Steady Linear Flow Research Articles

Second-order inertial forces and torques on a sphere in a viscous steady linear flow

We compute the full set of second-order inertial corrections to the instantaneous force and torque acting on a small spherical rigid particle moving unsteadily in a general steady linear flow. This is achieved by using matched asymptotic expansions and formulating the problem in a coordinate system co-moving with the background flow. Effects of unsteadiness and fluid-velocity gradients are assumed to be small, but to dominate in the far field over those of the velocity difference between the body and fluid, making the results essentially relevant to weakly positively or negatively buoyant particles. The outer solution (which at first order is responsible for the Basset–Boussinesq history force at short time and for shear-induced forces such as the Saffman lift force at long time) is expressed via a flow-dependent tensorial kernel. The second-order inner solution brings a number of different contributions to the force and torque. Some are proportional to the relative translational or angular acceleration between the particle and fluid, while others take the form of products of the rotation/strain rate of the background flow and the relative translational or angular velocity between the particle and fluid. Adding the outer and inner contributions, the known added-mass force or the spin-induced lift force are recovered, and new effects involving the velocity gradients of the background flow are revealed. The resulting force and torque equations provide a rational extension of the classical Basset–Boussinesq–Oseen equation incorporating all first- and second-order fluid inertia effects resulting from both unsteadiness and velocity gradients of the carrying flow.

Time-dependent lift and drag on a rigid body in a viscous steady linear flow

We compute the leading-order inertial corrections to the instantaneous force acting on a rigid body moving with a time-dependent slip velocity in a linear flow field, assuming that the square root of the Reynolds number based on the fluid-velocity gradient is much larger than the Reynolds number based on the slip velocity between the body and the fluid. As a first step towards applications to dilute sheared suspensions and turbulent particle-laden flows, we seek a formulation allowing this force to be determined for an arbitrarily shaped body moving in a general linear flow. We express the equations governing the flow disturbance in a non-orthogonal coordinate system moving with the undisturbed flow and solve the problem using matched asymptotic expansions. The use of the co-moving coordinates enables the leading-order inertial corrections to the force to be obtained at any time in an arbitrary linear flow field. We then specialize this approach to compute the time-dependent force components for a sphere moving in three canonical flows: solid-body rotation, planar elongation, and uniform shear. We discuss the behaviour and physical origin of the different force components in the short-time and quasi-steady limits. Last, we illustrate the influence of time-dependent and quasi-steady inertial effects by examining the sedimentation of prolate and oblate spheroids in a pure shear flow.

UIA: a uniform integrated advection algorithm for steady and unsteady piecewise linear flow field on structured and unstructured grids

Integration-based geometric method is widely used in vector field visualization. To improve the efficiency of integration advection-based visualization, we propose a uniform integrated advection (UIA) algorithm on steady and unsteady vector field according to common piecewise linear field data set analysis. UIA employs cell gradient-based interpolation along spatial and temporal direction, and transforms multi-step advection into single-step advection in association with fourth-order Runge–Kutta advection process. UIA can significantly reduce computational load, and is applicable on arbitrary grid type with cell-center/cell-vertex data structure. The experiments are performed on steady/unsteady vector fields with two-dimensional cell-center unstructured grids and three-dimensional cell-vertex grids, and also on unsteady field from fluid dynamics numerical simulation. The result shows that the proposed algorithm can significantly improve advection efficiency and reduce visualization computational time compared with fourth-order Runge–Kutta.

The orientation dynamics of small prolate and oblate spheroids in linear shear flows

The present work deals with the stable orientation of oblate and prolate spheroids in general steady linear flows and with the mode of convergence to these stable orientations. The orientation dynamics is governed by the Jeffery equation. The stable orientations are either fixed points or limit cycles in the orientation space. The type of stable orientation depends on whether the eigenvalues of the linear part of Jeffery equation are real or complex. We define prolate and oblate spheroids to be equivalent if the aspect ratio of one is the reciprocal of the other. We show that, in a given flow, equivalent oblate and prolate spheroids possess the same number of fixed points and limit cycles of which only one is stable. If they possess only fixed points, then their corresponding stable fixed points are orthogonal. If they possess one fixed point and one limit cycle each, then the stable fixed point of one is orthogonal to the plane of the limit cycle of the other. The rate of convergence to these attractors is important to consideration of the orientations in time-space varying flow fields. We show that non-normal growth (NNG) of the distance to these attractors may delay the convergence by several characteristic shear time scales. We derive conditions for occurrence of NNG and explicit expressions for the maximal duration of the growth. We consider a specific case of which the vorticity is a stable orientation of prolate spheroids. We analyze the conditions that imply monotonic or, conversely, non-monotonic convergence to this orientation due to NNG. We thereby find the corresponding conditions for convergence of the equivalent oblate spheroids to their attractors, normal to the vorticity. We show that the convergence is monotonic if the vorticity is parallel to the strain tensor’s largest eigenvector, but that NNG occurs if the vorticity is parallel to the strain tensor’s intermediate eigenvector. The NNG duration decreases with increasing vorticity-strain ratio and with the strain intermediate eigenvalue approaching the largest eigenvalue.

