No-slip Surface Research Articles

Enhanced electroosmotic flow of Herschel-Bulkley fluid in a channel patterned with periodically arranged slipping surfaces

In this paper, we consider the electroosmotic flow (EOF) of a viscoplastic fluid within a slit nanochannel modulated by periodically arranged uncharged slipping surfaces and no-slip charged surfaces embedded on the channel walls. The objective of the present study is to achieve an enhanced EOF of a non-Newtonian yield stress fluid. The Herschel-Bulkley model is adopted to describe the transport of the non-Newtonian electrolyte, which is coupled with the ion transport equations governed by the Nernst-Planck equations and the Poisson equation for electric field. A pressure-correction-based control volume approach is adopted for the numerical computation of the governing nonlinear equations. We have derived an analytic solution for the power-law fluid when the periodic length is much higher than channel height with uncharged free-slip patches. An agreement of our numerical results under limiting conditions with this analytic model is encouraging. A significant EOF enhancement and current density in this modulated channel are achieved when the Debye length is in the order of the nanochannel height. Flow enhancement in the modulated channel is higher for the yield stress fluid compared with the power-law fluid. Unyielded region develops adjacent to the uncharged slipping patches, and this region expands as slip length is increased. The impact of the boundary slip is significant for the shear thinning fluid. The results indicate that the channel can be cation selective and nonselective based on the Debye layer thickness, flow behavior index, yield stress, and planform length of the slip stripes.

MHD free convection in a vertical slit micro-channel with super-hydrophobic slip and temperature jump: non-linear Boussinesq approximation approach

An exact solution is obtained for natural convection flow of electrically conducting fluid in a vertical slit micro-channel with super-hydrophobic slip and temperature jump using non-linear Boussinesq approximation approach. The inner surface of one of the micro-channel surfaces is structurally changed so that it exhibited super-hydrophobic velocity slip and jump in temperature. A transverse magnetic field is applied to the flow direction. A unified solution is derived for two different physical situations. Case I, represents a physical situation when super-hydrophobic surface is heated and no-slip surface is unheated while case II represents a physical situation when no-slip surface is heated and super-hydrophobic surface is unheated. The velocity slip lowers the Nusselt number in Case I and raises it in Case II while the temperature jump co-efficient does the opposite. Velocity and flow rate are higher in case of non-linear density variation with temperature in comparison to linear density variation with temperature.

Stokes resistance of a solid cylinder near a superhydrophobic surface. Part 1. Grooves perpendicular to cylinder axis

An important class of canonical problems that is employed in quantifying the slipperiness of microstructured superhydrophobic surfaces is concerned with the calculation of the hydrodynamic loads on adjacent solid bodies whose size is large relative to the microstructure period. The effect of superhydrophobicity is most pronounced when the latter period is comparable to the separation between the solid probe and the superhydrophobic surface. We address the above distinguished limit, considering a simple configuration where the superhydrophobic surface is formed by a periodically grooved array, in which air bubbles are trapped in a Cassie state, and the solid body is an infinite cylinder. In the present part, we consider the case where the grooves are aligned perpendicular to the cylinder and allow for three modes of rigid-body motion: rectilinear motion perpendicular to the surface; rectilinear motion parallel to the surface, in the groove direction; and angular rotation about the cylinder axis. In this scenario, the flow is periodic in the direction parallel to the axis. Averaging over the small-scale periodicity yields a modified lubrication description where the small-scale details are encapsulated in two auxiliary two-dimensional cell problems which respectively describe pressure- and boundary-driven longitudinal flow through an asymmetric rectangular domain, bounded by a compound surface from the bottom and a no-slip surface from the top. Once the integral flux and averaged shear stress associated with each of these cell problems are calculated as a function of the slowly varying cell geometry, the hydrodynamic loads experienced by the cylinder are provided as quadratures of nonlinear functions of the latter distributions over a continuous sequence of cells.

