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The goal of this method is to implement the volume and mass conserving characteristic method. This method is appropriate to the equation with advections and diffusions. Characteristic mixed FEM with piecewise constant approximation is applied to the advection part, and the diffusion part is handled by FDM.

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The convective-diffusive transport of sub-micron aerosols in an oscillatory laminar flow within a 2-D single bifurcation is studied, using order-of-magnitude analysis and numerical simulation using a commercial software (FEMLAB®). Based on the similarity between momentum and mass transfer equations, various transient mass transport regimes are classified and scaled according to Strouhal and beta numbers. Results show that the mass transfer rate is highest at the carinal ridge and there is a phase-shift in diffusive transport time if the beta number is greater than one. It is also shown that diffusive mass transfer becomes independent of the oscillating outer flow if the Strouhal number is greater than one. / Singapore-MIT Alliance (SMA)

Convection set up in a fluid due to the presence of two components of differing diffusivities is known as double diffusive convection. Double diffusive convection is observed in nature, in oceans, in the formation of certain columnar rock structures and in stellar interiors. The major engineering applications of double diffusive convection are in the fields metallurgy and alloy solidification in casting processes. The two components may be any two substances which affect the density of the fluid, heat and salt being the pair found most commonly in nature. Depending upon the initial stratifications of the two components, double diffusive convection can be set up in either the diffusive mode or the finger mode. In this thesis, the linear stability of a double diffusive system prone to finger instability has been studied in the presence of temporally varying non-linear background profiles of temperature and salinity. The motivation for the present study is to bridge the gap between existing theories, which mainly concentrate on linear background profiles independent of time, on the one hand and experiments and numerical simulations, which have time dependent step-like non-linear background profiles, on the other. The general stability characteristics of a double diffusive system with step-like background profiles have been studied using the standard normal mode method. The background temperature and salinity profiles are assumed to follow the hyperbolic tangent function, since it has a step-like character. The sharpness of the step can be altered by changing a suitable parameter in the hyperbolic tangent function. It is found that changing the degree of non-linearity of the background profile of one of the components keeping the background profile of the other component linear affects the growth rate, Wave number and the form of the disturbances. In general, increasing the degree of nonlinearity of background salinity profile makes the system more unstable and results in a reduction in the vertical extent of the disturbances. On the other hand, increasing the degree of non-linearity of the background temperature profile with the salinity profile kept linear results in a reduction in the growth rate and increase in the wave number. The form of the disturbance may change due to enhanced modal competition between the gravest odd and even modes in this case. The method of normal modes inherently assumes that the background profiles of temperature and salinity are independent of time and hence, it cannot be used for studying the stability of systems with time varying background profiles. A pseudo-similarity method has been used to handle such background profiles. Initial steps of temperature and salinity diffuse according to the error function form, and hence, the case of error function background profiles has been studied in detail. Taking into account the time-dependence of background profiles has been shown to significantly change the wave number and the incipient flux ratio. The dependence of the critical wave number (kc) on the thermal Rayleigh number (RaT ) can be determined analytically and is found to change from kc ~ Ra T1/4 for linear background profiles to kc ~ Ra T1/3 for error function profiles. The region of instability in the Rp (density stability ratio) space is found to increase from 1 ≤ R ρ ≤ r−1 for linear background profiles to 1 ≤ Rρ < r−3/2 for error function background profiles, where T denotes the ratio of the diffusivity of the slower diffusing component to that of the faster diffusing one. A parametric study covering a wide range of parameter values has been carried out to determine the effect of the parameters density stability ratio (Rp), diffusivity ratio (ρ ) and Prandtl number (Pr) on the onset time, critical wavenumber and the incipient flux ratio. The wide range of governing parameters covered here is beyond the scope of experimental and numerical studies. Such a wide range can be covered by theoretical approaches alone. It has been shown that the time of onset of convection determines the thicknesses of the temperature and salinity boundary layers, which in turn determine the width of salt fingers. Finally, the theoretical predictions of salt finger widths have been shown to be in agreement with the results of two dimensional numerical simulations of thermohaline system.

The effects of gravity modulation on the instability of double-diffusive convections in a horizontal tank with aspect ratio (width/height) of 11 have been investigated by experiments and numerical simulations. The stably stratified fluid layer is set up with ethanol-water solution of 0.0 and 2.0% (by weight). The tank is fixed on a platform that can oscillate in the vertical direction. A constant temperature difference is maintained across the tank at thermal Rayleigh number . The fluid layer becomes unstable as the initially stable solute gradient slowly decreases due to the non-diffusive boundary conditions. The experiments determine that the instability onset under steady gravity is at with onset vortices of wavelength and oscillatory frequency . When the tank is oscillated at modulation frequency and amplitude , the fluid layer is destabilized slightly with a critical and onset vortices of and . A two-dimensional numerical simulation has accurately reproduced the experimental results of steady gravity, and demonstrated that the slight destability effect of gravity modulation is contributed by the asymmetry of the actual gravity modulation.Further simulations have yielded following results: (1) Under steady gravity, the kinetic energy and mechanical work components oscillate synchronously with . Under modulated gravity, they only oscillate synchronously with when is low, whereas not only synchronously with locally but also synchronously with globally when is high; (2) The resonance phenomenon predicted by Chen (2001) also exists under the present lab conditions. Such instability is in the sub-harmonic mode and the destability effect increases as increases. (3) The double-diffusive fluid layer may experience density-mode instability before the double-diffusive instability onset at certain and . Such density-mode instability is generally in the sub-harmonic mode, although it may be in the synchronous mode when is low and is large. This instability accelerates the mixing of the density gradient across the fluid layer and thus affects the succeeding double-diffusive instability; (4) When the background gravity is absent, the purely modulated gravity destabilizes the fluid layer when is low. On the contrary, it stabilizes the fluid layer when is high and the instability onset is in the synchronous mode.

