Slow-growth approximation for near-wall patch representation of wall-bounded turbulence

Wall-bounded turbulent shear flows are known to exhibit universal small-scale dynamics that are modulated by large-scale flow structures. Strong pressure gradients complicate this characterization, however; they can cause significant variation of the mean flow in the streamwise direction. For such situations, we perform asymptotic analysis of the Navier-Stokes equations to inform a model for the effect of mean flow growth on near-wall turbulence in a small domain localized to the boundary. The asymptotics are valid whenever the viscous length scale is small relative to the length scale over which the mean flow varies. To ensure the correct momentum environment, a dynamic procedure is introduced that accounts for the additional sources of mean momentum flux through the upper domain boundary arising from the asymptotic terms. Comparisons of the model's low-order, single-point statistics with those from direct numerical simulation and well-resolved large eddy simulation of adverse-pressure gradient turbulent boundary layers indicate the asymptotic model successfully accounts for the effect of boundary layer growth on the small-scale near-wall turbulence

Physics informed neural networks for elliptic equations with oscillatory differential operators

We consider standard physics informed neural network solution methods for elliptic partial differential equations with oscillatory coefficients. We show that if the coefficient in the elliptic operator contains frequencies on the order of $1/\epsilon$, then the Frobenius norm of the neural tangent kernel matrix associated to the loss function grows as $1/\epsilon^2$. Numerical examples illustrate the stiffness of the optimization problem

On the nature of the boundary resonance error in numerical homogenization and its reduction

Numerical homogenization of multiscale equations typically requires taking an average of the solution to a microscale problem. Both the boundary conditions and domain size of the microscale problem play an important role in the accuracy of the homogenization procedure. In particular, imposing naive boundary conditions leads to a $\mathcal{O}(\epsilon/\eta)$ error in the computation, where $\epsilon$ is the characteristic size of the microscopic fluctuations in the heterogeneous media, and $\eta$ is the size of the microscopic domain. This so-called boundary, or ``cell resonance" error can dominate discretization error and pollute the entire homogenization scheme. There exist several techniques in the literature to reduce the error. Most strategies involve modifying the form of the microscale cell problem. Below we present an alternative procedure based on the observation that the resonance error itself is an oscillatory function of domain size $\eta$. After rigorously characterizing the oscillatory behavior for one dimensional and quasi-one dimensional microscale domains, we present a novel strategy to reduce the resonance error. Rather than modifying the form of the cell problem, the original problem is solved for a sequence of domain sizes, and the results are averaged against kernels satisfying certain moment conditions and regularity properties. Numerical examples in one and two dimensions illustrate the utility of the approach

Low Mach number fluctuating hydrodynamics model for ionic liquids

We present a new mesoscale model for ionic liquids based on a low Mach number fluctuating hydrodynamics formulation for multicomponent charged species. The low Mach number approach eliminates sound waves from the fully compressible equations leading to a computationally efficient incompressible formulation. The model uses a Gibbs free-energy functional that includes enthalpy of mixing, interfacial energy, and electrostatic contributions. These lead to a new fourth-order term in the mass equations and a reversible stress in the momentum equations. We calibrate our model using parameters for [DMPI+][F6P-], an extensively studied room temperature ionic liquid (RTIL), and numerically demonstrate the formation of mesoscopic structuring at equilibrium in two and three dimensions. In simulations with electrode boundaries the measured double-layer capacitance decreases with voltage, in agreement with theoretical predictions and experimental measurements for RTILs. Finally, we present a shear electroosmosis example to demonstrate that the methodology can be used to model electrokinetic flows

Fluid dynamics alters liquid-liquid phase separation in confined aqueous two-phase systems

