Continuum Hydrodynamic Model Research Articles

Stratification in drying films: diffusiophoresis driven by nanoparticles and their counterions

Abstract Recent experimental work has highlighted that electrolyte-driven diffusiophoresis is likely to be the most significant phoretic motion in a mixture of silica nanoparticles and relatively large latex particles, which are commonly used in coatings. In this work, this diffusiophoretic effect, powered by gradients in the nanoparticles and their stabilizing cations, is modelled in drying films. A continuum hydrodynamic model is derived, and the resulting partial differential equations solved numerically. An asymptotic solution is found for high evaporation rate. It is found that the final film structure is governed by the relative magnitudes of the diffusive and diffusiophoretic terms. Two methods are discovered to control the resulting stratification: (i) setting the surface charge on the particles, and (ii) setting the background salt concentration. Either of these can be used to select either small- or large-on-top stratification or a homogenous film. The diffusiophoretic term promotes small-on-top stratification, and so may account for experimental observations of accumulated small particles at the top surface of dried films.

Pore-Scale Simulation of Calcite Matrix Acidizing with Hydrochloric Acid

Summary A continuum hydrodynamic model with immersed solid/fluid interface is developed for simulating calcite dissolution by hydrochloric acid (HCl) at the pore scale, and is most accurate for a mass-transfer-controlled dissolution regime under laminar flow conditions. The model uses averaged Navier-Stokes equations to model momentum transfer in porous media and adopts a theoretically developed mass-transfer formulation with assumptions. The model includes no fitting parameter and is validated using experimental results. The findings of previous research and existing models are briefly discussed and their shortcomings and advantages are elucidated. The present model is used in some pore-scale simulations on hypothetical but realistic cases, investigating the evolution of Darcy-scale permeability. Darcy-scale permeability exhibits totally different functionality of porosity in different dissolution regimes.

Ions' motion in water.

Over the decades, a great deal of attention has been focused on the solvation and transport properties of small rigid monatomic ions such as Na+, K+, Li+, Cl-, and Br- due to their importance in physical chemistry. Much less attention has been devoted to polyatomic ions although many polyatomic ions (such as nitrate, acetate, sulfate, and ammonium) are of great importance in biological and chemical processes. While the translational diffusion of smaller rigid ions shows the remarkable nonmonotonic dependence on inverse ion size (known as the "breakdown of Walden product"), the intermediate- to large-sized polyatomic ions (such as nitrate, acetate, and sulfate) exhibit different anomalies pointed out only recently. In this Perspective article, we provide an overview of how rotational diffusion and translational diffusion of these ions themselves are coupled to translational and rotational motions of water molecules. We discuss how diffusion of polyatomic ions is different from that of monatomic ions due to the rotational self-motion of the former that enhances diffusion in specific cases because of symmetry. While a continuum hydrodynamic model fails to describe the motion of polyatomic ions, we discuss how a mode-coupling theory approach can capture many aspects of this coupling between the solute ion and solvent water. We discuss how ionic mobility in water and other dipolar solvents are intimately connected to the dipolar solvation dynamics, in particular to its ultrafast component. We point out how the usual thinking on the relation between the diffusion and entropy needs to be modified in the case of ion diffusion.

Analytical solution of an Eulerian dilute particle-laden flow problem

Dilute particle-laden flows are encountered in various natural processes and manmade applications. To reduce the computational resources used to simulate such flows, the particle phase could be formulated using the Eulerian approach, resulting in a continuum hydrodynamic model. The aim of this paper is to present the analytical solution of a steady two-dimensional flow problem using that model. The dispersed solid particles, considered in this problem, are immersed in a uniform fluid flow field. The solid phase is coupled to the fluid using a linear Stokes-drag, which is valid for low slip velocities. The general solution of the solid phase velocity field is obtained by solving its quasi-linear hyperbolic momentum equations using the method of characteristics. For the case of a uniform inlet particle velocity, the solid phase velocity field is obtained in terms of Lambert W function. Subsequently, this velocity field is substituted in the solid phase continuity equation; transforming it to a semi-linear hyperbolic partial differential equation, which is solved to obtain the solid phase volume fraction field.

Hydrodynamic descriptions for surface roughness in fracture front propagation.

Fracture is ubiquitous in a crystalline material. Inspired by the observed phenomenological similarities between the spatial profile of a fractured surface and velocities in hydrodynamic turbulence, we set up a hydrodynamic description for the dynamics of fracture surface propagation mode I or opening fracture front. We consider several related continuum hydrodynamic models and use them to extract the similarities between the profile of a fractured surface and velocities in hydrodynamic turbulence. We conclude that a fractured surface should be generically self-similar with an underlying multifractal behaviour.This article is part of the theme issue 'Statistical physics of fracture and earthquakes'.

