Time-independent Velocity Research Articles

An implicit solution for Asay foil trajectories generated by separable, sustained-production ejecta source models

We present a simple implicit solution for the time-dependent trajectory of a thin Asay foil ejecta diagnostic for the general case where the impinging ejecta cloud is generated by a source function characterized by an arbitrary (sustained) time dependence and a time-independent (stationary) particle velocity distribution. In the limit that the source function time dependence becomes a delta function, this solution—which is amenable to rapid numerical calculations of arbitrary accuracy—exactly recovers a previously published solution for the special case of instantaneous ejecta production. We also derive simple expressions for the free-surface arrival (catch-up) time as well as the true ejecta areal mass accumulation on the accelerating foil and place bounds on the level of error incurred when applying instant-production mass solutions to a sustained-production trajectory. We demonstrate these solutions with example calculations for hypothetical source functions spanning a wide range of ejecta production durations, velocity distributions, and temporal behaviors. These calculations demonstrate how the foil trajectory is often insensitive to the temporal dependence of the source function, instead being dominated by the velocity distribution. We quantify this insensitivity using a “compatibility score” metric. Under certain conditions, one may capitalize upon this insensitivity to obtain a good approximation of the second integral of the velocity distribution from the observed foil trajectory.

Universal Poisson statistics of a passive tracer diffusing in dilute active suspensions.

The statistics of a passive tracer immersed in a suspension of active particles (swimmers) is derived from first principles by considering a perturbative expansion of the tracer interaction with the microscopic swimmer field. To first order in the swimmer density, the tracer statistics is shown to be exactly represented by a spatial Poisson process combined with independent tracer-swimmer scattering events, rigorously reducing the multiparticle dynamics to two-body interactions. The Poisson representation is valid in any dimension, for arbitrary interaction forces and for a large class of swimmer dynamics. The framework not only allows for the systematic calculation of the tracer statistics in various dynamical regimes but highlights in particular surprising universal features that are independent of the swimmer dynamics such as a time-independent velocity distribution.

Acoustic Streaming and Its Applications

Broadly speaking, acoustic streaming is generated by a nonlinear acoustic wave with a finite amplitude propagating in a viscid fluid. The fluid volume elements of molecules, d V , are forced to oscillate at the same frequency as the incident acoustic wave. Due to the nature of the nonlinearity of the acoustic wave, the second-order effect of the wave propagation produces a time-independent flow velocity (DC flow) in addition to a regular oscillatory motion (AC motion). Consequently, the fluid moves in a certain direction, which depends on the geometry of the system and its boundary conditions, as well as the parameters of the incident acoustic wave. The small scale acoustic streaming in a fluid is called “microstreaming”. When it is associated with acoustic cavitation, which refers to activities of microbubbles in a general sense, it is often called “cavitation microstreaming”. For biomedical applications, microstreaming usually takes place in a boundary layer at proximity of a solid boundary, which could be the membrane of a cell or walls of a container. To satisfy the non-slip boundary condition, the flow motion at a solid boundary should be zero. The magnitude of the DC acoustic streaming velocity, as well as the oscillatory flow velocity near the boundary, drop drastically; consequently, the acoustic streaming velocity generates a DC velocity gradient and the oscillatory flow velocity gradient produces an AC velocity gradient; they both will produce shear stress. The former is a DC shear stress and the latter is AC shear stress. It was observed the DC shear stress plays the dominant role, which may enhance the permeability of molecules passing through the cell membrane. This phenomenon is called “sonoporation”. Sonoporation has shown a great potential for the targeted delivery of DNA, drugs, and macromolecules into a cell. Acoustic streaming has also been used in fluid mixing, boundary cooling, and many other applications. The goal of this work is to give a brief review of the basic mathematical theory for acoustic microstreaming related to the aforementioned applications. The emphasis will be on its applications in biotechnology.

