Linear Velocity Field Research Articles

Drop deformation and emulsion rheology under the combined influence of uniform electric field and linear flow

Electrohydrodynamics of a leaky dielectric suspended drop subjected to the combined influence of a uniform electric field and linear velocity field is analysed analytically and numerically. In the limit of small charge convection and small shape deformation, an analytical solution is obtained for the deformed drop shape when the imposed linear flow is of uniaxial extensional type with the extensional component aligned in the direction of the electric field. This perturbation approach is then applied towards obtaining the effect of a uniform electric field on the effective extensional rheology of a dilute emulsion. Key results indicate that the magnitude and sense of drop deformation not only depends on the material properties of the drop and medium but is also governed by the strength of the applied electric field relative to the applied flow field. The interfacial charge convection is found to increase or decrease the drop deformation depending on the direction of electrohydrodynamic flow and relative strength of electric field. The electrohydrodynamic flow and drop deformation modulates the effective extensional viscosity of the emulsion. Importantly, the presence of the electric field leads to strain-rate-dependent effective extensional viscosity of the emulsion. The emulsion is found to exhibit strain-rate thinning/thickening behaviour depending on the drop to medium charge relaxation time scale. The analytically obtained drop shape and deformation are in excellent agreement with numerical simulations for the small deformation ranges.

Kinematics of a Fluid Ellipse in a Linear Flow

A four-parameter kinematic model for the position of a fluid parcel in a time-varying ellipse is introduced. For any ellipse advected by an arbitrary linear two-dimensional flow, the rates of change of the ellipse parameters are uniquely determined by the four parameters of the velocity gradient matrix, and vice versa. This result, termed ellipse/flow equivalence, provides a stronger version of the well-known result that a linear velocity field maps an ellipse into another ellipse. Moreover, ellipse/flow equivalence is shown to be a manifestation of Stokes’ theorem. This is done by deriving a matrix-valued extension of the classical Stokes’ theorem that involves a spatial integral over the velocity gradient tensor, thus accounting for the two strain terms in addition to the divergence and vorticity. General expressions for various physical properties of an elliptical ring of fluid are also derived. The ellipse kinetic energy is found to be composed of three portions, associated respectively with the circulation, the rate of change of the moment of inertia, and the variance of parcel angular velocity around the ellipse. A particular innovation is the use of four matrices, termed the I J K L basis, that greatly facilitate the required calculations.

Small-scale turbulent magnetic field: Growth vs. decay

Statistical moments of magnetic field in a viscous range of turbulence are calculated for arbitrary initial conditions. It is shown that the evolution of magnetic field in the case of finite initial distribution in a linear velocity field consists of two or three consecutive regimes: exponential growth is followed by exponential decay. This solves the apparent contradiction between “anti-dynamo” theorems and growth of magnetic field with statistically homogeneous initial conditions.

Solution with a linear velocity field for a submodel of one-dimensional gas motions

A method for finding exact solutions of the equations of gas dynamics with a linear velocity field is proposed. This method was used to find exact solutions for one submodel of the evolutionary type which was fully integrated for the case of a polytropic gas. Examples of particle motion for the obtain exact solutions are given.

Решение с линейным полем скоростей для подмодели одномерных движений газа

Решение с линейным полем скоростей для подмодели одномерных движений газа

Accurate and consistent particle tracking on unstructured grids

SummaryA new numerical method for particle tracking (Lagrangian particle advection) on 2‐D unstructured grids with triangular cells is presented and tested. This method combines key attributes of published methods, including streamline closure for steady flows and local mass conservation (uniformity preservation). The subgrid‐scale velocity reconstruction is linear, and this linear velocity field is integrated analytically to obtain particle trajectories. A complete analytic solution to the 2‐D system of ordinary differential equations (ODEs) governing particle trajectories within a grid cell is provided. The analytic solution to the linear system of locally mass‐conserving constraints that must be enforced to obtain the coefficients in the ODEs is also provided. Numerical experiments are performed to demonstrate that the new method has substantial advantages in accuracy over previously published methods and that it does not suffer from unphysical particle clustering. The method can be used not only in particle‐tracking applications but also as part of a semi‐Lagrangian advection scheme.Copyright © 2015 John Wiley & Sons, Ltd.

Large-scale peculiar velocities through the galaxy luminosity function at z ~ 0.1

AbstractPeculiar motion introduces systematic variations in the observed luminosity distribution of galaxies. This allows one to constrain the cosmic peculiar velocity field from large galaxy redshift surveys. Using around half a million galaxies from the SDSS Data Release 7 at z ~ 0.1, we demonstrate the applicability of this approach to large datasets and obtain bounds on peculiar velocity moments and σ8, the amplitude of the linear matter power spectrum. Our results are in good agreement with the ΛCDM model and consistent with the previously reported ~ 1% zero-point tilt in the SDSS photometry. Finally, we discuss the prospects of constraining the growth rate of density perturbations by reconstructing the full linear velocity field from the observed galaxy clustering in redshift space.

