Reactive Flows in Porous Media: Challenges in Theoretical and Numerical Methods.

We review theoretical and computational research, primarily from the past 10 years, addressing the flow of reactive fluids in porous media. The focus is on systems where chemical reactions at the solid-fluid interface cause dissolution of the surrounding porous matrix, creating nonlinear feedback mechanisms that can often lead to greatly enhanced permeability. We discuss insights into the evolution of geological forms that can be inferred from these feedback mechanisms, as well as some geotechnical applications such as enhanced oil recovery, hydraulic fracturing, and carbon sequestration. Until recently, most practical applications of reactive transport have been based on Darcy-scale modeling, where averaged equations for the flow and reactant transport are solved. We summarize the successes and limitations of volume averaging, which leads to Darcy-scale equations, as an introduction to pore-scale modeling. Pore-scale modeling is computationally intensive but offers new insights as well as tests of averaging theories and pore-network models. We include recent research devoted to validation of pore-scale simulations, particularly the use of visual observations from microfluidic experiments.

Electro-hydrodynamic extraction of DNA from mixtures of DNA and bovine serum albumin.

We report separation of genomic DNA (48 kbp) from bovine serum albumin (BSA) by the electro-hydrodynamic coupling between a pressure-driven flow and a parallel electric field. Electro-hydrodynamic extraction exploits this coupling to trap DNA molecules at the entrance of a microfluidic contraction channel, while allowing proteins and salts to be flushed from the device. Samples (10 µL) containing λ-DNA (1 ng) and BSA (0.3 mg) were injected directly into the device and convected to the contraction channel entrance by a flowing buffer solution. The DNA remains trapped in this region essentially indefinitely, while proteins and salts are eluted. The effectiveness of the concept has been assessed by fluorescence measurements of DNA and BSA concentrations. Electro-hydrodynamic extraction in a single-stage device was found to enhance the concentration of DNA 40-fold, while reducing the BSA concentration by four orders of magnitude. The relative concentrations of DNA to BSA at the contraction channel entrance can be as large as 1.5 : 1, corresponding to an A260/280 ratio of 1.9. The maximum yield of DNA from a salt-free solution is 50%, while salted (150 mM) solutions have a lower yield (38%).

Transverse migration and microfluidic concentration of DNA using Newtonian buffers.

We present experimental evidence that DNA can be concentrated due to an electrohydrodynamic coupling between a pressure-driven flow and a parallel electric field. The effects of buffer properties on the process were measured in a microfluidic channel. The concentration rates and the efficiency of trapping DNA were quantified as functions of the ion and polymer concentrations of the buffer solution. Buffers with large ion concentrations hindered the ability to trap DNA, reducing the short-time efficiency of the concentration process from nearly 100% to zero. Importantly, DNA was trapped in the microfluidic channel even when the buffer solution lacked any measurable viscoelastic response. These observations indicate that electrohydrodynamic migration drives the concentration of DNA. We found no evidence of viscoelastic migration in these experiments.

Trapping DNA with a high throughput microfluidic device.

Long strands of DNA can be trapped and concentrated near the inlet of a microfluidic channel by applying a pressure gradient and an opposing electric field. The mechanism for trapping involves a migration of DNA perpendicular to both the fluid flow and the electric field. Migration leads to a highly nonuniform distribution of DNA within a cross section of the channel, with the bulk of the DNA concentrated in a thin (10 µm) layer next to the walls of the channel. This highly concentrated layer generates an electrophoretic flux toward the inlet to the device, despite the much larger fluid flow in the opposite direction. In this paper, the extent to which DNA can be trapped and concentrated by this means has been characterized by fluorescence measurements. At short times (<2 hours) nearly all the incoming DNA remains trapped within the device until the electric field is turned off. The DNA largely accumulates near the inlet, but after 30-60 minutes additional DNA starts to accumulate deeper into the channel. Eventually DNA leaks from the device itself, but ≈80% of the incoming DNA can be retained for up to 5 hours. Optimizing the electric field strength can increase the amount of DNA that can be trapped, but the efficiency is not affected by the channel cross-section.

Electro-hydrodynamic concentration of genomic length DNA.