Maximal heat transfer between two parallel plates

The divergence-free time-independent velocity field has been determined so as to maximise heat transfer between two parallel plates with a constant temperature difference under the constraint of fixed total enstrophy. The present variational problem is the same as that first formulated by Hassanzadeh et al. (J. Fluid Mech., vol. 751, 2014, pp. 627–662); however, the search range for optimal states has been extended to a three-dimensional velocity field. A scaling of the Nusselt number $Nu$ with the Péclet number $Pe$ (i.e., the square root of the non-dimensionalised enstrophy with thermal diffusion time scale), $Nu\sim Pe^{2/3}$, has been found in the three-dimensional optimal states, corresponding to the asymptotic scaling with the Rayleigh number $Ra$, $Nu\sim Ra^{1/2}$, expected to appear in an ultimate state, and thus to the Taylor energy dissipation law in high-Reynolds-number turbulence. At $Pe\sim 10^{0}$, a two-dimensional array of large-scale convection rolls provides maximal heat transfer. A three-dimensional optimal solution emerges from bifurcation on the two-dimensional solution branch at $Pe\sim 10^{1}$, and the three-dimensional solution branch has been tracked up to $Pe\sim 10^{4}$ (corresponding to $Ra\approx 2.7\times 10^{6}$). At $Pe\gtrsim 10^{3}$, the optimised velocity fields consist of convection cells with hierarchical self-similar vortical structures, and the temperature fields exhibit a logarithmic-like mean profile near the walls.

Doppler-Based Acoustic Gyrator

Non-reciprocal phase shifters have been attracting a great deal of attention due to their important applications in filtering, isolation, modulation, and mode locking. Here, we demonstrate a non-reciprocal acoustic phase shifter using a simple acoustic waveguide. We show, both analytically and numerically, that when the fluid within the waveguide is biased by a time-independent velocity, the sound waves travelling in forward and backward directions experience different amounts of phase shifts. We further show that the differential phase shift between the forward and backward waves can be conveniently adjusted by changing the imparted bias velocity. Setting the corresponding differential phase shift to 180 degrees, we then realize an acoustic gyrator, which is of paramount importance not only for the network realization of two port components, but also as the building block for the construction of different non-reciprocal devices like isolators and circulators.

Adiabatic plasma rotations and symmetric magnetohydrodynamical stationary equilibria: analytical and semi-numerical solutions

Plasma rotation appears in many problems of fusion (neutral beam injection) and astrophysical (magnetic stars) interest. In the case of a time-independent plasma velocity it is possible to describe a stationary magnetohydrodynamical (MHD) equilibrium by a set of infinitely nested flux surfaces. Maschke and Perrin have obtained an equation describing stationary MHD equilibria for azimuthal (toroidal) plasma rotations, when the entropy is constant on magnetic flux surfaces. In the present paper we express their equation in curvilinear coordinates, both in orthogonal and non-orthogonal cases, provided there is an ignorable quantity. This equation is presented in some coordinate systems of plasma physics interest. We consider approximated analytical and semi-analytical solutions for plasma configurations in cylindrical and spherical coordinates.

An easy-to-use turbulent inlet boundary condition for large-eddy simulation of radial diffusers

When applying time-accurate computational fluid dynamics, large-eddy simulation (CFD LES) to room ventilation, a common problem is that the supply jets do not model properly if there is no realistic turbulence in the inflow boundary. In this paper we introduce a novel, easy-to-use method of generating a turbulent inlet boundary condition for simulating a radial diffuser. The basic idea is to apply a time-independent velocity distribution with realistic spatial statistics, and to add temporal fluctuations by rotation. The method is simple to use, requires only one extra parameter, and does not necessitate large storage files. It needs a momentary velocity distribution on the inflow surface with the desired volume flow rate obtained from a time-accurate internal simulation of the diffuser alone. In a room-scale simulation, this velocity distribution is put in rotating movement in order to mimic a turbulent inlet boundary. The method is verified by comparing the outcome with published measurements and with a simulation containing the interior of the diffuser.