Errors due to a practical Green function for steady ship waves

Errors that stem from a practical analytical approximation La to the local flow component L in the Green function associated with steady linear potential flow around a ship hull are considered. Although the approximation La is not very accurate, the flow potentials evaluated via the exact local flow component L or the approximation La for Froude numbers F=0.15, 0.3 and 0.5 cannot be distinguished, except at F=0.15 for which relatively small differences can be observed. Moreover, the sinkage, the trim angle and the wave drag predicted by the Neumann–Michell (NM) theory, with the local flow potential evaluated using L or La, are in very close agreement. Despite its remarkably simplicity, the analytical approximation La can then be used to compute the local flow component L in the Green function, in the entire flow region, within the framework of the NM linear potential flow theory and the related Hogner approximation. The practical analytical approximation La is far more practical than the basic integral representation of L, and is an important element of the NM theory. Indeed, the analytical approximation La, and other features of the NM theory, make it possible to compute the flow around a steadily advancing ship hull in a highly efficient way, as required for routine practical applications to design and hull-form optimization.

Application of a 2D harmonic polynomial cell (HPC) method to singular flows and lifting problems

Further developments and applications of the 2D harmonic polynomial cell (HPC) method proposed by Shao and Faltinsen [22] are presented. First, a local potential flow solution coupled with the HPC method and based on the domain decomposition strategy is proposed to cope with singular potential flow characteristics at sharp corners fully submerged in a fluid. The results are verified by comparing them with the analytical added mass of a double-wedge in infinite fluid. The effect of the singular potential flow is not dominant for added mass and damping, but the error is non-negligible when calculating mean wave loads using direct pressure integration. Then, the double-layer nodes technique is used to simulate a thin free shear layer shed from lifting bodies, across which the velocity potential is discontinuous. The results are verified by comparing them with analytical results for steady and unsteady lifting problems of a flat plate in infinite fluid. The latter includes the Wagner problem and the Theodorsen functions. Satisfactory agreement with other numerical results is documented for steady linear flow past a foil and beneath the free surface.

Wrinkling instability of vesicles in steady linear flow

We present experimental observations and numerical simulations of a wrinkling instability that occurs at sufficiently high strain rates in the trembling regime of vesicle dynamics in steady linear flow. Spectral and statistical analysis of the data shows similarities and differences with the wrinkling instability observed earlier for vesicles in transient elongation flow. The critical relevance of thermal fluctuations for this phenomenon is revealed by a simple model using coupled Langevin equations that reproduces the experimental observations quite well.

Orientational dynamics of weakly inertial axisymmetric particles in steady viscous flows

The orientational dynamics of weakly inertial axisymmetric particles in a steady flow is investigated. We derive an asymptotic equation of motion for the unit axial vector along the particle symmetry axis, valid for small Stokes number St, and for any axisymmetric particle in any steady linear viscous flow. This reduced dynamics is analysed in two ways, both pertain to the case of a simple shear flow. In this case inertia induces a coupling between precession and nutation. This coupling affects the dynamics of the particle, breaks the degeneracy of the Jeffery orbits, and creates two limiting periodic orbits. We calculate the leading-order Floquet exponents of the limiting periodic orbits and show analytically that prolate objects tend to a tumbling orbit, while oblate objects tend to a log-rolling orbit, in agreement with previous analytical and numerical results. Second, we analyse the role of the limiting orbits when rotational noise is present. We formulate the Fokker–Planck equation describing the orientational distribution of an axisymmetric particle, valid for small St and general Péclet number Pe. Numerical solutions of the Fokker–Planck equation, obtained by means of expansion in spherical harmonics, show that stationary orientational distributions are close to the inertia-free case when Pe St≪1, whereas they are determined by inertial effects, though small, when Pe≫1/St≫1.