Perspectives on scaling and multiscaling in passive scalar turbulence.

We revisit the well-known problem of multiscaling in substances passively advected by homogeneous and isotropic turbulent flows or passive scalar turbulence. To that end we propose a two-parameter continuum hydrodynamic model for an advected substance concentration θ, parametrized jointly by y and y[over ¯], that characterize the spatial scaling behavior of the variances of the advecting stochastic velocity and the stochastic additive driving force, respectively. We analyze it within a one-loop dynamic renormalization group method to calculate the multiscaling exponents of the equal-time structure functions of θ. We show how the interplay between the advective velocity and the additive force may lead to simple scaling or multiscaling. In one limit, our results reduce to the well-known results from the Kraichnan model for passive scalar. Our framework of analysis should be of help for analytical approaches for the still intractable problem of fluid turbulence itself.

Self-alignment driven by jump processes: Macroscopic limit and numerical investigation

In this paper, we are interested in studying self-alignment mechanisms described as jump processes. In the dynamics proposed, active particles are moving at a constant speed and align with their neighbors at random times following a Poisson process. This dynamics can be viewed as an asynchronous version of the so-called Vicsek model. Starting from this particle dynamics, we introduce the related kinetic description and then derive a continuum hydrodynamic model. We then introduce different discretization strategies for the hierarchy of proposed models, we numerically study the convergence of the schemes and compare the behaviors of the different systems for several test cases.

Experimental and computational study of damage pocess induced by 1064 nm nanosecond laser pulse on the exit surface of fused silica

Material response and the launch of laser plasma during the 1064 nm nanosecond laser pulse induced damage to the exit surface of fused silica are investigated. Employing a polarization-based two-frame shadowgraphy setup with ～ 60 fs probing resolution, the transient material responses from the rising part of nanosecond pumping pulse to several hundred nanosecond timescale are captured. Using a shearing interferometry setup, the evolution of transient phase shift of laser plasma in the expansion process to the ambient air is also investigated. Inhomogeneous distribution of phase shift caused by the electrons and neutrals in the plasma is quantitatively resolved by employing the fast Fourier transform based filtering algorism. To demonstrate the evolutions of important plasma parameters such as pressure, temperature and density, a continuum hydrodynamic model is numerically solved. The initial pressure of plasma is estimated according to the point-explosion model, and the initial plasma temperature is achieved by calculating the difference between simulating shockwave front radius and experimental value at the same delay. The optimal temperature is chosen when the radius difference is minimal. Main conclusions are as follows. 1) Abundant suprathermal electrons are excited in the early energy deposition process. Part of these electrons contribute to the thermal transport process and produce the laser supported solid-state absorption front (LSSAF) which propagates into the bulk silica. Other electrons escape to the air side and contribute to the formation of air plasma through the impact ionization process. Plasma expansion speed is about 20 km/s during this phase. 2) When the pump pulse is terminated, the LSSAF and air plasma lose their energy supplied and experience a rapid decline of the temperature and expansion velocity. As a result, the final damage crater depth exhibits seldomly no increase compared with the transient crater depth during this phase. Hot bulk plasma formed in this phase becomes the damage precursor and induces the ejection of abundant neutrals probably due to the phase explosion mechanism. Inhomogeneous distribution of stress is formed by Rayleigh-Taylor instability at the interface between hot bulk plasma and surrounding bulk material during the expansion of LSSAF. Radial and circumferential cracks are formed due to the release of stress. 3) Evolution of air plasma follows the conventional evolution process of laser-induced plasma, i. e. , internal pressure, temperature and density decrease quickly with time delay. The simulated transient highest pressure is about 600 MPa. Simulation also predicts the formation of the internal shockwave. Our work will be helpful in understanding the laser damage mechanism of the fused silica optical window.

Long-range ordered vorticity patterns in living tissue induced by cell division.

In healthy blood vessels with a laminar blood flow, the endothelial cell division rate is low, only sufficient to replace apoptotic cells. The division rate significantly increases during embryonic development and under halted or turbulent flow. Cells in barrier tissue are connected and their motility is highly correlated. Here we investigate the long-range dynamics induced by cell division in an endothelial monolayer under non-flow conditions, mimicking the conditions during vessel formation or around blood clots. Cell divisions induce long-range, well-ordered vortex patterns extending several cell diameters away from the division site, in spite of the system’s low Reynolds number. Our experimental results are reproduced by a hydrodynamic continuum model simulating division as a local pressure increase corresponding to a local tension decrease. Such long-range physical communication may be crucial for embryonic development and for healing tissue, for instance around blood clots.

