Parabolic Shear Flow Research Articles

Effect of latitudinal differential rotation on solar Rossby waves: Critical layers, eigenfunctions, and momentum fluxes in the equatorialβplane

Context.Retrograde-propagating waves of vertical vorticity with longitudinal wavenumbers between 3 and 15 have been observed on the Sun with a dispersion relation close to that of classical sectoral Rossby waves. The observed vorticity eigenfunctions are symmetric in latitude, peak at the equator, switch sign near 20°–30°, and decrease at higher latitudes.Aims.We search for an explanation that takes solar latitudinal differential rotation into account.Methods.In the equatorialβplane, we studied the propagation of linear Rossby waves (phase speedc < 0) in a parabolic zonal shear flow,U= −U̅ξ2< 0, whereU̅= 244 m s−1, andξis the sine of latitude.Results.In the inviscid case, the eigenvalue spectrum is real and continuous, and the velocity stream functions are singular at the critical latitudes whereU = c. We add eddy viscosity to the problem to account for wave attenuation. In the viscous case, the stream functions solve a fourth-order modified Orr-Sommerfeld equation. Eigenvalues are complex and discrete. For reasonable values of the eddy viscosity corresponding to supergranular scales and above (Reynolds number 100 ≤ Re ≤ 700), all modes are stable. At fixed longitudinal wavenumber, the least damped mode is a symmetric mode whose real frequency is close to that of the classical Rossby mode, which we call the R mode. ForRe ≈ 300, the attenuation and the real part of the eigenfunction is in qualitative agreement with the observations (unlike the imaginary part of the eigenfunction, which has a larger amplitude in the model).Conclusions.Each longitudinal wavenumber is associated with a latitudinally symmetric R mode trapped at low latitudes by solar differential rotation. In the viscous model, R modes transport significant angular momentum from the dissipation layers toward the equator.

The Spectral Web of stationary plasma equilibria. II. Internal modes

The new method of the Spectral Web to calculate the spectrum of waves and instabilities of plasma equilibria with sizeable flows, developed in the preceding Paper I [Goedbloed, Phys. Plasmas 25, 032109 (2018)], is applied to a collection of classical magnetohydrodynamic instabilities operating in cylindrical plasmas with shear flow or rotation. After a review of the basic concepts of the complementary energy giving the solution path and the conjugate path, which together constitute the Spectral Web, the cylindrical model is presented and the spectral equations are derived. The first example concerns the internal kink instabilities of a cylindrical force-free magnetic field of constant α subjected to a parabolic shear flow profile. The old stability diagram and the associated growth rate calculations for static equilibria are replaced by a new intricate stability diagram and associated complex growth rates for the stationary model. The power of the Spectral Web method is demonstrated by showing that the two associated paths in the complex ω-plane nearly automatically guide to the new class of global Alfvén instabilities of the force-free configuration that would have been very hard to predict by other methods. The second example concerns the Rayleigh–Taylor instability of a rotating theta-pinch. The old literature is revisited and shown to suffer from inconsistencies that are remedied. The most global n = 1 instability and a cluster sequence of more local but much more unstable n=2,3,…∞ modes are located on separate solution paths in the hydrodynamic (HD) version of the instability, whereas they merge in the MHD version. The Spectral Web offers visual demonstration of the central position the HD flow continuum and of the MHD Alfvén and slow magneto-sonic continua in the respective spectra by connecting the discrete modes in the complex plane by physically meaningful curves towards the continua. The third example concerns the magneto-rotational instability (MRI) thought to be operating in accretion disks about black holes. The sequence n=1,2,… of unstable MRIs is located on one continuous solution path, but also on infinitely many separate loops (“pancakes”) of the conjugate path with just one MRI on each of them. For narrow accretion disks, those sequences are connected with the slow magneto-sonic continuum, which is far away though from the marginal stability transition. In this case, the Spectral Web method is the first to effectively incorporate the MRIs into the general MHD spectral theory of equilibria with background flows. Together, the three examples provide compelling evidence of the computational power of the Spectral Web Method.

On the acoustic modes in a duct containing a parabolic shear flow

The exact solution of the acoustic wave equation in an unidirectional shear flow with a parabolic velocity profile is obtained, representing sound propagation in a plane, parallel walled duct, with two boundary layers over rigid or impedance walls. It is shown that there are four cases, depending on the critical level(s) where the Doppler shifted frequency vanishes: (i) for propagation upstream the critical levels are outside the duct (case II); (ii) for propagation downstream there may be two (case IV), one (case I) or no (case III) critical level inside the duct. The acoustic wave equation is transformed in each of the four cases to particular forms of the extended hypergeometric equation, which has power series solutions, some involving logarithmic singularities. In the cases where critical levels occur, at real or ‘imaginary’ distance, matching of two or three pairs of solutions, valid over regions each overlapping the next, is needed. The particular case of the parabolic velocity profile is used to address general properties of sound in unidirectional shear flows. For example, it is shown that for ducted shear flows, there exist a pair of even and odd eigenfunctions, in the absence of critical levels. It is also proved, in more than one instance, that there is no single set of eigenvalues and eigenfunctions valid across one or two shear layers. This leads to the general conjecture, considering the acoustics of shear flows in ducts, that critical levels separate regions with distinct sets of eigenvalues and eigenfunctions.