Buoyancy Arrest and Shelf–Ocean Exchange

Abstract When steady flow in a stratified ocean passes between the continental slope and open ocean, its ability to cross isobaths is potentially limited by buoyancy arrest. If the bottom Ekman transport vanishes and there are no interior stresses, then steady linear flow on an f plane must be geostrophic and follow isobaths exactly. The influence of arrest on cross-shelf transport is investigated here to establish 1) whether there are substantial penetration asymmetries between cases with upwelling and downwelling in the bottom boundary layer; 2) over what spatial scales, hence in what parameter regime, buoyancy arrest is important; and 3) the effects of depth-dependent interior flow. The problem is approached using scalings and idealized numerical models. The results show that there is little or no asymmetry introduced by bottom boundary layer behavior. Further, if the stratification is weak or moderate, as measured by a slope Burger number s = αN/f (where α is the bottom slope, N is buoyancy frequency, and f is the Coriolis parameter), buoyancy arrest does not exert a strong constraint on cross-isobath exchange.

Interaction of toroidal focal conic defects with shear flow

Toroidal focal conic defects form in smectic liquid crystals due to antagonistic surface anchoring conditions. The resulting layer structure, defect size, and the formation of ordered patterns of defects have been studied in detail. Here, we investigate the effect of shear flow on toroidal focal conic defects. The defects are formed and subjected to a steady linear Couette flow in a microscale shear cell. In situ visualization of the evolution of the polarization texture of individual defects reveals the impact of three distinct flow regimes on the layer structure. At low Ericksen numbers, individual defects exhibit an early time elastic regime in which the optical intensity of the layer structure does not change in the presence of the applied shear flow. At increasing applied strains, the defect layer structure deforms, resulting in a monotonically increasing optical intensity within its original footprint. At even larger applied strains, satellite defects are emitted from the anchored defect. Cessation of shear flow at moderate strains, prior to the formation of satellite defects, leads to an exponential decay of the polarization intensity of the defect, suggesting that relaxation of the layer structure occurs with a time constant that is consistent with that predicted by a permeation flow mechanism. This study establishes the relevant length and time scales involved in the interaction of shear stresses and elastic stresses in smectic liquid crystalline samples whose structure deviates significantly from an idealized aligned lamellar configuration.

Solving the steady‐state groundwater flow equation for finite linear aquifers using a generalized Fourier series approach in two‐dimensional domains

AbstractA method to solve steady linear groundwater flow problems using generalized Fourier Series is developed and particularized for multiple Fourier series in two‐dimensional domains. It leads to a linear vector equation whose solution provides a finite number of generalized Fourier coefficients approximating the hydraulic head field. Its implementation is shown and two relevant properties are found for the system matrix. It is always symmetric and, once computed, if additional Fourier terms are needed for a better approximation of the hydraulic head field, previously computed matrix elements remain invariant, i.e. only new rows and columns are added to the system matrix. The method is demonstrated in three simple cases with different geometries and transmissivity fields, where solutions are compared with analytical and finite element method results. Thus, the method is verified as an alternative to other flow solvers. Additionally, it provides a direct way to obtain the spectral form of the flow equation solution, given a spectral representation of transmissivity, and can be easily extended to obtain continuous velocity fields and their approximated spectral expressions. Copyright © 2009 John Wiley & Sons, Ltd.

Collective diffusion in sheared colloidal suspensions

Collective diffusivity in a suspension of rigid particles in steady linear viscous flows is evaluated by investigating the dynamics of the time correlation of long-wavelength density fluctuations. In the absence of hydrodynamic interactions between suspended particles in a dilute suspension of identical hard spheres, closed-form asymptotic expressions for the collective diffusivity are derived in the limits of low and high Péclet numbers, where the Péclet number ${\it Pe}\,{=}\,\gamdot a^2/D_0$ with $\gamdot$ being the shear rate and D0 = kBT/6πη a is the Stokes–Einstein diffusion coefficient of an isolated sphere of radius a in a fluid of viscosity η. The effect of hydrodynamic interactions is studied in the analytically tractable case of weakly sheared (Pe ≪ 1) suspensions.For strongly sheared suspensions, i.e. at high Pe, in the absence of hydrodynamics the collective diffusivity Dc = 6 Ds∞, where Ds∞ is the long-time self-diffusivity and both scale as $\phi \gamdot a^2$, where φ is the particle volume fraction. For weakly sheared suspensions it is shown that the leading dependence of collective diffusivity on the imposed flow is proportional to D0 φPeÊ, where Ê is the rate-of-strain tensor scaled by $\gamdot$, regardless of whether particles interact hydrodynamically. When hydrodynamic interactions are considered, however, correlations of hydrodynamic velocity fluctuations yield a weakly singular logarithmic dependence of the cross-gradient-diffusivity on k at leading order as ak → 0 with k being the wavenumber of the density fluctuation. The diagonal components of the collective diffusivity tensor, both with and without hydrodynamic interactions, are of O(φPe2), quadratic in the imposed flow, and finite at k = 0.At moderate particle volume fractions, 0.10 ≤ φ ≤ 0.35, Brownian Dynamics (BD) numerical simulations in which there are no hydrodynamic interactions are performed and the transverse collective diffusivity in simple shear flow is determined via time evolution of the dynamic structure factor. The BD simulation results compare well with the derived asymptotic estimates. A comparison of the high-Pe BD simulation results with available experimental data on collective diffusivity in non-Brownian sheared suspensions shows a good qualitative agreement, though hydrodynamic interactions prove to be important at moderate concentrations.