A theoretical study of permeability enhancement for ultrafiltration ceramic membranes with conical pores and slippage

Ceramic membranes are currently favourable in membrane filtration applications due to their excellent mechanical strength, thermal and chemical resistance, backflush capability, and thus a long-service cycle. Coated on top of a mesoporous support, the selective top layer of ultrafiltration ceramic membranes has pore size not exceeding a few tens of nanometers and thickness in the order of O10 μm. In fact, the permeability of an ultrafiltration ceramic membrane can be estimated by the permeability of the top layer due to its smallest pore size. Without impairing the filtration function but still improving the permeability, a gradient conical pore shape is proposed. Two formulae for the filtrate flow rate versus pressure drop relationship through a conical pore exhibiting surface slippage are established here by extending the Hagen-Poiseuille law and an analytical solution for the axisymmetric creeping flow. It is analytically proved that the surface slip length in a conical flow is proportional to a local pore radius by a slip coefficient that is unique for a given pore configuration at a prescribed flow rate. The permeability of a conical-pore membrane is enhanced for radius ratio not exceeding 6.5. The optimum configuration, achieved at a ratio of 2.3, produces an enhancement factor for a membrane permeability of 1.5 for a no-slip surface; this enhancement increases linearly with the slip coefficient if a surface slippage exists.

Viscous growth and rebound of a bubble near a rigid surface

Motivated by the dynamics of microbubbles near catalytic surfaces in bubble-powered microrockets, we consider theoretically the growth of a free spherical bubble near a flat no-slip surface in a Stokes flow. The flow at the bubble surface is characterised by a constant slip length allowing us to tune the hydrodynamic mobility of its surface and tackle in one formulation both clean and contaminated bubbles as well as rigid shells. Starting with a bubble of infinitesimal size, the fluid flow and hydrodynamic forces on the growing bubble are obtained analytically. We demonstrate that, depending on the value of the bubble slip length relative to the initial distance to the wall, the bubble will either monotonically drain the fluid separating it from the wall, which will exponentially thin, or it will bounce off the surface once before eventually draining the thin film. Clean bubbles are shown to be a singular limit which always monotonically get repelled from the surface. The bouncing events for bubbles with finite slip lengths are further analysed in detail in the lubrication limit. In particular, we identify the origin of the reversal of the hydrodynamic force direction as due to the change in the flow pattern in the film between the bubble and the surface and to the associated lubrication pressure. Last, the final drainage dynamics of the film is observed to follow a universal algebraic scaling for all finite slip lengths.

The effect of contamination on the bubble cluster formation in swarm of spherical bubbles rising along an inclined flat wall

Abstract The motion of mono-dispersed spherical bubbles rising along an inclined flat wall is investigated experimentally. An inclination angle of the flat wall is changed to control the bubble Reynolds number set to be 100. As an experimental condition, the boundary condition on the bubble surface (free-slip and no-slip conditions) is controlled, and its effect on the motion of the bubbles has been analyzed. MgSO4 solution and Triton X-100 solution are used to achieve free-slip and no-slip bubble surfaces, respectively. Bubble coalescences are almost inhibited by using these additives. The bubbles tend to be horizontally arranged in lines and such arrangements pile up to be remarkable bubble clusters. The radial pair distribution function and the conditional average of the relative velocity of two bubbles are calculated to evaluate the spatial distribution and bubble–bubble interaction quantitatively. The results show that the tandem configuration of bubble pair next to each other frequently appears due to the increment in the rising velocity of the trailing bubble; a horizontally-oriented (side-by-side) configuration is stable compared to a vertically-oriented (tandem) one, which leads to a horizontally arranged bubbles in a single line. A similar tendency for the bubble clustering is confirmed for both free-slip bubbles and no-slip bubbles. However, there are some quantitative differences such as the faster approaching velocity of the trailing bubble in wider wake region of the leading bubble for no-slip bubbles, etc. The bubble cluster velocity as a velocity of highly accumulated bubbles is also defined and evaluated.

Hydrodynamic interaction of a self-propelling particle with a wall : Comparison between an active Janus particle and a squirmer model.