Lift on a sphere moving near a wall in a parabolic flow

The lift on a solid sphere moving along a wall in a parabolic shear flow is obtained as a regular perturbation problem for low Reynolds number when the sphere is in the inner region of expansion. Comprehensive results are given for the 10 terms of the lift, which involve the sphere translation and rotation, the linear and quadratic parts of the shear flow and all binary couplings. Based on very accurate earlier results of a creeping flow in bispherical coordinates, precise results for these lift terms are obtained for a large range of sphere-to-wall distances, including the lubrication region for sphere-to-wall gaps down to 0.01 of a sphere radius. Fitting formulae are also provided in view of applications. The migration velocity of an inertialess spherical particle is given explicitly, for a non-rotating sphere with a prescribed translation velocity and for a freely moving sphere in a parabolic shear flow. Values of the lift and migration velocity are in good agreement with earlier results whenever available.

Analysis and modeling of edge fluctuations and transport mechanism in the Maryland Centrifugal Experiment

The Maryland Centrifugal Experiment [R. F. Ellis et al., Phys. Plasmas 12, 055704 (2005)] is a mirror machine designed to have a plasma axially confined by supersonic rotation and dominantly interchange stable by the radial shear in the azimuthal velocity. Nevertheless, residual fluctuations still persist. To investigate the presence of such fluctuations, an azimuthal array of 16 magnetic pickup coils at the edge region of the plasma has been employed. A comprehensive analysis of the magnetic fluctuations reveals that, under the imposed shear flow, only m=0 and m=2 modes are dominant; yet, the observed frequency spectrum is broadband. Using higher order spectral analysis, clear evidence of nonlinear mode coupling is detected. It is also observed that the amplification of magnetic fluctuations leads to enhanced transport consistent with the drop of the plasma density and voltage. As a result, the magnetic fluctuations start to decrease in amplitude as the central plasma pressure drops. In return, the anomalous radial particle and momentum transport are reduced; thus, the plasma confinement improves. As the plasma pressure starts to build up, the plasma voltage increases, destabilizing the m=2 interchange mode. The cycle of enhanced transport and intermittent fluctuations repeats itself. A two-dimensional magnetohydrodynamics code in slab geometry is employed to investigate the dynamics of the primary interchange instability and to assess the level of transport. For very low sheared rotation, a broad spatial spectrum of unstable modes is obtained. As the sheared rotation is increased, the high mode numbers become stabilized and low mode numbers dominate the spectrum. Both the experimental data obtained from the azimuthal array probes and the simulations in case of parabolic shear flow show clear evidence of nonlinear mode coupling, explaining the broadband frequency spectrum for low mode numbers. A detailed comparison of spatiotemporal dynamics of simulations with the experimental data is presented.

A computational study of leukocyte adhesion and its effect on flow pattern in microvessels

Three-dimensional computational modeling and simulation are presented on the adhesive rolling of deformable leukocytes over a P-selectin coated surface in parabolic shear flow in microchannels. The computational model is based on the immersed boundary method for cell deformation and Monte Carlo simulation for receptor/ligand interaction. The simulations are continued for at least 1 s of leukocyte rolling during which the instantaneous quantities such as cell deformation index, cell/substrate contact area, and fluid drag remain statistically stationary. The characteristic ‘stop-and-go’ motion of rolling leukocytes, and the ‘tear-drop’ shape of adherent leukocytes as observed in experiments are reproduced by the simulations. We first consider the role of cell deformation and cell concentration on rolling characteristics. We observe that compliant cells roll slower and more stably than rigid cells. Our simulations agree with previous in vivo observation that the hydrodynamic interactions between nearby leukocytes affect cell rolling, and that the rolling velocity decreases inversely with the separation distance, irrespective of cell deformability. We also find that cell deformation decreases, and the cells roll more stably with reduced velocity fluctuation, as the cell concentration is increased. However, the effect of nearby cells on the rolling characteristics is found to be more significant for rigid cells than compliant cells. We then address the effect of cell deformability and rolling velocity on the flow resistance due to, and the fluid drag on, adherent leukocytes. While several earlier computational works have addressed this problem, two key features of leukocyte adhesion, such as cell deformation and rolling, were often neglected. Our results suggest that neglecting cell deformability and rolling velocity may significantly overpredict the flow resistance and drag force. Increasing the cell concentration is shown to increase the flow resistance and reduce the fluid drag. The reduced drag then results in slower and more stable rolling of the leukocytes with longer pause time and shorter step distance. But the increase/decrease in the flow resistance/fluid drag due to the increase in the cell concentration is observed to be more significant in case of rigid cells than compliant cells.

