Tangential Surface Velocity Research Articles

Influence of heterogeneity or shape on the locomotion of a caged squirmer

The development of novel drug delivery systems, which are revolutionizing modern medicine, is benefiting from studies on microorganisms’ swimming. In this paper we consider a model microorganism (a squirmer) enclosed in a viscous droplet to investigate the effects of medium heterogeneity or geometry on the propulsion speed of the caged squirmer. We first consider the squirmer and droplet to be spherical (no shape effects) and derive exact solutions for the equations governing the problem. For a squirmer with purely tangential surface velocity, the squirmer is always able to move inside the droplet (even when the latter ceases to move as a result of large fluid resistance of the heterogeneous medium). Adding radial modes to the surface velocity, we establish a new condition for the existence of a co-swimming speed (where squirmer and droplet move at the same speed). Next, to probe the effects of geometry on propulsion, we consider the squirmer and droplet to be in Newtonian fluids. For a squirmer with purely tangential surface velocity, numerical simulations reveal a strong dependence of the squirmer's speed on shapes, the size of the droplet and the viscosity contrast. We found that the squirmer speed is largest when the droplet size and squirmer's eccentricity are small, and the viscosity contrast is large. For co-swimming, our results reveal a complex, non-trivial interplay between the various factors that combine to yield the squirmer's propulsion speed. Taken together, our study provides several considerations for the efficient design of future drug delivery systems.

Investigation on the evolution of flow field stability in a spiral separator

The liquid film in a spiral separator undergoes an evolution from unsteady to steady flow with increasing trough length. Understanding the process of the flow field evolution is a prerequisite for revealing the mechanism of particle segregation. In this study, a numerical investigation of the flow field evolution in a spiral separator with the cross-sectional profile of cubic parabola is conducted. The distribution profiles of the liquid film thickness, surface tangential velocity, and net radial flux along the trough length are used to characterize the stability of the liquid film shape, primary flow, and secondary flow, respectively. The results indicate that the flow field characteristics at the inner and middle troughs will reach a stable state earlier than that of the outer trough with the increase in trough length. Before reaching a fully developed steady state, the instability of the flow field characteristics at the outer trough exhibits damped oscillations. The instability of the flow field in the first turn is mainly caused by the collision between the fluid and the outer trough sidewall, which leads to large fluctuations in the flow field characteristics at the outer trough. Moreover, the unbalanced inward and outward flows generated from the collision may be the primary reason for the formation of the secondary flow. After the first turn, the flow field characteristics have approximately stabilized, but a certain transitional trough length is still required to reach a fully developed steady state. The secondary flow shows poorer stability compared with the liquid film shape and primary flow. These observations provide insight into the physics of the fluid flow in a spiral separator.

Wake Flow Measurements Behind Rotating Smooth Spheres and Baseballs Near Critical Reynolds Numbers

Abstract The flow field behind spinning baseballs at two different seam orientations was investigated, and compared with a smooth sphere, to isolate effects of seams on the Magnus effect at Reynolds numbers of 5×104 and 1×105. The rotational speed of the three spheres varied from 0 to 2400 rpm, which are typical of spin rates imparted to a thrown baseball. These spin rates are represented nondimensionally as a relative spin rate relating the surface tangential velocity to the freestream velocity, and varied between 0 and 0.94. Mean velocity profiles, streamline patterns, and power spectral density of the velocity signals were taken using hot-wire anemometry and/or stereoscopic particle image velocimetry in the wake region. The sphere wake orientation changed over a range of relative spin rates, indicating an inverse Magnus effect. Vortex shedding at a Strouhal number of 0.25 was present on the sphere at low relative spin rates. However, the seams on the baseball prevented any consequential change in wake orientation and, at most spin rates, suppressed the shedding frequency exhibited by the sphere. Instead, frequencies corresponding to the seam rotation rates were observed in the wake flow. It was concluded that the so-called inverse Magnus effect recorded by previous investigators at specific combinations of Reynolds number and relative spin rate on a sphere exists for a smooth sphere or an axisymmetrically dimpled sphere but not for a baseball near critical Reynolds numbers, where the wake flow pattern is strongly influenced by the raised seams.