Using lattice Boltzmann simulations we study the hydrodynamics of an active spherical particle near a no-slip wall. We develop a computational model for an active Janus particle, by considering different and independent mobilities on the two hemispheres and compare the behaviour to a standard squirmer model. We show that the topology of the far-field hydrodynamic nature of the active Janus particle is similar to the standard squirmer model, but in the near-field the hydrodynamics differ. In order to study how the near-field effects affect the interaction between the particle and a flat wall, we compare the behaviour of a Janus swimmer and a squirmer near a no-slip surface via extensive numerical simulations. Our results show generally a good agreement between these two models, but they reveal some key differences especially with low magnitudes of the squirming parameter [Formula: see text]. Notably the affinity of the particles to be trapped at a surface is increased for the active Janus particles when compared to standard squirmers. Finally, we find that when the particle is trapped on the surface, the velocity parallel to the surface exceeds the bulk swimming speed and scales linearly with [Formula: see text].

Minimal model for a hydrodynamic fingering instability in microroller suspensions

We derive a minimal continuum model to investigate the hydrodynamic mechanism behind the fingering instability recently discovered in a suspension of microrollers near a floor [Driscoll et al. Nature Physics, 2016]. Our model, consisting of two continuous lines of rotlets, exhibits a linear instability driven only by hydrodynamics interactions, and reproduces the lengthscale selection observed in large scale particle simulations and in experiments. By adjusting only one parameter, the distance between the two lines, our dispersion relation exhibits quantitative agreement with the simulations and qualitative agreement with experimental measurements. Our linear stability analysis indicate that this instability is caused by the combination of the advective and transverse flows generated by the microrollers near a no-slip surface. Our simple model offers an interesting formalism to characterize other hydrodynamic instabilities that have not been yet well understood, such as size scale selection in suspensions of particles sedimenting adjacent to a wall, or the recently observed formations of traveling phonons in systems of confined driven particles.

Axisymmetric motion of two spherical particles in a Brinkman medium with slip surfaces

An effective-medium approach, based on the Brinkman equation is used to study the axisymmetric quasi-steady motion of two spherical particles embedded in a porous medium. The particles are in general of different sizes and translating with different velocities along the line connecting their centers, and allowing for the hydrodynamic slip at their surfaces. Under the Stokes flow approximation, a general solution is constructed using superposition of the basic solutions in two moving spherical coordinate systems based on the centers of the particles. A collocation technique is used to satisfy the boundary conditions on the surfaces of the particles. Numerical results for the normalized drag force acting on each particle are obtained with rapid convergence for various values of slip coefficients, size ratio, separation parameter, velocity ratio of the particles, and permeability parameter. The normalized drag force on each particle reach the single particle limit as the distance between centers grows large enough and each particle then may be translated independent of each other. The accuracy of the numerical technique has been tested against known solutions for two spheres with no-slip surfaces and when the porous medium becomes a clear fluid.

Turbulent boundary layer over a divergent convergent superhydrophobic surface

A direct numerical simulation of a spatially developing turbulent boundary layer over a divergent and convergent superhydrophobic surface (SHS) was performed over the range 800 < Reθ < 1200. The surface patterns were aligned along the streamwise direction. The SHS was modeled as a pattern of free-slip and no-slip surfaces. The gas fraction of the divergent and convergent SHS was the same as that for the straight SHS for a given slip area. The divergent and convergent SHS gave 21% more drag reduction than the straight SHS. Although the maximum value of the streamwise slip velocity was larger over the divergent and convergent SHS, the average slip velocity (Uslip/U∞) was larger over the straight SHS. The greater drag reduction was attributed to the manipulation of the secondary flow in the y–z plane and the changes in the turbulence structure. The streamwise vortices generated by the secondary flow over the divergent and convergent SHS were diminished which reduced drag relative to the flow over the straight SHS. The ejection and sweep motions were weak, and the vortical structure was attenuated near the wall over the divergent and convergent SHS. The skin friction contributions were explored using the velocity–vorticity correlation. The vortex stretching contribution dominated the skin friction budget. The reduced skin friction over the divergent and convergent SHS resulted mainly from reduced vortex stretching.

