The direct problem of water-wave equations is the problem of determining the surface and its velocity potential, in time T > 0, for a given initial profile and velocity potential, where the profile of the bottom, the bathymetry, is known. In this paper, we study the inverse problem of recovering the shape of the solid bottom boundary of an inviscid, irrotational, incompressible fluid from measurements of a portion of the free surface. In particular, given the water-wave height and its velocity potential on an open set, together with the first time derivative of the free surface, on a single time, we address the identifiability problem. Moreover we compute the derivatives with respect to the shape of the bottom, which allows us to obtain the optimality conditions for this inverse problem.

In this study, a numerical study has been performed on the two-phase heat transfer of a new nanostructured heat transfer fluid: Water-in-Polyalphaolefin (PAO) Nanoemulsion Fluid inside a mini-channel heat exchanger using ANSYS FLUENT. Nanoemulsion fluids are liquid suspensions of nanosized droplets, which are part of a broad class of colloidal dispersions. The nanoemulsion fluid can be formed spontaneously by self-assembly, in which these nanodroplets are in fact swollen micelles. To simplify the complexity of the numerical model, the nanoemulsion fluid was then treated as a homogenous fluid during single-phase and only the water vaporizes during the phase change. The volume of fraction (VOF) model with Pressure-Velocity coupling based Semi Implicit Method for Pressure Linked Equations (SIMPLE) iterative algorithm is employed to solve the continuity, momentum, energy equations in two dimensional domains. The thermophysical properties of the nanoemulsion fluid were measured and used for the current simulation. The results were verified using the experimental results and has shown good agreement. This study has demonstrated the feasibility of simplyig the simulation of flow boiling heat transfer of this new heat transfer fluid through treating it as a homogenous fluid during single-phase convective heat transfer and separating the vapor phase of the nano-micelles during flow boiling. This study has also shown that this Water-in-PAO nanoemulsion could function as a good and alternative conventional working fluid in heat transfer applications.

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The present study experimentally and numerically investigates the flow and heat transfer characteristics of a novel nanostructured heat transfer fluid, namely, ethanol/polyalphaolefin nanoemulsion, inside a conventionally manufactured minichannel of circular cross section and a microchannel heat exchanger of rectangular cross section manufactured additively using the Direct Metal Laser Sintering (DMLS) process. The experiments were conducted for single-phase flow of pure polyalphaolefin (PAO) and ethanol/PAO nanoemulsion fluids with two ethanol concentrations of 4 wt% and 8 wt% as well as for two-phase flow boiling of nanoemulsion fluids to study the effect of ethanol nanodroplets on the convective flow and heat transfer characteristics. Furthermore, the effects of flow regime of the working fluids on the heat transfer performance for both the minichannel and microchannel heat exchangers were examined within the laminar and transitional flow regimes. It was found that the ethanol/PAO nanoemulsion fluids can improve convective heat transfer compared to that of the pure PAO base fluid under both single- and two-phase flow regimes. While the concentration of nanoemulsion fluids did not reflect a remarkable distinction in single-phase heat transfer performance within the laminar regime, a significant heat transfer enhancement was observed using the nanoemulsion fluids upon entering the transitional flow regime. The heat transfer enhancement at higher concentrations of nanoemulsion within the transitional regime is mainly attributed to the enhanced interaction and interfacial thermal transport between ethanol nanodroplets and PAO base fluid. For two-phase flow boiling, heat transfer coefficients of ethanol/PAO nanoemulsion fluids were further enhanced when the ethanol nanodroplets underwent phase change. A comparative study on the flow and heat transfer characteristics was also implemented between the traditionally fabricated minichannel and additively manufactured microchannel of similar dimensions using the same working fluid of pure PAO and the same operating conditions. The results revealed that although the DMLS fabricated microchannel posed a higher pressure loss, a substantial heat transfer enhancement was achieved as compared to the minichannel heat exchanger tested under the same conditions. The non-post processed surface of the DMLS manufactured microchannel is likely to be the main contributor to the augmented heat transfer performance. Further studies are required to fully appreciate the possible mechanisms behind this phenomenon as well as the convective heat transfer properties of nanoemulsion fluids.

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