Analytical solution of boundary integral equations for 2-D steady linear wave problems

Based on the Fourier transform, the analytical solution of boundary integral equations formulated for the complex velocity of a 2-D steady linear surface flow is derived. It has been found that before the radiation condition is imposed, free waves appear both far upstream and downstream. In order to cancel the free waves in far upstream regions, the eigensolution of a specific eigenvalue, which satisfies the homogeneous boundary integral equation, is found and superposed to the analytical solution. An example, a submerged vortex, is used to demonstrate the derived analytical solution. Furthermore, an analytical approach to imposing the radiation condition in the numerical solution of boundary integral equations for 2-D steady linear wave problems is proposed.

A LATTICE BOLTZMANN APPROACH FOR IMMISCIBLE FLUIDS WITH VERY DIFFERENT VISCOSITIES

In this article, a lattice Boltzmann model for two immiscible fluids with same density but a large viscosity difference is developed. The effect of surface tension is included. An indicator function is used to model the interface motion and is updated by a lattice Boltzmann scheme. The macroscopic equation for this function is derived using multi-scale analysis. Numerical results are compared with theoretical and experimental ones for the problem of drop deformation under steady linear flows. A qualitative comparison with results given by a level contour reconstruction method for drop collision in three dimensions is also shown. Other numerical applications, such as flow past obstacles and a falling drop, are presented.

On estimating stress in free-draining Kramers chain simulations using stochastic filtering

For free-draining Kramers chains, several possible algorithms for calculating the stress tensor are examined, when Liu’s algorithm is used for the dynamics. Previous methods assumed that Stratonovich calculus was necessary to interpret the stochastic differential equation and calculate the stress, which is incorrect. Here, we consider these previous methods and derive two new ones using Ito calculus. All methods considered give correct averages. However, differences are seen in the statistical error, computational costs, and simplicity. The explicit expression derived by Liu is the most expensive to calculate, and most complicated. A version using ‘stochastic filtering’, proposed by Grassia and Hinch [P. Grassia, E.J. Hinch, Computer simulations of polymer chain relaxation via Brownian motion, J. Fluid Mech. 308 (1996) 255–288], and independently by Doyle et al. [P.S. Doyle, E.S.G. Shaqfeh, A.P. Gast, Dynamic simulation of freely draining flexible polymers in steady linear flows, J. Fluid Mech. 334 (1997) 251–291] yields the same statistical error, but requires less computational cost than Liu’s expression. A new algorithm is proposed, which yields uncertainties identical to those found by Grassia and Hinch, but is simpler, and requires smaller memory storage. An even simpler expression is proposed which, unlike that using stochastic filtering, requires no additional information from that required for the Brownian dynamics simulations. This last expression yields a larger statistical error than the method proposed here, and is found not to be advantageous for computational costs. Hence, the simplified stress calculator introduced here is the recommended one.

Computational studies of the shear flow behaviour of a model for nematic liquid crystalline polymers

We present an accurate and efficient numerical method to the solve the microstructure-flow problem that constitutes the coupling of the molecular-based Doi-Marrucci-Greco model to the Cauchy equation of motion and continuity equation. We also provide a general introduction to the problem of the flow of complex fluids. Preliminary investigations of the predictive capabilities of the Doi-Marrucci-Greco model show that with increasing shear rate, in accordance with both experimental observations and other theoretical predictions, the model exhibits three flow regimes: steady linear shear flow at low shear rate; steady roll cells at intermediate shear rates; and irregular flow and orientation patterns at high shear rates. Given sufficiently high shear rates, the irregular flow structure is accompanied by the formation of ?1 and ?1/2 strength disclinations. Furthermore, at shear rates large enough to inhibit the tumbling of the average molecular orientation, the solution no longer contains disclinations or roll cells, but retains some structure in the flow direction that has the visual appearance of stripes.