We report a method of concentrating genomic length DNA within a microfluidic device, using a novel mechanism that combines polyelectrolyte migration with electrophoretic recirculation. Suitable combinations of geometry, pressure and voltage will trap long DNA molecules (>10 kbp) within a small volume (approximately 1 nL), amplifying the local concentration at rates in excess of 1000 fold per minute. The rate at which DNA accumulates is length dependent, while charged particles of similar size pass freely through the device. Experimental observations confirm that the rapid accumulation of DNA at the inlet is caused by an outward migration of the polyelectrolyte towards the capillary boundaries, followed by electrophoresis of DNA within the stagnant fluid layer next to the wall.

Transient Pinning and Pulling: A Mechanism for Bending Microtubules.

Microtubules have a persistence length of the order of millimeters in vitro, but inside cells they bend over length scales of microns. It has been proposed that polymerization forces bend microtubules in the vicinity of the cell boundary or other obstacles, yet bends develop even when microtubules are polymerizing freely, unaffected by obstacles and cell boundaries. How these bends are formed remains unclear. By tracking the motions of microtubules marked by photobleaching, we found that in LLC-PK1 epithelial cells local bends develop primarily by plus-end directed transport of portions of the microtubule contour towards stationary locations (termed pinning points) along the length of the microtubule. The pinning points were transient in nature, and their eventual release allowed the bends to relax. The directionality of the transport as well as the overall incidence of local bends decreased when dynein was inhibited, while myosin inhibition had no observable effect. This suggests that dynein generates a tangential force that bends microtubules against stationary pinning points. Simulations of microtubule motion and polymerization accounting for filament mechanics and dynein forces predict the development of bends of size and shape similar to those observed in cells. Furthermore, simulations show that dynein-generated bends at a pinning point near the plus end can cause a persistent rotation of the tip consistent with the observation that bend formation near the tip can change the direction of microtubule growth. Collectively, these results suggest a simple physical mechanism for the bending of growing microtubules by dynein forces accumulating at pinning points.

Transverse migration of polyelectrolytes in microfluidic channels induced by combined shear and electric fields.

If a dilute solution of a polyelectrolyte such as DNA is forced through a microcapillary by an electric field, while simultaneously driven by a pressure gradient, then the polymer will migrate in directions transverse to the field lines. Here we investigate the sharp increase in concentration in the center of the channel that arises when the flow and electric field drive the polymer in the same direction. We report the first systematic investigation of the effects of flow velocity, electric field, and ionic strength on the degree of migration. Our experiments show that migration increases with increasing shear and electric field as predicted by kinetic theory [Butler et al., Phys. Fluids, 2007, 19, 113101], but eventually saturates as suggested by computer simulations [Kekre et al., Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2010, 82, 050803(R)]. The addition of salt reduces the strength of the migration, consistent with a screening of long-range hydrodynamic flow fields by added salt. However, increasing the ionic strength of a Tris-acetate-EDTA buffer solution has much less effect on the degree of migration.

Effects of dynein on microtubule mechanics and centrosome positioning.

To determine forces on intracellular microtubules, we measured shape changes of individual microtubules following laser severing in bovine capillary endothelial cells. Surprisingly, regions near newly created minus ends increased in curvature following severing, whereas regions near new microtubule plus ends depolymerized without any observable change in shape. With dynein inhibited, regions near severed minus ends straightened rapidly following severing. These observations suggest that dynein exerts a pulling force on the microtubule that buckles the newly created minus end. Moreover, the lack of any observable straightening suggests that dynein prevents lateral motion of microtubules. To explain these results, we developed a model for intracellular microtubule mechanics that predicts the enhanced buckling at the minus end of a severed microtubule. Our results show that microtubule shapes reflect a dynamic force balance in which dynein motor and friction forces dominate elastic forces arising from bending moments. A centrosomal array of microtubules subjected to dynein pulling forces and resisted by dynein friction is predicted to center on the experimentally observed time scale, with or without the pushing forces derived from microtubule buckling at the cell periphery.

Comparison of lattice-Boltzmann and brownian-dynamics simulations of polymer migration in confined flows.