Global mantle flow retrodictions for the early Cenozoic using an adjoint method: evolving dynamic topographies, deep mantle structures, flow trajectories and sublithospheric stresses

SUMMARY During the Cenozoic, the Earth experienced multiple first-order geological events that are likely mantle flow related. These include the termination of large-scale marine inundation in North America in the Palaeocene, the late Tertiary rise of Africa relative to other continents and the long-wavelength tilting of Australia since the late Cretaceous, which occurred when the continent approached the southeast Asia subduction systems on its northward passage from Antartica. Here we explore a suite of eight high-resolution, compressible, global mantle flow retrodictions going back to 50 Ma, using an adoint method with $\approx$670 million finite elements. These retrodictions show for the first time that these events emerge jointly as part of global Cenozoic mantle flow histories. Our retrodictions involve the dynamic effects from an upper mantle low-viscosity zone, assimilate a past plate-motion model for the tangential surface velocity field, probe the influence of two different present-day mantle state estimates derived from seismic tomography, and acknowledge the rheological uncertainties of dynamic Earth models by taking in four different realizations for the radial mantle viscosity profile, two of which were published previously. We find the retrodicted mantle flow histories are sensitive to the present-day mantle state estimate and the rheological properties of the Earth model, meaning that this input information is testable with inferences gleaned from the geological record. For a deep mantle viscosity of $1.7\times 10^{22}$ Pa s and a purely thermal interpretation of seismic structure, lower mantle flow velocities exceed 7 cm yr–1 in some regions, meaning they are difficult to reconcile with the existence of a hotspot reference frame. Conversely, a deep mantle viscosity of $10^{23}$ Pa s yields modest flow velocities (< 3 cm yr–1 ) and stability of deep mantle heterogeneity for much of the retrodiction time, albeit at the expense that African uplift is delayed into the latest Neogene. Retrodictions allow one to track material back in time from any given sampling location, making them potentially useful, for example, to geochemical studies. Our results call for improved estimates on non-isostatic vertical motion of the Earth’s surface—provided, for instance, by basin analysis, seismic stratigraphy, landform studies, thermochronological data or the sedimentation record—to constrain the recent mantle flow history and suggest that mantle flow retrodictions may yield synergies across different Earth science disciplines.

Tribo-dynamics model of a spur gear pair with gyroscopic effect and flexible shaft

This study proposes a tribo-dynamic model of a spur gear by coupling an elastic dynamic model and an isothermal elastohydrodynamic lubrication (EHL) model. In this model, the gear shafts are modelled as flexible bodies by the resultant elements of Timoshenko beam and bar elements, and the gyroscopic effect of gear rotors and shafts is also included. The transient and non-Newtonian effects are considered in the EHL model to predict tooth friction forces and moments. Then, an example is provided to analyze the interaction between gear tribology and dynamic behavior. Results indicate that the fluctuations of mesh force and surface tangential velocity caused by gear vibration considerably affect the pressure and thickness of the lubricant film. The gyroscopic effect exerts a significant influence on the dynamic and tribology properties of the gear drive at high speed. The proposed model has important guidance for the design of lubrication systems and dynamic properties in gear drives.

Potential flow through a cascade of aerofoils: direct and inverse problems.

The potential flow through an infinite cascade of aerofoils is considered as both a direct and inverse problem. In each case, a perturbation expansion about a background uniform flow is assumed where the size of the perturbation is comparable to the aspect ratio of the aerofoils. This perturbation must decay far upstream and also satisfy particular edge conditions, including the Kutta condition at each trailing edge. In the direct problem, the flow field through a cascade of aerofoils of known geometry is calculated. This is solved analytically by recasting the situation as a Riemann-Hilbert problem with only imaginary values prescribed on the chords. As the distance between aerofoils is taken to infinity, the solution is seen to converge to a known analytic expression for a single aerofoil. Analytic expressions for the surface velocity, lift and deflection angle are presented as functions of aerofoil geometry, angle of attack and stagger angle; these show good agreement with numerical results. In the inverse problem, the aerofoil geometry is calculated from a prescribed tangential surface velocity along the chords and upstream angle of attack. This is found via the solution of a singular integral equation prescribed on the chords of the aerofoils.