The plane isothermal motions of an ideal gas without expansions

A general solution of the equations of plane isothermal motions of an ideal gas without expansions is obtained. A method of reducing the overdetermined system of differential equations to an involution is proposed, which consists of obtaining integrable relations. All representations of the solution in time are obtained: polynomial, harmonic and biharmonic, and representations of linearly and exponentially increasing harmonics. All the representations give solutions of the overdetermined system, among which there are all the solutions with linear velocity fields, obtained by Ovsyannikov.

Equilibrium figures with polar rings

AbstractWe study the dynamics of a spheroidal galaxy with a gravitating polar ring whose plane is perpendicular to the equator. We consider this problem in the tidal approximation and derive a closed set of equations, which we reduce to a seventh‐order differential equation. We solve the resulting equation to find that perturbations from the polar ring deform the spheroid into a non‐stationary rotating triaxial ellipsoid with linear internal velocity field. (© 2013 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Microstructurally induced computational and material instabilities

A dislocation-density based finite strain crystal plasticity finite element model and an hourglass strain measure orthogonal to the linear velocity field were used to investigate microstructural effects on deformation and hourglass instability. An aluminum aggregate was modeled in quasi-static plane strain tension and compression, and the effects of the presence of manganese-based dispersed particles were investigated. Both local plastic deformation and the presence of these dispersed particles can trigger hourglass instabilities, which can be mistaken for deformation or failure modes. These hourglass modes occur due to the difference in the deformability of neighboring elements or regions and the associated energy modes associated with hourglassing, and the proposed computational methodology can be used to delineate between failure modes and numerical instabilities.

Integrable hydrodynamic submodels with a linear velocity field

Invariant and partially invariant solutions to the equations of gas dynamics with a linear velocity field are defined by a matrix satisfying a homogeneous integrable Riccati equation. The classification is carried out of solutions by the acceleration vector in the Lagrangian coordinates. Some example is given of an invariant solution for which the selected volume “collapses” to an interval.

Combination of 3D unsplit forward and backward volume-of-fluid transport and coupling to the level set method

This paper presents a novel methodology for interface capturing in two-phase flows by combining a Lagrangian–Eulerian Volume-of-Fluid method with a Level Set transport. While the Volume-of-Fluid transport relies on a robust and accurate polyhedral library, any high-order Level Set transport may be used. The method is shown to be less restrictive in terms of CFL conditions than split Volume-of-Fluid methods. Various geometric integration schemes are proposed and tested. For linear velocity fields, mass error is shown to vanish, and to be third order otherwise. The method is validated on 2D and 3D test cases. Conservation properties are shown to be excellent, while geometrical accuracy remains satisfactory even for complex flows such as the primary breakup of a liquid jet.

The linear velocity field of 2MASS Redshift Survey, Ks= 11.75 galaxies: constraints on β and bulk flow from the luminosity function

Using the nearly full sky Ks=11.75 2MASS Redshift Survey [2MRS]of ~45,000 galaxies we reconstruct the underlying peculiar velocity field and constrain the cosmological bulk flow within ~100. These results are obtained by maximizing the probability to estimate the absolute magnitude of a galaxy given its observed apparent magnitude and redshift. At a depth of ~60 Mpc/h we find a bulk flow Vb=(90\pm65,-230\pm65,50\pm65) km/s in agreement with the theoretical predictions of the LCDM model. The reconstructed peculiar velocity field that maximizes the likelihood is characterized by the parameter beta=0.323 +/- 0.08. Both results are in agreement with those obtained previously using the ~23,000 galaxies of the shallower Ks=11.25 2MRS survey. In our analysis we find that the luminosity function of 2MRS galaxies is poorly fitted by the Schechter form and that luminosity evolves such that objects become fainter with increasing redshift according to L(z)=L(z=0)(1+z)^(+2.7 +/-0.15).

Submodels of gas motion with a linear velocity field in the degenerate case

We look for solutions in the form of a linear velocity field for the gas dynamics equations with an arbitrary state equation. We find all possible state equations and the corresponding differential equations of submodels when the auxiliary matrix is degenerate.

Straightening expansions of gas from vortex with linear velocity field

In work one submodel of flow of gas with the linear field of speeds is considered. It is formed by system of non-linear differential equations with initial data. Several first integrals of such system are found. For express initial data of a task approximate solution of differential equations of a submodel is found. To such decision there correspond the world lines describing the becoming straight scattering of particles of gas from a whirlwind. Trajectories of driving of particles of gas are constructed.