Steady States of Sheared Active Nematics

AbstractA continuum hydrodynamic model has been used to characterize flowing active nematics. The behavior of such a system subjected to a weak steady shear is analyzed. We explore the director structures and flow behaviors of the system in flow-aligning and flow tumbling regimes. Combining asymptotic analysis and numerical simulations, we extend previous studies to give a complete characterization of the steady states for both contractile and extensile particles in flow-aligning and flow-tumbling regimes. Another key prediction of this work is the role of the system size on the steady states of an active nematic system: if the system size is small, the velocity and the director angle files for both flow-tumbling contractile and extensile systems are similar to those of passive nematics; if the system is big, the velocity and the director angle files for flow-aligning contractile systems and tumbling extensile systems are akin to sheared passive cholesterics while they are oscillatory for flow-aligning extensile and tumbling contractile systems.

Cell Divisions Cause Long-Range Well-Ordered Vorticity Patterns in Endothelial Tissue

Endothelial cells line the blood vessels of the circulatory system and adhere tightly to their neighbors to prevent leakage, thus causing the cells to move within the endothelial monolayer i a highly collectively fashion. In healthy blood vessels with a laminar blood flow, the endothelial cell division rate is low, only sufficient to replace apoptotic cells. The division rate significantly increases during embryonic development and under halted or turbulent flow. Using PIV analysis we investigated the long-range dynamics induced by cell division in an endothelial monolayer under non-flow conditions, thus mimicking the conditions during vessel formation or around blood clots. A cell division locally injects energy into the system and strongly influences monolayer dynamics. We show that cell divisions induce long-range, well-ordered vortex patterns extending several cell diameters away from the division site. This is rather surprising considering the system's low Reynolds number. Our experimental results are reproduced by a hydrodynamic continuum model simulating division as a local pressure increase in a non-nematic, meso-scale turbulent state. The long-range physical communication may be crucial for embryonic development and for healing endothelial tissue around blood clots.View Large Image | View Hi-Res Image | Download PowerPoint Slide

EVAPORATIVE DROPLETS IN ONE-COMPONENT FLUIDS DRIVEN BY THERMAL GRADIENTS ON SOLID SUBSTRATES

A continuum hydrodynamic model is presented for one-component liquid–gas flows on nonisothermal solid substrates. Numerical simulations are carried out for evaporative droplets moving on substrates with thermal gradients. For droplets in one-component fluids on heated/cooled substrates, the free liquid–gas interfaces are nearly isothermal. Consequently, a thermal singularity occurs at the contact line while the Marangoni effect due to interfacial temperature variation is suppressed. Through evaporation/condensation near the contact line, the thermal singularity makes the contact angle increase with the increasing substrate temperature. Due to this effect, droplets will move toward the cold end on substrates with thermal gradients. The droplet migration velocity is found to be proportional to the change of substrate temperature across the droplet. It follows that for two droplets of different sizes on a substrate with temperature gradient, the larger droplet moves faster and will catch up with the smaller droplet ahead. As soon as they touch, they coalesce rapidly into an even larger droplet that will move even faster.

Lateral diffusion of lipids separated from rotational and translational diffusion of a fluid large unilamellar vesicle

A new method to separate lateral diffusion of lipids in spherical large unilamellar vesicles from the rotational and the translational diffusion of the vesicle as a whole is proposed. The lateral diffusion coefficient DL is obtained as a time-dependent part of the observed diffusion coefficient in vesicles of 800-nm diameters, by systematically changing the diffusion time interval of the high-field-gradient NMR measurement. Although the lipid is in a confined space, the DL of 1,2-dipalmitoyl-sn-glycero-3-phosphocholine is (1.5±0.6)×10−11m2s−1 in the fluid state at 45°C, more than one order of magnitude faster than the rotational and the translational diffusion coefficients of the vesicle by the hydrodynamic continuum model. The method provides a potential for quantifying the lateral diffusion of lipids and proteins in fluid bilayer vesicles as model cell membranes in a natural manner.

Hydrodynamic drag coefficient for soft core–shell nanoparticles in hydrogels

Nanoparticle transport in porous materials, including nanoparticle gel electrophoresis, hinges on knowledge of the hydrodynamic drag coefficient. Here, we compute the drag force on a soft core–shell sphere translating in a porous medium from a continuum hydrodynamic model in which slow viscous flow in the soft shell and embedding porous medium are modelled using Brinkman's equations. This model unifies the Brinkman force for bare spheres in porous media and the drag force of Masliyah et al. (1987) for soft core–shell spheres in pure viscous fluids. We compare the drag force with the Brinkman force evaluated using the composite sphere hydrodynamic radius, showing that this convenient approximation is reasonable when the Brinkman screening length of the embedding porous medium is large compared to the core radius; otherwise, the Brinkman force can significantly underestimate the force.