This paper compares results from lattice-Boltzmann and brownian-dynamics simulations of polymer migration in confined flows bounded by planar walls. We have considered both a uniform shear rate and a constant pressure gradient. Lattice-Boltzmann simulations of the center-of-mass distribution agree quantitatively with brownian-dynamics results, contradicting previously published results. The mean end-to-end distance of the extended polymer is more sensitive to grid resolution Δx and time-step Δt. Nevertheless, for sufficiently small Δx and Δt, convergent results for the polymer stretch are obtained which agree with brownian dynamics within statistical uncertainties. The brownian-dynamics simulations incorporate a mobility matrix for a confined polymer that is both symmetric and positive definite for all physically accessible configurations.

Mechanics of Vorticella contraction.

Vorticella convallaria is one of a class of fast-moving organisms that can traverse its body size in less than a millisecond by rapidly coiling a slender stalk anchoring it to a nearby surface. The stalk houses a fiber called the spasmoneme, which winds helically within the stalk and rapidly contracts in response to calcium signaling. We have developed a coupled mechanical-chemical model of the coiling process, accounting for the coiling of the elastic stalk and the binding of calcium to the protein spasmin. Simulations of the model describe the contraction and recovery processes quantitatively. The stalk-spasmoneme system is shown to satisfy geometric constraints, which explains why the cell body sometimes rotates during contraction. The shape of the collapsing and recovering stalk bounds its effective bending stiffness. Simulations suggest that recovery from the contracted state is driven by the stalk at a rate controlled by dissociation of calcium from spasmin.

Role of hydrodynamic interactions in the migration of polyelectrolytes driven by a pressure gradient and an electric field.

Experiments have shown that DNA molecules in capillary electrophoresis migrate across field lines if a pressure gradient is applied simultaneously. We suggest that this migration results from an electrically driven flow field around the polyelectrolyte, which generates additional contributions to the center-of-mass velocity if the overall polymer conformation is asymmetric. This hypothesis leads to a coarse-grained polymer model, without explicit charges, that quantitatively explains the experimentally observed migration. The simulations contradict the widely held notion that charge neutrality eliminates the effects of hydrodynamic interactions in electrically driven flows of polyelectrolytes. We predict a measurable increase in the electrophoretic velocity of a sheared polyelectrolyte that depends on chain length.

Comparison of the static and dynamic properties of a semiflexible polymer using lattice Boltzmann and Brownian-dynamics simulations.

The aim of this paper is to compare results from lattice Boltzmann and Brownian-dynamics simulations of linear chain molecules. We have systematically varied the parameters that may affect the accuracy of the lattice Boltzmann simulations including grid resolution, temperature, polymer mass, and fluid viscosity. The effects of the periodic boundary conditions are minimized by an analytic correction for the different long-range interactions in periodic and unbounded systems. Lattice Boltzmann results for the diffusion coefficient and Rouse mode relaxation times were found to be insensitive to temperature, which suggests that effects of hydrodynamic retardation are small. By increasing the resolution of the lattice Boltzmann grid with respect to the polymer size, convergent results for the diffusion coefficient and relaxation times were obtained; these results agree with Brownian dynamics to within 1%-2%.

A symplectic integration method for elastic filaments.

A new method is proposed for integrating the equations of motion of an elastic filament. In the standard finite-difference and finite-element formulations the continuum equations of motion are discretized in space and time, but it is then difficult to ensure that the Hamiltonian structure of the exact equations is preserved. Here we discretize the Hamiltonian itself, expressed as a line integral over the contour of the filament. This discrete representation of the continuum filament can then be integrated by one of the explicit symplectic integrators frequently used in molecular dynamics. The model systematically approximates the continuum partial differential equations, but has the same level of computational complexity as molecular dynamics and is constraint-free. Numerical tests show that the algorithm is much more stable than a finite-difference formulation and can be used for high aspect ratio filaments, such as actin.

Statistical mechanics of the fluctuating lattice Boltzmann equation.

