Mesoscopic modeling of a two-phase flow in the presence of boundaries: the Contact Angle

We present a mesoscopic model, based on the Boltzmann Equation, for the interaction between a solid wall and a non-ideal fluid. We present an analytic derivation of the contact angle in terms of the surface tension between the liquid-gas, the liquid-solid and the gas-solid phases. We study the dependency of the contact angle on the two free parameters of the model, which determine the interaction between the fluid and the boundaries, i.e. the equivalent of the wall density and of the wall-fluid potential in Molecular Dynamics studies. We compare the analytical results obtained in the hydrodynamical limit for the density profile and for the surface tension expression with the numerical simulations. We compare also our two-phase approach with some exact results for a pure hydrodynamical incompressible fluid based on Navier-Stokes equations with boundary conditions made up of alternating slip and no-slip strips. Finally, we show how to overcome some theoretical limitations connected with a discretized Boltzmann scheme and we discuss the equivalence between the surface tension defined in terms of the mechanical equilibrium and in terms of the Maxwell construction.Comment: 29 pages, 12 figure

Thermal fluctuations and boundary conditions in the lattice Boltzmann method

The lattice Boltzmann method is a popular approach for simulating hydrodynamic interactions in soft matter and complex fluids. The solvent is represented on a discrete lattice whose nodes are populated by particle distributions that propagate on the discrete links between the nodes and undergo local collisions. On large length and time scales, the microdynamics leads to a hydrodynamic flow field that satisfies the Navier-Stokes equation. In this thesis, several extensions to the lattice Boltzmann method are developed. In complex fluids, for example suspensions, Brownian motion of the solutes is of paramount importance. However, it can not be simulated with the original lattice Boltzmann method because the dynamics is completely deterministic. It is possible, though, to introduce thermal fluctuations in order to reproduce the equations of fluctuating hydrodynamics. In this work, a generalized lattice gas model is used to systematically derive the fluctuating lattice Boltzmann equation from statistical mechanics principles. The stochastic part of the dynamics is interpreted as a Monte Carlo process, which is then required to satisfy the condition of detailed balance. This leads to an expression for the thermal fluctuations which implies that it is essential to thermalize all degrees of freedom of the system, including the kinetic modes. The new formalism guarantees that the fluctuating lattice Boltzmann equation is simultaneously consistent with both fluctuating hydrodynamics and statistical mechanics. This establishes a foundation for future extensions, such as the treatment of multi-phase and thermal flows. An important range of applications for the lattice Boltzmann method is formed by microfluidics. Fostered by the “lab-on-a-chip” paradigm, there is an increasing need for computer simulations which are able to complement the achievements of theory and experiment. Microfluidic systems are characterized by a large surface-to-volume ratio and, therefore, boundary conditions are of special relevance. On the microscale, the standard no-slip boundary condition used in hydrodynamics has to be replaced by a slip boundary condition. In this work, a boundary condition for lattice Boltzmann is constructed that allows the slip length to be tuned by a single model parameter. Furthermore, a conceptually new approach for constructing boundary conditions is explored, where the reduced symmetry at the boundary is explicitly incorporated into the lattice model. The lattice Boltzmann method is systematically extended to the reduced symmetry model. In the case of a Poiseuille flow in a plane channel, it is shown that a special choice of the collision operator is required to reproduce the correct flow profile. This systematic approach sheds light on the consequences of the reduced symmetry at the boundary and leads to a deeper understanding of boundary conditions in the lattice Boltzmann method. This can help to develop improved boundary conditions that lead to more accurate simulation results

Numerical Simulations of Polymers at the Nanoscale

In this thesis we study a variety of nanoscale phenomena in certain polymer systems using a combination of numerical simulation methods and mathematical modelling. The problems considered are: (a) the mixing behaviour of polymeric fluids in micro- and nanofluidic devices, (b) capillary absorption of polymer droplets into narrow capillaries, and (c) modelling the phase separation and self-assembly behaviour in polymer systems with freely deforming boundaries. These problems are significant in nanotechnological applications of polymer-based systems. First, the mixing behaviour of a polymeric melt over two parallely patternedslip surfaces is considered. Using molecular dynamics (MD) simulations, it is shown that mixing is enhanced when the polymer chain size is smaller than the wavelength of the chemical pattern of the surfaces. An off-set in the upper and lowerwall patterns improved themixing in the centre of the channel. Application of a sinusoidally varying body force in addition to the patterned-slip conditions is shown to enhance mixing further, compared to a constant body force case, with some limitations. Simulation findings for the constant body force cases are in qualitative agreement with the continuum theory of Pereira [1]. However, in the case of a sinusoidally varying body force our simulations do not agree with the continuum theory. We explain the reasons for the discrepancy between the two and point out the deficiencies in the continuum theory in predicting the correct behaviour. Second, the capillary phenomena of polymer droplets in narrow capillaries is studied using MD simulations. It is demonstrated that droplets composed of longer chains require wider tubes for absorption and this result is in agreement with our continuum modelling. The observed capillary dynamics deviate significantly from the standard Lucas-Washburn description thus questioning its validity at the nanoscale. The metastable states during the capillary absorption in some cases cannot be explained using the existing models of capillary dynamics. Lastly, the phase separation process in polymer blends between both confined and unconfined boundaries is studied using Smoothed Particle Hydrodynamics (SPH). The SPH technique has the advantage of not using a grid to discretize the spatial domain, which makes it appealing when dealing with problems where the spatial domain can change with time. The applicability of the SPH method in describing phase separation in these systems is demonstrated. In particular, its ability to model freely deforming polymer blends is shown

Non-Linear Lattice

The development of mathematical techniques, combined with new possibilities of computational simulation, have greatly broadened the study of non-linear lattices, a theme among the most refined and interdisciplinary-oriented in the field of mathematical physics. This Special Issue mainly focuses on state-of-the-art advancements concerning the many facets of non-linear lattices, from the theoretical ones to more applied ones. The non-linear and discrete systems play a key role in all ranges of physical experience, from macrophenomena to condensed matter, up to some models of space discrete space-time

A study of pseudopotential lattice Boltzmann method with applications to thermal bubble nucleation

The nature of the work dealt with in this thesis is mathematical modelling of multiphase flows. The main objective of this doctoral work was to study multiphase lattice Boltzmann models (LBM) and to develop an advanced pseudopotential model. Specifically, advanced thermal lattice Boltzmann models were applied to study bubble nucleation in nucleate pool boiling at subatmospheric pressures. The numerical investigations carried out as part of this work follow the format well-established in the literature and allow further studies in more complex geometries. The work carried out contributes to current discussions in the literature and fulfils the recommendations of a number of authors. Fluid-fluid interactions in the Yuan-Schaefer, multipseudopotential interaction and piecewise linear equation of state methods were investigated. Multipseudopotential interaction was established as a practicable method of multiphase simulations by combination with the multiple relaxation time collision operator, surface tension modification methods and with modified temperature double distribution function and hybrid (4th order Runge-Kutta) thermal LBM models. Thermal LBM simulations were found to agree well with experimental findings on the influence of subatmospheric pressure on bubble nucleation. It was found that as pressure is lowered in LBM simulations the size of bubbles nucleated increases, according to bubble diameter ~ pressure-1 , with results falling in between experimental data for brass and stainless steel tubes