A novel approach is developed to describe the dynamics of proteins, the coarse-grained Langevin Equation for Protein Dynamics (LE4PD). The approach describes proteins as fundamentally semiflexible objects collapsed into the free energy well representing the folded state. This is a multi-scale approach, where structural correlations are used as input to an effectively linear description, which can be solved in diffusive modes. The mode solution to this Langevin equation naturally separates into global modes describing the fully anisotropic tumbling of the macromolecule as a whole, and internal modes which describe local fluctuations about the folded structure. A protein in solution populates a structural ensemble of metastable configurations around the global fold, and we propose a simulation-free coarse-grained approach which utilizes knowledge of the important metastable folded states of the protein to predict the protein dynamics. The accuracy of the LE4PD is verified by analyzing the predicted dynamics across a set of seven different proteins for which both relaxation data and NMR solution structures are available. Using experimental NMR conformers as the input structural ensembles, LE4PD predicts quantitatively accurate results, with correlation coefficient $\rho=.93$ to NMR backbone relaxation measurements for the seven proteins. The NMR solution structure derived ensemble and predicted dynamical relaxation is compared with molecular dynamics simulation-derived structural ensembles and LE4PD predictions, and are consistent in the timescale of the simulations. The use of the experimental NMR conformers frees the approach from computationally demanding simulations. The biological function of proteins is encoded in their structure and expressed through the mediation of their dynamics. We present here a study of how local fluctuation relates to binding and function. This study indicates how local fluctuations are likely to initiate biologically relevant pathways as they cooperatively enhance the dynamics in specific spatial regions of the protein. The picture that emerges is a dynamically heterogenous protein where biologically active regions provide energetically-comparable conformational states that can be trapped by a reacting partner. The slowest, most collective motion localizes directly to highly conserved regions involved in binding partner recognition and active-site regulation. We analyze this possible relation between dynamics and binding mechanism as we calculate the dynamics of monomeric and dimerized HIV protease, free Insulin Growth Factor II Receptor (IGF2R) domain 11 and its IGF2R:IGF2 complex. The long-time dynamics of proteins is controlled by an activated regime where the dynamics of the large amplitude diffusive modes becomes dominated by the presence of energy barriers. We explicitly study the atomistic simulation-derived free energy landscape projected from the diffusive modes of the linear Langevin description of the protein, and obtain a general scaling between the fluctuation lengthscale and complexity. This hierarchical property of the free energy landscape of proteins is shown to be general across a set of six different single-domain monomeric proteins. As a consequence microscopic timescales of sub-angstrom sized fluctuations rapidly propagate out to folding timescales at the nanometer lengthscale of globular single-domain proteins. This glassy activated regime extending from the nanosecond timescale we propose to be set by cooperative rearrangements of the protein-water and protein-protein hydrogen-bonding network. This results in metastable protein configurations with large changes in the protein-solvent hydrogen-bonding network coupled to subtle changes in the protein-protein hydrogen-bonding network. The Langevin formalism predictions are shown to agree with molecular dynamic simulations from the picosecond out to the millisecond timescale.

The efficiency of one fluid displacing another in a permeable medium depends on the pore-scale dynamics at the main wetting front. Experiments have shown that the frontal dynamics can result in two different flow regimes: a sharp and a diffuse front. In the sharp front regime, the displacing fluid occupies nearly all the pores and throats behind the main wetting front and the saturation changes abruptly. In contrast, in the diffuse front regime, pores are filled gradually at the main wetting front, and the saturation change is gradual in space. The different fronts can greatly alter the relative permeability curves, the trapping mechanisms, and the displacement efficiency. Directly measuring the sharpness of the front is difficult. Instead, here we correlate the front sharpness to saturation overshoot, which occurs for moderate to high flux vertical displacements of low density fluid by a higher density fluid in 1-D homogeneous permeable media. We hypothesize the sharpness of wetting front can be explained by competition between two different pore - filling mechanisms (called snap-off and piston-like) with the competition controlled by the velocity of the front and thus the injected flux. We conduct series of infiltration experiments to determine the saturation profile as a function of flux for seven different fluids. We find that for each fluid there is a flux (called overshoot flux) below which saturation overshoot ceases and the front is diffuse. We find that the overshoot flux depends inversely on the invading fluid’s viscosity, and shows little or no dependence on the invading fluid’s surface tension, vapor pressure, and its miscibility with water / text

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