A new method of solving the Navier-Stokes equations e ciently by reducing their number of modes is proposed in the present paper. It is based on the Karhunen-Lo eve decomposition which is a technique of obtaining empirical eigenfunctions from the experimental or numerical data of a system. Employing these empirical eigenfunctions as basis functions of a Galerkin procedure, one can a priori limit the function space considered to the smallest linear subspace that is su cient to describe the observed phenomena, and consequently reduce the Navier-Stokes equation deÿned on a complicated geometry to a set of ordinary di erential equations with a minimum degree of freedom. The present algorithm is well suited for the problems of ow control or optimization, where one has to compute the ow ÿeld repeatedly using the Navier-Stokes equation but one can also estimate the approximate solution space of the ow ÿeld based on the range of control variables. The low-dimensional dynamic model of viscous uid ow derived by the present method is shown to produce accurate ow ÿelds at a drastically reduced computational cost when compared with the ÿnite di erence solution of the Navier-Stokes equation. ?

Many lab-on-a-chip based microsystems process biofluids such as blood and DNA solutions. These fluids are viscoelastic and show extraordinary flow behaviors, not existing in Newtonian fluids. Adopting appropriate constitutive equations these exotic flow behaviors can be modeled and predicted reasonably using various numerical methods. In the present paper, we investigate viscoelastic electroosmotic flows through a rectangular straight microchannel with and without pressure gradient. It is shown that the volumetric flow rates of viscoelastic fluids are significantly different from those of Newtonian fluids under the same external electric field and pressure gradient. Moreover, when pressure gradient is imposed on the microchannel there appear appreciable secondary flows in the viscoelastic fluids, which is never possible for Newtonian laminar flows through straight microchannels. The retarded or enhanced volumetric flow rates and secondary flows affect dispersion of solutes in the microchannel nontrivially.

SUMMARYWe consider an inverse problem of identifying the boundary shape of a domain, where the temperature ÿeld is dominated by natural convection, from temperature measurements on the other boundary. The potential applications of the present investigations are the determination of a phase change isotherm in the Bridgman crystal growth or the thermal tomography which detects aws in materials nondestructively. After mapping the irregular domains into a reference one employing a set of parameters, the inverse problem is formulated as a parameter optimization problem which is solved by a conjugate gradient method. The present method is found to identify the domains reasonably accurately even with noisy temperature measurements.

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Electrokinetic flows through hydrophobic microchannels experience velocity slip at the microchannel wall, which affects volumetric flow rate and solute retention time. The usual method of predicting the volumetric flow rate and velocity profile for hydrophobic microchannels is to solve the Navier-Stokes equation and the Poisson-Boltzmann equation for the electric potential with the boundary condition of velocity slip expressed by the Navier slip coefficient, which is computationally demanding and defies analytic solutions. In the present investigation, we have devised a simple method of predicting the velocity profiles and volumetric flow rates of electrokinetic flows by extending the concept of the Helmholtz-Smoluchowski velocity to microchannels with Navier slip. The extended Helmholtz-Smoluchowski velocity is simple to use and yields accurate results as compared to the exact solutions. Employing the extended Helmholtz-Smoluchowski velocity, the analytical expressions for volumetric flow rate and velocity profile for electrokinetic flows through rectangular microchannels with Navier slip have been obtained at high values of zeta potential. The range of validity of the extended Helmholtz-Smoluchowski velocity is also investigated.