Asymptotic analysis of flow near a glacier terminus

AbstractThe non-linearly viscous ice flow in the vicinity of a glacier terminus, an observation region, depends crucially on the upstream flow as well as on the local surface and bed conditions. The former requires a likely complex solution of the balance laws and boundary conditions for the complete glacier. However, if the profile and downstream surface tangential velocity in the observation region are measured at an observation time t = 0, and a two-dimensional flow approximation is satisfactory, the complete stress and velocity fields satisfying local reduced model equations in the observation region at time t = 0 can be determined by asymptotic expansions in upstream distance from the (moving) terminus. Thus the full strain-rate and stress tensors are determined without prescribing the basal conditions. The terminus velocity is determined in terms of the net accumulation or melt flux and surface velocity at the terminus, with bounds for advance or retreat. The analysis and illustration are presented for a plane flow approximation.

Retrodictions of Mid Paleogene mantle flow and dynamic topography in the Atlantic region from compressible high resolution adjoint mantle convection models: Sensitivity to deep mantle viscosity and tomographic input model

Although mantle convection at Earth-like vigor is a chaotic process, it has been shown by conceptual studies that one can constrain its flow history back in time for periods comparable to a mantle overturn, ≈100million years, if knowledge of the tangential surface velocity field and estimates on the present-day heterogeneity state are available. Such retrodictions, enabled through computationally demanding adjoint methods, are a promising tool to improve our understanding of deep Earth processes, and to link uncertain geodynamic modeling parameters to geologic observables. Here we present the first mantle flow retrodictions for geodynamically plausible, compressible, high resolution Earth models with ≈670 million finite elements, going back in time to the Mid Paleogene. Our retrodictions involve the dynamic effects from a low viscosity zone (LVZ) in the upper mantle, assimilate a past plate motion model for the tangential surface velocity field, and probe the influence from uncertain modeling parameters by using two different estimates for the present-day heterogeneity state of the mantle as imaged by two recent seismic tomographic studies, and two different values for deep mantle viscosity. Focusing on the African hemisphere, we find that our retrodictions produce a spatially and temporally highly variable asthenosphere flow with faster-than-plate velocities, and a history of dynamic topography characterized by local doming events, in agreement with considerations on plate driving forces, and regional scale uplifts reported in the geologic literature. Our results suggest that improved constraints on non-isostatic vertical motion of Earth's surface — provided, for instance, by basin analysis, seismic stratigraphy, landform studies, or the sedimentation record — will play an important role in our understanding of the recent mantle flow history.

Modeling a spheroidal microswimmer and cooperative swimming in a narrow slit.

We propose a hydrodynamic model for a spheroidal microswimmer with two tangential surface velocity modes. This model is analytically solvable and reduces to Lighthill's and Blake's spherical squirmer model in the limit of equal major and minor semi-axes. Furthermore, we present an implementation of such a spheroidal squirmer by means of particle-based mesoscale hydrodynamics simulations using the multiparticle collision dynamics approach. We investigate its properties as well as the scattering of two spheroidal squirmers in a slit geometry. Thereby we find a stable fixed point, where two pullers swim cooperatively forming a wedge-like conformation with a small constant angle.

Locomotion of a microorganism in weakly viscoelastic liquids.

In the present work we study the motion of microorganisms swimming by an axisymmetric distribution of surface tangential velocity in a weakly viscoelastic fluid. The second-order fluid constitutive equation is used to model the suspending fluid, while the well-known "squirmer model" [M. J. Lighthill, Comm. Pure Appl. Math. 5, 109 (1952); J. R. Blake, J. Fluid Mech. 46, 199 (1971)] is employed to describe the organism propulsion mechanism. A regular perturbation expansion up to first order in the Deborah number is performed, and the generalized reciprocity theorem from Stokes flow theory is then used, to derive analytical formulas for the squirmer velocity. Results show that "neutral" squirmers are unaffected by viscoelasticity, whereas "pullers" and "pushers" are slowed down and hastened, respectively. The power dissipated by the swimming microorganism and the "swimming efficiency" are also analytically quantified.