Droplet motion in one-component fluids on solid substrates with wettability gradients

Droplet motion on solid substrates has been widely studied not only because of its importance in fundamental research but also because of its promising potentials in droplet-based devices developed for various applications in chemistry, biology, and industry. In this paper, we investigate the motion of an evaporating droplet in one-component fluids on a solid substrate with a wettability gradient. As is well known, there are two major difficulties in the continuum description of fluid flows and heat fluxes near the contact line of droplets on solid substrates, namely, the hydrodynamic (stress) singularity and thermal singularity. To model the droplet motion, we use the dynamic van der Waals theory [Phys. Rev. E 75, 036304 (2007)] for the hydrodynamic equations in the bulk region, supplemented with the boundary conditions at the fluid-solid interface. In this continuum hydrodynamic model, various physical processes involved in the droplet motion can be taken into account simultaneously, e.g., phase transitions (evaporation or condensation), capillary flows, fluid velocity slip, and substrate cooling or heating. Due to the use of the phase field method (diffuse interface method), the hydrodynamic and thermal singularities are resolved automatically. Furthermore, in the dynamic van der Waals theory, the evaporation or condensation rate at the liquid-gas interface is an outcome of the calculation rather than a prerequisite as in most of the other models proposed for evaporating droplets. Numerical results show that the droplet migrates in the direction of increasing wettability on the solid substrates. The migration velocity of the droplet is found to be proportional to the wettability gradients as predicted by Brochard [Langmuir 5, 432 (1989)]. The proportionality coefficient is found to be linearly dependent on the ratio of slip length to initial droplet radius. These results indicate that the steady migration of the droplets results from the balance between the (conservative) driving force due to the wettability gradient and the (dissipative) viscous drag force. In addition, we study the motion of droplets on cooled or heated solid substrates with wettability gradients. The fast temperature variations from the solid to the fluid can be accurately described in the present approach. It is observed that accompanying the droplet migration, the contact lines move through phase transition and boundary velocity slip with their relative contributions mostly determined by the slip length. The results presented in this paper may lead to a more complete understanding of the droplet motion driven by wettability gradients with a detailed picture of the fluid flows and phase transitions in the vicinity of the moving contact line.

A Variational Model for Two-Phase Immiscible Electroosmotic Flow at Solid Surfaces

AbstractWe develop a continuum hydrodynamic model for two-phase immiscible flows that involve electroosmotic effect in an electrolyte and moving contact line at solid surfaces. The model is derived through a variational approach based on the On-sager principle of minimum energy dissipation. This approach was first presented in the derivation of a continuum hydrodynamic model for moving contact line in neutral two-phase immiscible flows (Qian, Wang, and Sheng, J. Fluid Mech. 564, 333-360 (2006)). Physically, the electroosmotic effect can be formulated by the Onsager principle as well in the linear response regime. Therefore, the same variational approach is applied here to the derivation of the continuum hydrodynamic model for charged two-phase immiscible flows where one fluid component is an electrolyte exhibiting electroosmotic effect on a charged surface. A phase field is employed to model the diffuse interface between two immiscible fluid components, one being the electrolyte and the other a nonconductive fluid, both allowed to slip at solid surfaces. Our model consists of the incompressible Navier-Stokes equation for momentum transport, the Nernst-Planck equation for ion transport, the Cahn-Hilliard phase-field equation for interface motion, and the Poisson equation for electric potential, along with all the necessary boundary conditions. In particular, all the dynamic boundary conditions at solid surfaces, including the generalized Navier boundary condition for slip, are derived together with the equations of motion in the bulk region. Numerical examples in two-dimensional space, which involve overlapped electric double layer fields, have been presented to demonstrate the validity and applicability of the model, and a few salient features of the two-phase immiscible electroosmotic flows at solid surface. The wall slip in the vicinity ofmoving contact line and the Smoluchowski slip in the electric double layer are both investigated.