An upper bound-based solution for the shape factors of bearing capacity of footings under drained conditions using a parallelized mixed f.e. formulation with quadratic velocity fields

A strict upper-bound limit analysis finite element formulation is used to estimate shape factors sγ and sq for determining the bearing capacity of shallow foundations using the classic bearing capacity formula. The finite element formulation uses a quadratic approximation for the velocity field, an extension of a previously published Augmented Lagrangian formulation with a linear velocity field, and was implemented for a parallel processing environment. Results from determining the limit loads under three-dimensional conditions are presented and compared with previously published data. The results obtained allow a strict upper-bound determination of the shape factors. Furthermore, a practical proposal for these factors is made and compared with other proposals made by other authors.

Particle dispersion by random waves in the rotating Boussinesq system

We present a theoretical and numerical study of horizontal particle dispersion due to random waves in the three-dimensional rotating and stratified Boussinesq system, which serves as a simple model to study the dispersion of tracers in the ocean by the internal wave field. Specifically, the effective one-particle diffusivity in the sense of Taylor (Proc. Lond. Math. Soc., vol. 20, 1921, p. 196) is computed for a small-amplitude internal gravity wave field modelled as a stationary homogeneous and horizontally isotropic Gaussian random field whose frequency spectrum is bounded away from zero. Dispersion in this system does not arise simply because of a Stokes drift effect, as in the case of surface gravity waves, but in addition it is driven by the nonlinear, second-order corrections to the linear velocity field, which can be computed using the methods of wave–mean interaction theory. A formula for the one-particle diffusivity as a function of the spectrum of the random wave field is presented. It is shown that this diffusivity is much smaller than might be expected from heuristic arguments based on the magnitude of the Stokes drift or the pseudomomentum. This appears to stem from certain incompressibility constraints for the Stokes drift and the second-order velocity field. Finally, the theory is applied to oceanic conditions described by a typical model wave spectrum, the Garrett–Munk spectrum, and also by detailed field observations from the North Atlantic tracer release experiment.

Kinematic strain localization

Deformation within a steady-state compressional orogen, i.e., where tectonic accretion is, on geological time scales, balanced by surface erosion, can best be described by a stationary velocity field. Instantaneous deformation results from spatial gradients in the velocity field, whereas total accumulated strain results from the integration of this instantaneous deformation along material paths following the flow lines defined by the velocity field. We have synthesized the net strain distributions for rocks exposed at the surface of such an orogenic system using simple, linear velocity fields corresponding to (a) simple shear within a dipping shear zone and (b) pure vertical shear. In both cases we demonstrate the development of surface patterns of finite strain accumulation that do not reflect the geometry of the assumed velocity field in a simple manner. Large gradients in finite strain arise as a consequence of the geographic variation in particle residence time imposed by the surface boundary, even for the limiting case where no instantaneous strain gradient exists. Such patterns of deformation are often recognised in exhumed orogenic systems, but have commonly been assumed to reflect more complex velocity fields resulting from nonlinear, localizing crustal rheologies. We therefore demonstrate that caution should be exercised in interpreting observed strain patterns because a proportion of the observed strain localization must be attributed to this purely kinematic (or geometric) effect — and this proportion may be significant in many systems. Such kinematic effects should be quantified, and subtracted from observed strain distributions before they are used to infer the rheological behavior of crustal rocks. We also suggest that in a simple shear (thrust) setting, kinematic strain localization may in fact nucleate strain softening on the side of the deforming region that is stable or fixed with respect to the Earth's surface and thus be responsible for the asymmetry that characterizes the large majority of thrust systems.

Classification of submodels with a linear velocity field in gas dynamics

Some solutions as linear velocity fields are found to the gas dynamics equations with an arbitrary state equation. A classification of the submodels with respect to state equations is given in the case that an auxiliary matrix is nondegenerate.

Particle dispersion by random waves in rotating shallow water

We present a theoretical and numerical study of wave-induced particle dispersion due to random waves in the rotating shallow-water system, as part of an ongoing study of particle dispersion in the ocean. Specifically, the effective particle diffusivities in the sense of Taylor (Proc. Lond. Math. Soc., vol. 20, 1921, p. 196) are computed for a small-amplitude wave field modelled as a stationary homogeneous isotropic Gaussian random field whose frequency spectrum is bounded away from zero. In this case, the leading-order diffusivity depends crucially on the nonlinear, second-order corrections to the linear velocity field, which can be computed using the methods of wave–mean interaction theory. A closed-form analytic expression for the effective diffusivity is derived and carefully tested against numerical Monte Carlo simulations. The main conclusions are that Coriolis forces in shallow water invariably decrease the effective particle diffusivity and that there is a peculiar choking effect for the second-order particle flow in the limit of strong rotation.