Effective slip lengths for immobilized superhydrophobic surfaces

Analytical solutions are found for both longitudinal and transverse shear flow, at zero Reynolds number, over immobilized superhydrophobic surfaces comprising a periodic array of near-circular menisci penetrating into a no-slip surface and where the menisci are no longer shear-free but are taken to be no-slip zones. Explicit formulae for the associated longitudinal and transverse effective slip lengths are derived; these are then compared with analogous results for superhydrophobic surfaces of the same characteristic geometry but where the menisci are shear-free. The new formulae give results that are consistent with recent experimental observations that have prompted suggestions that menisci that are assumed to be free of shear have in fact been immobilized. Significantly, for transverse shear flow, it is found that at critical downward meniscus protrusion angles of around$47^{\circ }$, for many surface geometries, it is impossible to distinguish, purely from the effective slip length, between a no-shear and a no-slip boundary condition. We also find that immobilized menisci bowing into the grooves at supercritical angles just below$90^{\circ }$can be almost twice as slippery to transverse shear as no-shear menisci. The results are relevant to recent discussion as to whether surface immobilization, due to contamination by surfactants or other physical mechanisms, is compromising drag reduction properties expected from an assumed no-shear condition.

Symplectic and antiplectic waves in an array of beating cilia attached to a closed body.

By taking into account the hydrodynamic interactions in a one dimensional array of model cilia attached to a no-slip cylinderical surface, we investigate their synchronized motion. We show how the emergence of metachronal waves depends on the initial state of the system and investigate the conditions under which the formation of symplectic and antiplectic waves are possible.

Analysis of the interface configuration and flow characteristics in tanks in a multiphase liquid–gas system using numerical simulation

ABSTRACTA multiphase study was conducted using a turbulence model of large eddy simulation to investigate the interaction between the gaseous phase and the interface and its respective behaviour until the liquid phase movement was established, first in the near interface, as well as the presence of turbulent structures in the study of transport between phases. The results are shown for three surface configurations: a surface with waves in which the Reynolds number and friction velocity of the gaseous phase are, respectively, 210 and 0.25 m/s; a surface with small undulations, 86 and 0.10 m/s; and a flat surface, 43 and 0.05 m/s. Coherent structures are detected on both sides of the interface; these are intensified and less elongated for larger Reynolds numbers. Additionally, the interface exhibits distinct behaviour with regard to the examined phases. For the gaseous phase, it behaves like a no-slip surface.

Dynamics and structure of planar gravity currents propagating down an inclined surface

Planar, Boussinesq, compositional gravity currents formed by the release of a fixed volume of heavier fluid from a closed lock and advancing on an inclined no-slip bottom surface in a reservoir with a horizontal free surface are investigated based on 3-D large eddy simulation. The initial region containing the lock fluid has a rectangular shape. Simulations are conducted for bottom slope angles, θ, between 0° and 60°. The running length of the inclined bottom was sufficiently long to allow a detailed study of the evolution, front dynamics and structure of the current during the latter stages of the deceleration phase (front velocity reduces with time). Results show that currents advancing over inclined surfaces with θ > 10° are characterized by the formation of an intensified mixed vortex (IMV) at the back of the head. The IMV forms faster and its coherence, circulation, and size increase monotonically with increasing bottom slope angle. The paper discusses how the buoyancy in the head varies with varying bottom slope angle and with time. In particular, for θ ≥ 30°, the current reaches a regime where the total buoyancy of the head and IMV is close to a constant and the value of this constant increases with increasing θ. During this regime, the head mainly loses buoyancy to the IMV. For θ ≥ 40°, a close to linear decay of the head buoyancy with time is observed during the later stages of this regime. Simulation results show that, while for relatively small bottom slope angles most of the sediment is entrained beneath the head, for θ > 20° the IMV has a much larger capacity to entrain the sediment compared to the head region past the initial stages of the propagation of the current. This means that sediment entrainment patterns of currents propagating over highly inclined surfaces are qualitatively very different from the widely studied case of currents propagating over horizontal surfaces. The paper also discusses the different regimes observed in the temporal evolution of the front velocity and the applicability of theoretical models derived based on the data obtained for relatively small bottom slope angles and a relatively short evolution of the current to describe the evolution of currents propagating over large bottom slope angles and/or at large times after the start of the deceleration phase. While it is found that mixing increases monotonically with increasing θ, the largest total kinetic energy for a given front position is observed for θ = 30°-40°. Results also show that the largest magnitude of the bed friction velocity is induced for θ = 30°-40°, which means that the currents with the largest capacity to entrain sediment are those with the largest rate of increase of total kinetic energy with the propagation distance.