Mass transfer around slender drops in an extensional flow: Inertial effects at large peclet numbers

AbstractThe exchange of mass between a slender drop and a viscous liquid in an axisymmetric extensional flow and at large Peclet numbers is examined. Four cases are discussed: an inviscid drop and a viscous drop, under both creeping flow conditions and when the external flow has a weak amount of inertia. Solutions are presented as a function of the capillary number (Ca ≫ 1), the viscosity ratio (κ ≪ 1), the external Reynolds number (Re ≪ 1), and the Peclet number (Pe ≫ 1). The results suggest that the steady mass transfer rate, to or from the slender drop, is proportional to the elongation rate to the (relative) high power of 3/2. In general, as inertia increases, the drop becomes thinner and longer, and the average tangential surface velocity and the surface area increase, resulting in larger mass transfer rates, especially near the breakup point.

A unified view of energetic efficiency in active drag reduction, thrust generation and self-propulsion through a loss coefficient with some applications

An analysis of the energy budget for the general case of a body translating in a stationary fluid under the action of an external force is used to define a power loss coefficient. This universal definition of power loss coefficient gives a measure of the energy lost in the wake of the translating body and, in general, is applicable to a variety of flow configurations including active drag reduction, self-propulsion and thrust generation. The utility of the power loss coefficient is demonstrated on a model bluff body flow problem concerning a two-dimensional elliptical cylinder in a uniform cross-flow. The upper and lower boundaries of the elliptic cylinder undergo continuous motion due to a prescribed reflectionally symmetric constant tangential surface velocity. It is shown that a decrease in drag resulting from an increase in the strength of tangential surface velocity leads to an initial reduction and eventual rise in the power loss coefficient. A maximum in energetic efficiency is attained for a drag reducing tangential surface velocity which minimizes the power loss coefficient. The effect of the tangential surface velocity on drag reduction and self-propulsion of both bluff and streamlined bodies is explored through a variation in the thickness ratio (ratio of the minor and major axes) of the elliptical cylinders.

Minimum power consumption for drag reduction on a circular cylinder by tangential surface motion

AbstractWe investigate the effect of a prescribed tangential velocity on the drag force on a circular cylinder in a spanwise uniform cross flow. Using a combination of theoretical and numerical techniques we make an attempt at determining the optimal tangential velocity profiles which will reduce the drag force acting on the cylindrical body while minimizing the net power consumption characterized through a non-dimensional power loss coefficient (${C}_{\mathit{PL}} $). A striking conclusion of our analysis is that the tangential velocity associated with the potential flow, which completely suppresses the drag force, is not optimal for both small and large, but finite Reynolds number. When inertial effects are negligible ($\mathit{Re}\ll 1$), theoretical analysis based on two-dimensional Oseen equations gives us the optimal tangential velocity profile which leads to energetically efficient drag reduction. Furthermore, in the limit of zero Reynolds number ($\mathit{Re}\ensuremath{\rightarrow} 0$), minimum power loss is achieved for a tangential velocity profile corresponding to a shear-free perfect slip boundary. At finite $\mathit{Re}$, results from numerical simulations indicate that perfect slip is not optimum and a further reduction in drag can be achieved for reduced power consumption. A gradual increase in the strength of a tangential velocity which involves only the first reflectionally symmetric mode leads to a monotonic reduction in drag and eventual thrust production. Simulations reveal the existence of an optimal strength for which the power consumption attains a minima. At a Reynolds number of 100, minimum value of the power loss coefficient (${C}_{\mathit{PL}} = 0. 37$) is obtained when the maximum in tangential surface velocity is about one and a half times the free stream uniform velocity corresponding to a percentage drag reduction of approximately 77 %; ${C}_{\mathit{PL}} = 0. 42$ and $0. 50$ for perfect slip and potential flow cases, respectively. Our results suggest that potential flow tangential velocity enables energetically efficient propulsion at all Reynolds numbers but optimal drag reduction only for $\mathit{Re}\ensuremath{\rightarrow} \infty $. The two-dimensional strategy of reducing drag while minimizing net power consumption is shown to be effective in three dimensions via numerical simulation of flow past an infinite circular cylinder at a Reynolds number of 300. Finally a strategy of reducing drag, suitable for practical implementation and amenable to experimental testing, through piecewise constant tangential velocities distributed along the cylinder periphery is proposed and analysed.