Simulation of nucleation dynamics at the cylinder-to-lamellar transition in a diblock copolymer melt

We examine the dynamical evolution of a stable lamellar phase nucleating from a metastable cylinder phase in a diblock copolymer melt, through large-scale simulations of the time-dependent Landau–Brazovskii model. Ellipsoidal nuclei form, whose minor axis is parallel to the cylinder axis. We use our observation of both shrinking and growing droplets to determine the critical nucleus size as a function of undercooling, and find that the critical size grows as we approach coexistence. The nucleus shape and critical size agree, near coexistence, with the predictions of an approximate theory. This supports the idea that the underlying microstructure produces an anisotropic droplet interfacial tension, and that the interplay between this interfacial tension and a reduction in bulk free-energy is central to the nucleation process. The nucleus interface moves with a time-independent velocity that depends on the interface orientation in a manner that preserves the ellipsoidal droplet shape into the late stages of growth. Near coexistence, the magnitude of the interfacial velocity varies linearly with undercooling, consistent with theoretical predictions and experimental observations.

Analytic Solution for Oscillatory Flow in a Channel for Jeffrey Fluid

AbstractThis paper describes the study of two-dimensional oscillatory flow inside a rectangular channel for Jeffrey fluid with small suction. The main objective is to investigate the viscoelastic behavior of non-Newtonian fluids subject to time harmonic oscillations. The fluid flow in the channel is generated by suction at the porous walls while time harmonic pressure waves give rise to oscillations in the velocity field. Analytical solution of the time-independent axial velocity and time-dependent oscillatory axial velocity in the channel is obtained. To understand the characteristics of non-Newtonian fluids, the effects of retardation time, and Deborah number on the oscillatory velocity, its amplitude and penetration depth are analyzed. The methodologies adopted to solve this physically complex problem are perturbation, Wentzel-Kramers-Brillouin, and variation-of-parameter. The paper will help in understanding the characteristics of non-Newtonian fluids that are subject to applied oscillatory harmonic s...

Oscillatory channel flow for non-Newtonian fluid

This paper deals with an analytical solution of oscillatory flow of an incompressible second grade fluid in a channel. The flow in the channel is driven by suction at the permeable walls, whereas small amplitude time harmonic pressure waves are responsible for oscillations in the velocity field. The time independent axial velocity and the time dependent oscillatory axial velocity are calculated analytically. The important physical quantities like the velocity profile, amplitude of the oscillation and penetration depth of the wave are given special emphasis. The effects of second grade parameter and suction parameter on these quantities are particularly examined. A comparative study of the oscillatory flow for second grade with viscous fluid is also made. Key words: Oscillatory channel flow, second grade fluid, shear stress, perturbation solution, Wentzel-Kramers-Brillouin (WKB) approximation.

The Interaction of a Circularly Orbiting Electromagnetic Harmonic Wave with an Electron Having a Constant Time Independent Drift Velocity

A circularly orbiting electromagnetic harmonic wave may appear when a 1S electron encounters a decelerating stopping positively charged hole inside a semiconductor. The circularly orbiting electromagnetic harmonic wave can have an interaction with a conducting electron which has a constant time independent drift velocity.

Transparent boundary conditions as dissipative subgrid closures for the spectral representation of scalar advection by shear flows

We consider the evolution of a passive scalar in a shear flow in its representation as a system of lattice differential equations in wave number space. When the velocity field has small support, the interaction in wave number space is local and can be studied in terms of dispersive linear lattice waves. We close the restriction of the system to a finite set of wave numbers by implementing transparent boundary conditions for lattice waves. This closure is studied numerically in terms of energy dissipation rate and energy spectrum, both for a time-independent velocity field and for a time-dependent synthetic velocity field whose Fourier coefficients follow independent Ornstein-Uhlenbeck stochastic processes.

