A mesoscopic model for microscale hydrodynamics and interfacial phenomena: Slip, films, and contact angle hysteresis

We present a model based on the lattice Boltzmann equation that is suitable for the simulation of dynamic wetting. The model is capable of exhibiting fundamental interfacial phenomena such as weak adsorption of fluid on the solid substrate and the presence of a thin surface film within which a disjoining pressure acts. Dynamics in this surface film, tightly coupled with hydrodynamics in the fluid bulk, determine macroscopic properties of primary interest: the hydrodynamic slip; the equilibrium contact angle; and the static and dynamic hysteresis of the contact angles. The pseudo- potentials employed for fluid-solid interactions are composed of a repulsive core and an attractive tail that can be independently adjusted. This enables effective modification of the functional form of the disjoining pressure so that one can vary the static and dynamic hysteresis on surfaces that exhibit the same equilibrium contact angle. The modeled solid-fluid interface is diffuse, represented by a wall probability function which ultimately controls the momentum exchange between solid and fluid phases. This approach allows us to effectively vary the slip length for a given wettability (i.e. the static contact angle) of the solid substrate

Self-similar blow-up solutions in the generalized Korteweg-de Vries equation: Spectral analysis, normal form and asymptotics

In the present work we revisit the problem of the generalized Korteweg-de Vries equation parametrically, as a function of the relevant nonlinearity exponent, to examine the emergence of blow-up solutions, as traveling waveforms lose their stability past a critical point of the relevant parameter $p$, here at $p=5$. We provide a {\it normal form} of the associated collapse dynamics and illustrate how this captures the collapsing branch bifurcating from the unstable traveling branch. We also systematically characterize the linearization spectrum of not only the traveling states, but importantly of the emergent collapsing waveforms in the so-called co-exploding frame where these waveforms are identified as stationary states. This spectrum, in addition to two positive real eigenvalues which are shown to be associated with the symmetries of translation and scaling invariance of the original (non-exploding) frame features complex patterns of negative eigenvalues that we also fully characterize. We show that the phenomenology of the latter is significantly affected by the boundary conditions and is far more complicated than in the corresponding symmetric Laplacian case of the nonlinear Schr{\"o}dinger problem that has recently been explored. In addition, we explore the dynamics of the unstable solitary waves for $p>5$ in the co-exploding frame.Comment: 33 pages, 16 figure

Localized Breathing Modes in Granular Crystals with Defects

We investigate nonlinear localized modes at light-mass impurities in a one-dimensional, strongly-compressed chain of beads under Hertzian contacts. Focusing on the case of one or two such "defects", we analyze the problem's linear limit to identify the system eigenfrequencies and the linear defect modes. We then examine the bifurcation of nonlinear defect modes from their linear counterparts and study their linear stability in detail. We identify intriguing differences between the case of impurities in contact and ones that are not in contact. We find that the former bears similarities to the single defect case, whereas the latter features symmetry-breaking bifurcations with interesting static and dynamic implications

Population balances in case of crossing characteristic curves: Application to T-cells immune response

The progression of a cell population where each individual is characterized by the value of an internal variable varying with time (e.g. size, weight, and protein concentration) is typically modeled by a Population Balance Equation, a first order linear hyperbolic partial differential equation. The characteristics described by internal variables usually vary monotonically with the passage of time. A particular difficulty appears when the characteristic curves exhibit different slopes from each other and therefore cross each other at certain times. In particular such crossing phenomenon occurs during T-cells immune response when the concentrations of protein expressions depend upon each other and also when some global protein (e.g. Interleukin signals) is also involved which is shared by all T-cells. At these crossing points, the linear advection equation is not possible by using the classical way of hyperbolic conservation laws. Therefore, a new Transport Method is introduced in this article which allowed us to find the population density function for such processes. The newly developed Transport method (TM) is shown to work in the case of crossing and to provide a smooth solution at the crossing points in contrast to the classical PDF techniques.Comment: 18 pages, 10 figure