Potential vorticity dynamics in a domain with closed geostrophic contours. I: Steady linear flows

Potential vorticity dynamics in a domain with closed geostrophic contours. I: Steady linear flows

Hydrodynamic pair diffusion in isotropic random velocity fields with application to turbulent coagulation

Diffusion and coagulation are investigated in a random, isotropic flow in the presence of hydrodynamic interactions, interparticle forces, and Brownian diffusion. Different strain and rotation rate time scales characterize the velocity field and the particles are assumed small compared with the characteristic length of the flow, so that the velocity field is linear in the vicinity of the particles. The pair probability equation for the relative motion of two particles is written in terms of a diffusion tensor and a drift velocity. This technique is valid in the limit of small strain, i.e., when the product of the characteristic velocity gradient and time scale of the fluctuating velocity gradient is small. A consequence of the drift velocity is that, at steady state in a noncoagulating system, the pair probability distribution is nonuniform when hydrodynamic interactions are included, and there is a higher probability of particle pairs at close proximity. The pair probability conservation equation is used to determine the coagulation rate both without and with consideration of interparticle interactions. The stability factor, W, is the ratio of the coagulation rate in the absence and presence of interparticle forces, and W is calculated numerically for different size particles influenced by van der Waals attraction, electrostatic repulsion, hydrodynamic interactions, and Brownian motion. A semi-analytical expression is derived that is valid for large particles that are not influenced by Brownian motion and that experience weak van der Waals attraction. The analysis shows that colloidal stability increases with increasing particle size and shear rate as a result of the hydrodynamic resistance to particle–particle collision. Double layer repulsion can lead to stable colloidal suspensions, but increasing the fluid shear can reduce this effect. Colloid stability for the randomly varying flow considered here is comparable to that obtained for steady linear flows, such as simple shear when only van der Waals attraction is considered. Compared with the steady linear flows, double layer repulsion imparts additional resistance to aggregation in the randomly varying flow. The relevance of applying this analysis to coagulation in isotropic turbulent flows is discussed.

The Vertical-Scale Cascade in Atmospheric Tracers due to Large-Scale Differential Advection

Abstract Reduction in the vertical scale of atmospheric tracer fields occurs as a result of quasi-horizontal stirring by the large-scale flow, provided that the flow varies in the vertical. This scale reduction and the implications for dissipation and mixing are here analyzed using simple mathematical models. The first is based on a steady linear flow and gives simple insight into the interaction between horizontal strain, vertical shear, and diffusion. The second is a simple random-straining model, with random vertical shear being added to a horizontal random strain field. Analytical progress is possible in the limit where the correlation time for the flow is small or large compared to the inverse strain itself. Numerical integration allows investigation of the intermediate case. In all cases the vertical scale decreases exponentially fast, at the same rate as the horizontal scale, with the rate, say S, being controlled by the statistics of the horizontal strain field. As this exponential decrease occurs...

Dynamic simulation of freely draining flexible polymers in steady linear flows

We present a study of the rheological and optical behaviour of Kramers bead–rod chains in dilute solution using stochastic computer simulations. We consider two model linear flows, steady shear and uniaxial extensional flow, in which we calculate the non-Newtonian Brownian and viscous stress contribution of the polymers, their birefringence and a stress-optic coefficient. We have developed a computer algorithm to differentiate the Brownian from the viscous stress contributions which also avoids the order (δt)−1/2 noise associated with the Brownian forces. The strain or shear rate is made dimensionless with a molecular relaxation time determined by simulated relaxation of the birefringence and stress after a strong flow is applied. The characteristic long relaxation time obtained from the birefringence and stress are equivalent and shown to scale with N2 where N is the number of beads in the chain. For small shear or extension rates the viscous contribution to the effective viscosity is constant and scales as N. We obtain an analytic expression which explains the scaling and magnitude of this viscous contribution. In uniaxial extensional flow we find an increase in the extensional viscosity with the dimensionless flow strength up to a plateau value. Moreover, the Brownian stress also reaches a plateau and we develop an analytic expression which shows that the Brownian stress in an aligned bead–rod chain scales as N3. Using scaling arguments we show that the Brownian stress dominates in steady uniaxial extensional flow until large Wi, Wi ≈ 0.06N2, where Wi is the chain Weissenberg number. In shear flow the viscosity decays as Wi−1/2 and the first normal stress as Wi−4/3 at moderate Wi. We demonstrate that these scalings can be understood through a quasi-steady balance of shear forces with Brownian forces. For small Wi (in shear and uniaxial extensional flow) and for long times (in stress relaxation) the stress-optic law is found to be valid. We show that the rheology of the bead–rod chain can be qualitatively described by an equivalent FENE dumbbell for small Wi.