We propose a derivation of the fluctuating lattice Boltzmann equation that is consistent with both equilibrium statistical mechanics and fluctuating hydrodynamics. The formalism is based on a generalized lattice-gas model, with each velocity direction occupied by many particles. We show that the most probable state of this model corresponds to the usual equilibrium distribution of the lattice Boltzmann equation. Thermal fluctuations about this equilibrium are controlled by the mean number of particles at a lattice site. Stochastic collision rules are described by a Monte Carlo process satisfying detailed balance. This allows for a straightforward derivation of discrete Langevin equations for the fluctuating modes. It is shown that all nonconserved modes should be thermalized, as first pointed out by Adhikari et al. [Europhys. Lett. 71, 473 (2005)]; any other choice violates the condition of detailed balance. A Chapman-Enskog analysis is used to derive the equations of fluctuating hydrodynamics on large length and time scales; the level of fluctuations is shown to be thermodynamically consistent with the equation of state of an isothermal, ideal gas. We believe this formalism will be useful in developing new algorithms for thermal and multiphase flows.

Transverse migration of a confined polymer driven by an external force.

We demonstrate that a polymer confined to a narrow channel migrates towards the center when driven by an external force parallel to the channel walls. This migration results from asymmetric hydrodynamic interactions between polymer segments and the confining walls. A weak pressure-driven flow, applied in the same direction as the external force, enhances the migration. However, when the pressure gradient and the external force act in opposite directions the polymer can migrate towards the boundaries. Nevertheless, for sufficiently strong forces the polymer always migrates towards the center. A dumbbell kinetic theory explains these results qualitatively. A comparison of our results with experimental measurements on DNA suggests that hydrodynamic interactions in polyelectrolytes are only partially screened. We propose new experiments and analysis to investigate the extent of the screening in polyelectrolyte solutions.

Axial segregation of a settling suspension in a rotating cylinder.

A rotating suspension of nonbuoyant particles can develop striking inhomogeneities in particle concentration, with regular bands of high and low concentration along the symmetry axis. We report Stokes-flow simulations showing that the formation of axial bands is correlated with an inhomogeneous particle distribution in the radial plane. An order parameter, based on the average angular velocity of the particles, characterizes two distinct phases: a low-frequency segregated phase and a high-frequency dispersed phase. The axial band structure develops during the transition between these two phases.

Lattice-Boltzmann simulations of the dynamics of polymer solutions in periodic and confined geometries.

A numerical method to simulate the dynamics of polymer solutions in confined geometries has been implemented and tested. The method combines a fluctuating lattice-Boltzmann model of the solvent [Ladd, Phys. Rev. Lett. 70, 1339 (1993)] with a point-particle model of the polymer chains. A friction term couples the monomers to the fluid [Ahlrichs and Dunweg, J. Chem. Phys. 111, 8225 (1999)], providing both the hydrodynamic interactions between the monomers and the correlated random forces. The coupled equations for particles and fluid are solved on an inertial time scale, which proves to be surprisingly simple and efficient, avoiding the costly linear algebra associated with Brownian dynamics. Complex confined geometries can be represented by a straightforward mapping of the boundary surfaces onto a regular three-dimensional grid. The hydrodynamic interactions between monomers are shown to compare well with solutions of the Stokes equations down to distances of the order of the grid spacing. Numerical results are presented for the radius of gyration, end-to-end distance, and diffusion coefficient of an isolated polymer chain, ranging from 16 to 1024 monomers in length. The simulations are in excellent agreement with renormalization group calculations for an excluded volume chain. We show that hydrodynamic interactions in large polymers can be systematically coarse-grained to substantially reduce the computational cost of the simulation. Finally, we examine the effects of confinement and flow on the polymer distribution and diffusion constant in a narrow channel. Our results support the qualitative conclusions of recent Brownian dynamics simulations of confined polymers [Jendrejack et al., J. Chem. Phys. 119, 1165 (2003) and Jendrejack et al., J. Chem. Phys. 120, 2513 (2004)].

Axial segregation in a cylindrical centrifuge.

We propose a theory for axial segregation of suspensions of non-neutrally buoyant particles in a rotating cylinder. The cylinder is oriented in the horizontal plane, so that any axial forces must arise from interparticle interactions. We show that the hydrodynamic interaction between pairs of particles produces a relative motion in the axial direction, independent of the gravitational force. If the particles are denser than the suspending fluid, differential centrifuging between particles at different radial positions leads to an at-tractive interaction, inducing a rapid growth of axial density perturbations. We suggest that this mecha-nism can explain the origin of band formation in rotating suspensions of non-neutrally buoyant particles.