Lateral Diffusion of Membrane Proteins: Consequences of Hydrophobic Mismatch and Lipid Composition

Biological membranes are composed of a large number lipid species differing in hydrophobic length, degree of saturation, and charge and size of the headgroup. We now present data on the effect of hydrocarbon chain length of the lipids and headgroup composition on the lateral mobility of the proteins in model membranes. The trimeric glutamate transporter (GltT) and the monomeric lactose transporter (LacY) were reconstituted in giant unilamellar vesicles composed of unsaturated phosphocholine lipids of varying acyl chain length (14–22 carbon atoms) and various ratios of DOPE/DOPG/DOPC lipids. The lateral mobility of the proteins and of a fluorescent lipid analog was determined as a function of the hydrophobic thickness of the bilayer (h) and lipid composition, using fluorescence correlation spectroscopy. The diffusion coefficient of LacY decreased with increasing thickness of the bilayer, in accordance with the continuum hydrodynamic model of Saffman-Delbrück. For GltT, the mobility had its maximum at diC18:1 PC, which is close to the hydrophobic thickness of the bilayer in vivo. The lateral mobility decreased linearly with the concentration of DOPE but was not affected by the fraction of anionic lipids from DOPG. The addition of DOPG and DOPE did not affect the activity of GltT. We conclude that the hydrophobic thickness of the bilayer is a major determinant of molecule diffusion in membranes, but protein-specific properties may lead to deviations from the Saffman-Delbrück model.

Stick-Slip Motion of Moving Contact Line on Chemically Patterned Surfaces

Based on our continuum hydrodynamic model for immiscible two-phase flows at solid surfaces, the stick-slip motion has been predicted for moving contact line at chemically patterned surfaces [Wang et al., J. Fluid Mech., 605 (2008), pp. 59-78]. In this paper we show that the continuum predictions can be quantitatively verified by molecular dynamics (MD) simulations. Our MD simulations are carried out for two immiscible Lennard-Jones fluids confined by two planar solid walls in Poiseuille flow geometry. In particular, one solid surface is chemically patterned with alternating stripes. For comparison, the continuum model is numerically solved using material parameters directly measured in MD simulations. From oscillatory fluid-fluid interface to intermittent stick-slip motion of moving contact line, we have quantitative agreement between the continuum and MD results. This agreement is attributed to the accurate description down to molecular scale by the generalized Navier boundary condition in our continuum model. Numerical results are also presented for the relaxational dynamics of fluid-fluid interface, in agreement with a theoretical analysis based on the Onsager principle of minimum energy dissipation. PACS: 68.08.-p, 83.50.Rp, 83.10.Mj, 83.10.Ff

Modeling and simulations for molecular scale hydrodynamics of the moving contact line inimmiscible two-phase flows

This paper starts with an introduction to the Onsager principle of minimum energydissipation which governs the optimal paths of deviation and restoration to equilibrium.Then there is a review of the variational approach to moving contact line hydrodynamics.To demonstrate the validity of our continuum hydrodynamic model, numerical results frommodel calculations and molecular dynamics simulations are presented for immiscibleCouette and Poiseuille flows past homogeneous solid surfaces, with remarkableoverall agreement. Our continuum model is also used to study the contact linemotion on surfaces patterned with stripes of different contact angles (i.e. surfaces ofvarying wettability). Continuum calculations predict the stick–slip motion forcontact lines moving along these patterned surfaces, in quantitative agreement withmolecular dynamics simulation results. This periodic motion is tunable throughpattern period (geometry) and contrast in wetting property (chemistry). Theconsequence of stick–slip contact line motion on energy dissipation is discussed.

Reorientations of Aromatic Amino Acids and Their Side Chain Models: Anisotropy Measurements and Molecular Dynamics Simulations

We present a study of reorientation dynamics of the three aromatic amino acids and their side chain models in aqueous solution. Experimentally, anisotropy decay measurements with picosecond time resolution were performed for blocked tryptophan, tyrosine, and phenyalanine and model compounds p-cresol and 3-methylindole. Computationally, rotational diffusion was modeled by molecular dynamics simulations for the three aromatic residues and their side chain models: benzene, toluene, phenol, p-cresol, indole, and 3-methylindole in explicit water. Our simulations used the CHARMM protein force field and associated TIP3P water model and tend to underestimate the rotational correlation times. However, the simulations yield several interesting qualitative insights into reorientational motions that complement the experimental measurements. The effects of substituent and temperature on reorientations of the parent compounds are well reproduced computationally. Additionally, simulations indicate strongly anisotropic reorientations for most of the studied compounds and a separation of time scales between conformational dynamics and rotational diffusion. Comparison with continuum hydrodynamic models suggests that we may consider that the blocked amino acids move under stick boundary conditions, while the dynamics for most of the model compounds falls between stick and slip conditions. Our systematic treatment of blocked amino acids, starting from the parent compounds (benzene, phenol, and indole) provides a baseline for understanding the anisotropy decay signals of more complicated peptide systems.