Influence of hydrodynamic slip on convective transport in flow past a circular cylinder

The presence of a finite tangential velocity on a hydrodynamically slipping surface is known to reduce vorticity production in bluff body flows substantially while at the same time enhancing its convection downstream and into the wake. Here, we investigate the effect of hydrodynamic slippage on the convective heat transfer (scalar transport) from a heated isothermal circular cylinder placed in a uniform cross-flow of an incompressible fluid through analytical and simulation techniques. At low Reynolds (Re > 1) numbers, our theoretical analysis based on Oseen and thermal boundary layer equations allows for an explicit determination of the dependence of the thermal transport on the non-dimensional slip length l(s). In this case, the surface-averaged Nusselt number, Nu transitions gradually between the asymptotic limits of Nu similar to Pe(1/3) and Nu similar to Pe(1/2) for no-slip (l(s) -> 0) and shear-free (l(s) -> infinity) boundaries, respectively. Boundary layer analysis also shows that the scaling Nu similar to Pe(1/2) holds for a shear-free cylinder surface in the asymptotic limit of Re >> 1 so that the corresponding heat transfer rate becomes independent of the fluid viscosity. At finite Re, results from our two-dimensional simulations confirm the scaling Nu similar to Pe(1/2) for a shear-free boundary over the range 0.1 <= Re <= 10(3) and 0.1 <= Pr <= 10. A gradual transition from the lower asymptotic limit corresponding to a no-slip surface, to the upper limit for a shear-free boundary, with l(s), is observed in both the maximum slip velocity and the Nu. The local time-averaged Nusselt number Nu(theta) for a shear-free surface exceeds the one for a no-slip surface all along the cylinder boundary except over the downstream portion where unsteady separation and flow reversal lead to an appreciable rise in the local heat transfer rates, especially at high Re and Pr. At a Reynolds number of 10(3), the formation of secondary recirculating eddy pairs results in appearance of additional local maxima in Nu(theta) at locations that are in close proximity to the mean secondary stagnation points. As a consequence, Nu exhibits a non-monotonic variationwith l(s) increasing initially from its lowermost value for a no-slip surface and then decreasing before rising gradually toward the upper asymptotic limit for a shear-free cylinder. A non-monotonic dependence of the spanwise-averaged Nu on l(s) is observed in three dimensions as well with the three-dimensional wake instabilities that appear at sufficiently low l(s), strongly influencing the convective thermal transport from the cylinder. The analogy between heat transfer and single-component mass transfer implies that our results can directly be applied to determine the dependency of convective mass transfer of a single solute on hydrodynamic slip length in similar configurations through straightforward replacement of Nu and Pr with Sherwood and Schmidt numbers, respectively.

Exact solutions for hydrodynamic interactions of two squirming spheres

We provide exact solutions of the Stokes equations for a squirming sphere close to a no-slip surface, both planar and spherical, and for the interactions between two squirmers, in three dimensions. These allow the hydrodynamic interactions of swimming microscopic organisms with confining boundaries, or with each other, to be determined for arbitrary separation and, in particular, in the close proximity regime where approximate methods based on point-singularity descriptions cease to be valid. We give a detailed description of the circular motion of an arbitrary squirmer moving parallel to a no-slip spherical boundary or flat free surface at close separation, finding that the circling generically has opposite sense at free surfaces and at solid boundaries. While the asymptotic interaction is symmetric under head–tail reversal of the swimmer, in the near field, microscopic structure can result in significant asymmetry. We also find the translational velocity towards the surface for a simple model with only the lowest two squirming modes. By comparing these to asymptotic approximations of the interaction we find that the transition from near- to far-field behaviour occurs at a separation of approximately two swimmer diameters. These solutions are for the rotational velocity about the wall normal, or common diameter of two spheres, and the translational speed along that same direction, and are obtained using the Lorentz reciprocal theorem for Stokes flows in conjunction with known solutions for the conjugate Stokes drag problems, the derivations of which are demonstrated here for completeness. The analogous motions in the perpendicular directions, i.e. parallel to the wall, currently cannot be calculated exactly since the relevant Stokes drag solutions needed for the reciprocal theorem are not available.