Anomalous scaling of a passive scalar advected by the turbulent velocity field with finite correlation time: two-loop approximation.

The renormalization group and operator product expansion are applied to the model of a passive scalar quantity advected by the Gaussian self-similar velocity field with finite, and not small, correlation time. The inertial-range energy spectrum of the velocity is chosen in the form E(k) proportional, variant k(1-2 epsilon ), and the correlation time at the wave number k scales as k(-2+eta). Inertial-range anomalous scaling for the structure functions and other correlation functions emerges as a consequence of the existence in the model of composite operators with negative scaling dimensions, identified with anomalous exponents. For eta> epsilon, these exponents are the same as in the rapid-change limit of the model; for eta< epsilon, they are the same as in the limit of a time-independent (quenched) velocity field. For epsilon =eta (local turnover exponent), the anomalous exponents are nonuniversal through the dependence on a dimensionless parameter, the ratio of the velocity correlation time, and the scalar turnover time. The nonuniversality reveals itself, however, only in the second order of the epsilon expansion and the exponents are derived to order epsilon (2), including anisotropic contributions. It is shown that, for moderate order of the structure function n, and the space dimensionality d, finite correlation time enhances the intermittency in comparison with both the limits: the rapid-change and quenched ones. The situation changes when n and/or d become large enough: the correction to the rapid-change limit due to the finite correlation time is positive (that is, the anomalous scaling is suppressed), it is maximal for the quenched limit and monotonically decreases as the correlation time tends to zero.

Acoustic streaming in closed thermoacoustic devices

A derivation of acoustic streaming in a steady-state thermoacoustic device is presented in the case of zero second-order time-averaged mass flux across the resonator section (nonlooped device). This yields analytical expressions for the time-independent second-order velocity, pressure gradient, and time-averaged mass flux in a fluid supporting a temperature gradient and confined between widely to closely separated solid boundaries, both in the parallel plate and in the cylindrical tube geometries (two-dimensional problem). From this, streaming can be evaluated in a thermoacoustic stack, regenerator, pulse tube, main resonator of a thermoacoustic device, or in any closed tube that supports a mean temperature gradient, providing only that the acoustic pressure, the longitudinal derivative of the pressure, and the mean temperature variation are known.

Spatio-temporal patterns in excitable media with non-solenoidal flow straining

The propagation of spiral waves in excitable media with the Belousov–Zhabotinsky reactions in a non-solenoidal, time-independent velocity field is studied numerically as a function of the amplitude and frequency of the velocity. It is shown that the spiral wave is slightly distorted for small amplitudes and low frequencies, whereas it breaks-up into new spiral waves which merge and form periodic, cusped fronts at moderate amplitudes and small frequencies. For larger amplitudes but still small frequencies, the spiral wave undergoes a second transition to thick fronts characterized by small curvature, and the radius of curvature increases as the amplitude of the velocity field is increased. It is also shown that an increase in the frequency of the velocity field results in front distorsion and corrugations which are due to the increase in the number of stagnation points as the frequency is increased, straining of the front at stagnation points and the non-solenoidal velocity field employed in the paper. An explanation of these corrugations in terms of the straining, gradient of the transverse velocity along the normal to the front and compressibility is provided.

Transport properties of highly aligned polymer light-emitting diodes

We investigate hole transport in polymer light-emitting-diodes in which the emissive layer is made of liquid-crystalline polymer chains aligned perpendicular to the direction of transport. Calculations of the current as a function of time via a random-walk model show excellent qualitative agreement with experiments conducted on electroluminescent polyfluorene demonstrating non-dispersive hole transport. The current exhibits a constant plateau as the charge carriers move with a time-independent drift velocity, followed by a long tail when they reach the collecting electrode. Variation of the parameters within the model allows the investigation of the transition from non-dispersive to dispersive transport in highly aligned polymers. It turns out that large inter-chain hopping is required for non-dispersive hole transport and that structural disorder obstructs the propagation of holes through the polymer film.