Boundary elements solution of stokes flow between curved surfaces with nonlinear slip boundary condition

AbstractThis work presents a boundary integral equation formulation for Stokes nonlinear slip flows based on the normal and tangential projection of the Green's integral representational formulae for the velocity field. By imposing the surface tangential velocity discontinuity (slip velocity) in terms of the nonlinear slip flow boundary condition, a system of nonlinear boundary integral equations for the unknown normal and tangential components of the surface traction is obtained. The Boundary Element Method is used to solve the resulting system of integral equations using a direct Picard iteration scheme to deal with the resulting nonlinear terms. The formulation is used to study flows between curved rotating geometries: i.e., concentric and eccentric Couette flows and single rotor mixers, under nonlinear slip boundary conditions. The numerical results obtained for the concentric Couette flow is validated again a semianalytical solution of the problem, showing excellent agreements. The other two cases, eccentric Couette and single rotor mixers, are considered to study the effect of different nonlinear slip conditions in these flow configurations. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013

Mesoscale simulations of hydrodynamic squirmer interactions

The swimming behavior of self-propelled microorganisms is studied by particle-based mesoscale simulations. The simulation technique includes both hydrodynamics and thermal fluctuations that are both essential for the dynamics of microswimmers. The swimmers are modeled as squirmers, i.e., spherical objects with a prescribed tangential surface velocity, where the focus of thrust generation can be tuned from pushers to pullers. For passive squirmers (colloids), we show that the velocity autocorrelation function agrees quantitatively with the Boussinesq approximation. Single active squirmers show a persistent random-walk behavior, determined by forward motion, lateral diffusion, and orientational fluctuations, in agreement with theoretical predictions. For pairs of squirmers, which are initially swimming in parallel, we find an attraction for pushers and a repulsion for pullers, as expected. The hydrodynamic force between squirmer pairs is calculated as a function of the center-to-center distances d(cm) and is found to be consistent with a logarithmic distance dependence for d(cm) less than about two sphere diameters; here, the force is considerably stronger than expected from the far-field expansion. The dependence of the force strength on the asymmetry of the polar surface velocity is obtained. During the collision process, thermal fluctuations turn out to be very important and to strongly affect the postcollision velocity directions of both squirmers.

Collective Behaviour of Swimming Micro-organisms

The occurrence and mechanism of bioconvection patterns in suspensions of upswimming micro-organisms are described, and the linear stability theory for bioconvection in an otherwise uniform dilute suspension is outlined for the case of algae which swim upwards because they are bottom-heavy. Particular attention is paid to the effect of the stress generated in the suspension by the force-dipole exerted by each swimming cell. Coherent structures are also observed in concentrated suspensions, of bacteria in particular, and it is postulated that these come about solely through the hydrodynamic interactions between cells. We examine the deterministic swimming of model organisms which interact hydrodynamically but do not exhibit intrinsic randomness except in their initial positions and orientations. A micro-organism is modelled as a squirming, inertia-free sphere with prescribed tangential surface velocity. Pairwise interactions have been computed using the boundary element method, and the results stored in a database. The movement of a number of identical squirmers is computed by the Stokesian Dynamics method, with the help of the database of interactions. It is found that the spreading in three dimensions is correctly described as a diffusive process after a sufficiently long time. Scaling arguments are used to estimate this time-scale and the diffusitivities. These depend strongly on volume fraction and mode of squirming. However, in two dimensions the squirmers show a definite tendency to aggregation.

Implementing surfactant mass balance in 2D FEM–ALE models

Practical aspects of implementing surfactant mass balance computation in finite elements models, where the model geometry shape change is captured by utilizing the arbitrary Lagrange–Eulerian method are discussed briefly. The discussion and the reported simulations are carried out in two-dimensional Cartesian coordinates. Two alternative approaches to formulating the governing equation of surfactant mass balance for solving it computationally are presented and discussed. One of the approaches is based on computing the boundary curvature and boundary tangential velocity, as well as their differentials on the boundary, directly. The other approach is based on reformulating the governing equation in order to track the proportional rate of change of local surface area. As a conclusion, it is found that though both of the presented approaches can be configured to perform adequately in terms of surfactant mass conservation, surface differentials that are necessary to compute the surface curvature and surface tangential velocity in the first one of the methods evoke numerical oscillations near those points of the boundary where it is not smooth. The text is accompanied by example simulations and figures.

