   Abstract. In this paper we prove the existence and uniqueness of strong solutions for the stochastic Navier—Stokes equation in bounded and unbounded domains. These solutions are stochastic analogs of the classical Lions—Prodi solutions to the deterministic Navier—Stokes equation. Local monotonicity of the nonlinearity is exploited to obtain the solutions in a given probability space and this significantly improves the earlier techniques for obtaining strong solutions, which depended on pathwise solutions to the Navier—Stokes martingale problem where the probability space is also obtained as a part of the solution.     

Bubble‐free electrokinetic flow with propylene carbonate

For electroosmotic pumping, a large direct‐current (DC) electric field (10+ V/cm) is applied across a liquid, typically an aqueous electrolyte. At these high voltages, water undergoes electrolysis to form hydrogen and oxygen, generating bubbles that can block the electrodes, cause pressure fluctuations, and lead to pump failure. The requirement to manage these gases constrains system designs. This article presents an alternative polar liquid for DC electrokinetic pumping, propylene carbonate (PC), which remains free of bubbles up to at least 10 kV/cm. This offers the opportunity to create electrokinetic devices in closed configurations, which we demonstrate with a fully sealed microfluidic hydraulic actuator. Furthermore, the electroosmotic velocity of PC is similar to that of water in PDMS microchannels. Thus, water could be substituted by PC in existing electroosmotic pumps.     

An Acoustically Driven Microliter Flow Chamber on a Chip (μFCC) for Cell–Cell and Cell–Surface Interaction Studies

A novel method for pumping very small volumes of liquid by using surface acoustic waves is employed to create a microfluidic flow chamber on a chip. It holds a volume of only a few μl and its planar design provides complete architectural freedom. This allows for the reconstruction of even complex flow scenarios (e.g. curvatures, bifurcations and stenosis). Addition of polymer walls to the planar fluidic track enables cell culturing on the chip surface and the investigation of cell–cell adhesion dynamics under flow. We demonstrate the flexibility of the system for application in many areas of microfluidic investigations including blood clotting phenomena under various flow conditions and the investigation of different stages of cell adhesion.     

Deterministic and Stochastic Control of Navier—Stokes Equation with Linear, Monotone, and Hyperviscosities

This paper deals with the optimal control of space—time statistical behavior of turbulent fields. We provide a unified treatment of optimal control problems for the deterministic and stochastic Navier—Stokes equation with linear and nonlinear constitutive relations. Tonelli type ordinary controls as well as Young type chattering controls are analyzed. For the deterministic case with monotone viscosity we use the Minty—Browder technique to prove the existence of optimal controls. For the stochastic case with monotone viscosity, we combine the Minty—Browder technique with the martingale problem formulation of Stroock and Varadhan to establish existence of optimal controls. The deterministic models given in this paper also cover some simple eddy viscosity type turbulence closure models. Accepted 7 June 1999     

Large deviations for the stochastic shell model of turbulence

In this work, we first prove the existence and uniqueness of a strong solution to stochastic GOY model of turbulence with a small multiplicative noise. Then using the weak convergence approach, Laplace principle for solutions of the stochastic GOY model is established in certain Polish space. Thus a Wentzell–Freidlin type large deviation principle is established utilizing certain results by Varadhan and Bryc.