Application of the Modified Force-Coupling Method of Tracing the Trajectories of Spherical Bubbles with Solid-Like and Slip Surfaces

The force-coupling method (FCM) developed by Maxey and Patel (2001) was modified and applied to trace the trajectories of spherical bubbles with solid-like and slip surfaces. Careful comparison was made to the experimental results of Takemura et al. (2000, 2002a, 2002b). First, the result obtained by use of the original version of the FCM was compared to the experimental results. It was found that the original FCM was not feasible for tracing spherical bubble trajectories. Then, a correction was made to the FCM calculation of the bubble velocity by renormalization in terms of the bubble Reynolds number, which could very well trace the trajectory of the bubble with a solid-like, no-slip surface, but not that of a bubble with a slip surface. Finally, a substantial correction was made to the monopole term of the FCM, which could trace the trajectory of a bubble with a solid-like or slip surface very well even for the Reynolds number up to 20.

Numerical Study of Magnetic Field on Rough Porous Narrow Journal Bearing with Heterogeneous Surface

This paper examines theoretical study of effect of magnetic field on infinitely short rough porous journal bearing with the heterogeneous slip/ no-slip surface. Using Christensen stochastic model, Reynolds equation for uni-directional longitudinal roughness pattern is derived. The expressions for pressure and load carrying capacity are obtained as a function of various parameters like Hartmann number, roughness parameter, slip parameter, permeability parameter. It is observed that inclusion of roughness effect and Hartmann number boosts the load carrying capacity.

Force-induced elastic matrix-mediated interactions in the presence of a rigid wall.

We consider a soft elastic matrix that contains particulate inclusions and is bounded by a rigid wall, e.g., the substrate. Such a situation arises in elastic composite materials. They may serve as soft actuators when forces are imposed on or induced between the embedded particles. We investigate how the presence of the rigid wall affects the interactions between the inclusions. First, for no-slip boundary conditions, we transfer Blake's derivation of a corresponding Green's function from low-Reynolds-number hydrodynamics to the linearly elastic case. Then, we list the general expressions to describe the situation for point-like particles in the presence of no-slip and free-slip surface conditions. To compare the effect of the different surface conditions to each other and to the bulk behavior, we address the example situation of pairwise interactions between two embedded particles. The axis through both particle centers is either aligned parallel or perpendicular to the surface. Our results suggest that walls with free-slip surface conditions are preferred when they serve as substrates for soft actuators made from elastic composite materials. As we further demonstrate, the presence of a rigid wall can qualitatively change the interactions between the inclusions. In effect, it can switch attractive interactions into repulsive ones (and vice versa). It should be straightforward to observe the effects in future experiments and to combine our results, e.g., with the modeling of biological cells and tissue on rigid surfaces.

Hydrodynamics of micro-swimmers in films

One of the principal mechanisms by which surfaces and interfaces affect microbial life is by perturbing the hydrodynamic flows generated by swimming. By summing a recursive series of image systems, we derive a numerically tractable approximation to the three-dimensional flow fields of a stokeslet (point force) within a viscous film between a parallel no-slip surface and a no-shear interface and, from this Green’s function, we compute the flows produced by a force- and torque-free micro-swimmer. We also extend the exact solution of Liron &amp; Mochon (J. Engng Maths, vol. 10 (4), 1976, pp. 287–303) to the film geometry, which demonstrates that the image series gives a satisfactory approximation to the swimmer flow fields if the film is sufficiently thick compared to the swimmer size, and we derive the swimmer flows in the thin-film limit. Concentrating on the thick-film case, we find that the dipole moment induces a bias towards swimmer accumulation at the no-slip wall rather than the water–air interface, but that higher-order multipole moments can oppose this. Based on the analytic predictions, we propose an experimental method to find the multipole coefficient that induces circular swimming trajectories, allowing one to analytically determine the swimmer’s three-dimensional position under a microscope.