Liquid-liquid phase separation is key to understanding aqueous two-phase systems (ATPS) arising throughout cell biology, medical science, and the pharmaceutical industry. Controlling the detailed morphology of phase-separating compound droplets leads to new technologies for efficient single-cell analysis, targeted drug delivery, and effective cell scaffolds for wound healing. We present a computational model of liquid-liquid phase separation relevant to recent laboratory experiments with gelatin-polyethylene glycol mixtures. We include buoyancy and surface-tension-driven finite viscosity fluid dynamics with thermally induced phase separation. We show that the fluid dynamics greatly alters the evolution and equilibria of the phase separation problem. Notably, buoyancy plays a critical role in driving the ATPS to energy-minimizing crescent-shaped morphologies and shear flows can generate a tenfold speedup in particle formation. Neglecting fluid dynamics produces incorrect minimum-energy droplet shapes. The model allows for optimization of current manufacturing procedures for structured microparticles and improves understanding of ATPS evolution in confined and flowing settings important in biology and biotechnology.Comment: 9 pages, 8 figures, 3 supplementary movies, to appear in Proceedings of the National Academy of Sciences, accompanying code and parameters to generate data available at https://github.com/ericwhester/multiphase-fluids-cod

Kinetic and structural mechanism for DNA unwinding by a non-hexameric helicase

UvrD, a model for non-hexameric Superfamily 1 helicases, utilizes ATP hydrolysis to translocate stepwise along single-stranded DNA and unwind the duplex. Previous estimates of its step size have been indirect, and a consensus on its stepping mechanism is lacking. To dissect the mechanism underlying DNA unwinding, we use optical tweezers to measure directly the stepping behavior of UvrD as it processes a DNA hairpin and show that UvrD exhibits a variable step size averaging ~3 base pairs. Analyzing stepping kinetics across ATP reveals the type and number of catalytic events that occur with different step sizes. These single-molecule data reveal a mechanism in which UvrD moves one base pair at a time but sequesters the nascent single strands, releasing them non-uniformly after a variable number of catalytic cycles. Molecular dynamics simulations point to a structural basis for this behavior, identifying the protein-DNA interactions responsible for strand sequestration. Based on structural and sequence alignment data, we propose that this stepping mechanism may be conserved among other non-hexameric helicases

Modeling Electrokinetic Flows with the Discrete Ion Stochastic Continuum Overdamped Solvent Algorithm

In this article we develop an algorithm for the efficient simulation of electrolytes in the presence of physical boundaries. In previous work the Discrete Ion Stochastic Continuum Overdamped Solvent (DISCOS) algorithm was derived for triply periodic domains, and was validated through ion-ion pair correlation functions and Debye-H{\"u}ckel-Onsager theory for conductivity, including the Wien effect for strong electric fields. In extending this approach to include an accurate treatment of physical boundaries we must address several important issues. First, the modifications to the spreading and interpolation operators necessary to incorporate interactions of the ions with the boundary are described. Next we discuss the modifications to the electrostatic solver to handle the influence of charges near either a fixed potential or dielectric boundary. An additional short-ranged potential is also introduced to represent interaction of the ions with a solid wall. Finally, the dry diffusion term is modified to account for the reduced mobility of ions near a boundary, which introduces an additional stochastic drift correction. Several validation tests are presented confirming the correct equilibrium distribution of ions in a channel. Additionally, the methodology is demonstrated using electro-osmosis and induced charge electro-osmosis, with comparison made to theory and other numerical methods. Notably, the DISCOS approach achieves greater accuracy than a continuum electrostatic simulation method. We also examine the effect of under-resolving hydrodynamic effects using a `dry diffusion' approach, and find that considerable computational speedup can be achieved with a negligible impact on accuracy.Comment: 27 pages, 15 figure

A Low Mach Number Fluctuating Hydrodynamics Model For Ionic Liquids

We present a new mesoscale model for ionic liquids based on a low Mach number fluctuating hydrodynamics formulation for multicomponent charged species. The low Mach number approach eliminates sound waves from the fully compressible equations leading to a computationally efficient incompressible formulation. The model uses a Gibbs free energy functional that includes enthalpy of mixing, interfacial energy, and electrostatic contributions. These lead to a new fourth-order term in the mass equations and a reversible stress in the momentum equations. We calibrate our model using parameters for [DMPI+][F6P-], an extensively-studied room temperature ionic liquid (RTIL), and numerically demonstrate the formation of mesoscopic structuring at equilibrium in two and three dimensions. In simulations with electrode boundaries the measured double layer capacitance decreases with voltage, in agreement with theoretical predictions and experimental measurements for RTILs. Finally, we present a shear electroosmosis example to demonstrate that the methodology can be used to model electrokinetic flows