Spectrum of the fokker-planck operator representing diffusion in a random velocity field

We study spectral properties of the Fokker-Planck operator that represents particles moving via a combination of diffusion and advection in a time-independent random velocity field, presenting in detail work outlined elsewhere [J. T. Chalker and Z. J. Wang, Phys. Rev. Lett. 79, 1797 (1997)]. We calculate analytically the ensemble-averaged one-particle Green function and the eigenvalue density for this Fokker-Planck operator, using a diagrammatic expansion developed for resolvents of non-Hermitian random operators, together with a mean-field approximation (the self-consistent Born approximation) which is well controlled in the weak-disorder regime for dimension d>2. The eigenvalue density in the complex plane is nonzero within a wedge that encloses the negative real axis. Particle motion is diffusive at long times, but for short times we find a novel time dependence of the mean-square displacement, <r(2)> approximately t(2/d) in dimension d>2, associated with the imaginary parts of eigenvalues.

Anomalous scaling regimes of a passive scalar advected by the synthetic velocity field.

The field theoretic renormalization group (RG) is applied to the problem of a passive scalar advected by the Gaussian self-similar velocity field with finite correlation time and in the presence of an imposed linear mean gradient. The energy spectrum in the inertial range has the form E(k) proportional to (1-epsilon), and the correlation time at the wave number k scales as k(-2+eta). It is shown that, depending on the values of the exponents epsilon and eta, the model in the inertial-convective range exhibits various types of scaling regimes associated with the infrared stable fixed points of the RG equations: diffusive-type regimes for which the advection can be treated within ordinary perturbation theory, and three nontrivial convection-type regimes for which the correlation functions exhibit anomalous scaling behavior. The explicit asymptotic expressions for the structure functions and other correlation functions are obtained; the anomalous exponents, determined by the scaling dimensions of the scalar gradients, are calculated to the first order in epsilon and eta in any space dimension. For the first nontrivial regime the anomalous exponents are the same as in the rapid-change version of the model; for the second they are the same as in the model with time-independent (frozen) velocity field. In these regimes, the anomalous exponents are universal in the sense that they depend only on the exponents entering into the velocity correlator. For the last regime the exponents are nonuniversal (they can depend also on the amplitudes); however, the nonuniversality can reveal itself only in the second order of the RG expansion. A brief discussion of the passive advection in the non-Gaussian velocity field governed by the nonlinear stochastic Navier-Stokes equation is also given.

A regularizing effect of nonlinear transport equations

We consider the semigroup on L 1 ( R n ) {L^1}\left ( {\mathbb {R}^n} \right ) defined by the nonlinear transport equation for the scalar s s , \[ ∂ t s + d i v ( f ( s ) u ) = 0 i n ( 0 , ∞ ) × R n {\partial _t}s + div\left ( f\left ( s \right )u \right ) = 0 \qquad in \left ( 0, \infty \right ) \times {\mathbb {R}^n} \] for given velocity field u u . We show that this nonlinear semigroup is Hölder continuous for t &gt; 0 t &gt; 0 in the uniform operator topology, provided the graph of f f has no linear segments. This continuity property—which expresses a regularizing effect of the nonlinearity in the transport equation—is robust with respect to the spatial behaviour of the time-independent velocity field u u .

Field-theoretic approximations for normal diffusion in random velocity fields.

The motion of a classical Brownian particle entrained in a fluid with random, time-independent velocity fields is discussed. Two approaches to calculating the disorder-averaged Green's function are presented, both based upon a functional-integral formulation. A renormalization group (RG) approach proves to be less satisfactory than does a self-consistent perturbation theory approach, which reproduces the well-known direct interaction approximation. An explicit comparison of the results with Monte Carlo data is made in two dimensions. The relationship between the present results and the case of diffusion in a solid, where RG provides a superior answer, is discussed in physical terms.