Development of coherent structures in concentrated suspensions of swimming model micro-organisms

A swimming micro-organism is modelled as a squirming sphere with prescribed tangential surface velocity and referred to as a squirmer. The centre of mass of the sphere may be displaced from the geometric centre, and the effects of inertia and Brownian motion are neglected. The well-known Stokesian dynamics method is modified in order to simulate squirmer motions in a concentrated suspension. The movement of 216 identical squirmers in a concentrated suspension without any imposed flow is simulated in a cubic domain with periodic boundary conditions, and the coherent structures within the suspension are investigated. The results show that (a) a weak aggregation of cells appears as a result of the hydrodynamic interaction between cells; (b) the cells generate collective motions by the hydrodynamic interaction between themselves; and (c) the range and duration of the collective motions depend on the volume fraction and the squirmers' stresslet strengths. These tendencies show good qualitative agreement with previous experiments.

Diffusion of swimming model micro-organisms in a semi-dilute suspension

The diffusive behaviour of swimming micro-organisms should be clarified in order to obtain a better continuum model for cell suspensions. In this paper, a swimming micro-organism is modelled as a squirming sphere with prescribed tangential surface velocity, in which the centre of mass of the sphere may be displaced from the geometric centre (bottom-heaviness). Effects of inertia and Brownian motion are neglected, because real micro-organisms swim at very low Reynolds numbers but are too large for Brownian effects to be important. The three-dimensional movement of 64 or 27 identical squirmers in a fluid otherwise at rest, contained in a cube with periodic boundary conditions, is dynamically computed, for random initial positions and orientations. The computation utilizes a database of pairwise interactions that has been constructed by the boundary element method. In the case of (non-bottom-heavy) squirmers, both the translational and the orientational spreading of squirmers is correctly described as a diffusive process over a sufficiently long time scale, even though all the movements of the squirmers were deterministically calculated. Scaling of the results on the assumption that the squirmer trajectories are unbiased random walks is shown to capture some but not all of the main features of the results. In the case of (bottom-heavy) squirmers, the diffusive behaviour in squirmers' orientations can be described by a biased random walk model, but only when the effect of hydrodynamic interaction dominates that of the bottom-heaviness. The spreading of bottom-heavy squirmers in the horizontal directions show diffusive behaviour, and that in the vertical direction also does when the average upward velocity is subtracted. The rotational diffusivity in this case, at a volume fractionc=0.1, is shown to be at least as large as that previously measured in very dilute populations of swimming algal cells (Chlamydomonas nivalis).

The rheology of a semi-dilute suspension of swimming model micro-organisms

The rheological properties of a cell suspension may play an important role in the flow field generated by populations of swimming micro-organisms (e.g. in bioconvection). In this paper, a swimming micro-organism is modelled as a squirming sphere with prescribed tangential surface velocity, in which the centre of mass of the sphere may be displaced from the geometric centre (bottom-heaviness). Effects of inertia and Brownian motion are neglected, because real micro-organisms swim at very low Reynolds numbers but are too large for Brownian effects to be important. The three-dimensional movement of 64 identical squirmers in a simple shear flow field, contained in a cube with periodic boundary conditions, is dynamically computed, for random initial positions and orientations. The computation utilizes a database of pairwise interactions that has been constructed by the boundary element method. The restriction to pairwise additivity of forces is expected to be justified if the suspension is semi-dilute. The results for non-bottom-heavy squirmers show that the squirming does not have a direct influence on the apparent viscosity. However, it does change the probability density in configuration space, and thereby causes a slight decrease in the apparent viscosity atO(c2), wherecis the volume fraction of spheres. In the case of bottom-heavy squirmers, on the other hand, the stresslet generated by the squirming motion directly contributes to the bulk stress atO(c), and the suspension shows strong non-Newtonian properties. When the background simple shear flow is directed vertically, the apparent viscosity of the semi-dilute suspension of bottom-heavy squirmers becomes smaller than that of inert spheres. When the shear flow is horizontal and varies with the vertical coordinate, on the other hand, the apparent viscosity becomes larger than that of inert spheres. In addition, significant normal stress differences appear for all relative orientations of gravity and the shear flow, in the case of bottom-heavy squirmers.