Pattern formation in crystal growth under parabolic shear flow. II.

Wavy pattern of ice with a specific wavelength occurs during ice growth from a thin layer of undercooled water flowing down the surface of icicles or inclined plane. In a previous paper [Phys. Rev. E 68, 021603 (2003)]], we have found that restoring forces due to gravity and surface tension is a factor for stabilization of morphological instability of the solid-liquid interface. However, the mechanism for the morphological instability and stability of the solid-liquid interface has not been well understood. In the present paper, it is shown that a phase difference between fluctuation of the solid-liquid interface and distribution of heat flux at the deformed solid-liquid interface, which depends on the magnitude of the restoring forces, is a cause of the instability and stability of the interface. This mechanism is completely different from the usual Mullins-Sekerka instability due to diffusion and stabilization due to the Gibbs-Thomson effect.

Liquid convection effects on the pushing-engulfment transition of insoluble particles by a solidifying interface: Part II. Numerical calculation of drag and lift forces on a particle in parabolic shear flow

In this work, lift and drag forces acting on a particle in the close vicinity of a wall are calculated by numerically solving the incompressible two-dimensional Navier-Stokes equations. The flow field computations are done using the well-known Marker and Cell (MAC) method on a staggered grid. A parabolic shear flow at the inlet is assumed. The particle is assumed to be nonrotating and Magnus forces are not considered. The numerical results are compared to those obtained from analytical/empirical expressions for drag and lift forces from different theoretical models. Reasonably good agreement has been found between the two approaches.

Pattern formation in crystal growth under parabolic shear flow.

Morphological instability of the solid-liquid interface occurring in a crystal growing from an undercooled thin liquid bounded on one side by a free surface and flowing down inclined plane, is investigated by a linear stability analysis under shear flow. It is found that restoring forces due to gravity and surface tension is an important factor for stabilization of the solid-liquid interface on long length scales. This is a stabilizing effect different from the Gibbs-Thomson effect. A particular long wavelength mode of about 1 cm of wavy pattern, observed on the surface of icicles covered with a thin layer of flowing water is obtained from the dispersion relation, including the effect of flow and restoring forces.

The inertial lift on a spherical particle in a plane Poiseuille flow at large channel Reynolds number

The inertial migration of a small rigid sphere translating parallel to the walls within a channel flow at large channel Reynolds numbers is investigated. The method of matched asymptotic expansions is used to solve the equations governing the disturbance flow past a particle at small particle Reynolds number and to evaluate the lift. Both neutrally and non-neutrally buoyant particles are considered. The wall-induced inertia is significant in the thin layers near the walls where the lift is close to that calculated for linear shear flow, bounded by a single wall. In the major portion of the flow, excluding near-wall layers, the wall effect can be neglected, and the outer flow past a sphere can be treated as unbounded parabolic shear flow. The effect of the curvature of the unperturbed velocity profile is significant, and the lift differs from the values corresponding to a linear shear flow even at large Reynolds numbers.

On the exact solution of the acoustic wave equation in a parabolic velocity profile in a hard-walled duct

The exact solution of the acoustic wave equation in a parabolic shear flow profile is obtained; the only exact solution given in the literature [Goldstein and Rice (1963); Jones (1977), (1978); Scott (1979); Koutsoyannis (1980)] is for a linear velocity profile, and the solution for the exponential shear flow is given elsewhere [Campos and Serrão (1998)]. The wave equation has three singularities, like the Gaussian hypergeometric equation, but it is of an extended type, since the singularity at infinity is irregular (all three singularities of the Gaussian hypergeometric equation are regular). The other two singularities of the present equation are regular, and one lies at the axis of the duct, and the other at the dicritical layers, where the Doppler-shifted frequency vanishes. The critical layers do not exist (they lie outside the duct) for longitudinal wave vector antiparallel to the mean flow (case I), i.e., propagation upstream in a local frame of reference. In the opposite case of longitudinal wave vector parallel to the mean flow (case II) there are three subcases, depending on whether the Doppler-shifted frequency on the axis of the duct is: (case II A) positive, i.e., critical layers are at imaginary ‘‘distance;’’ (case II B) zero, i.e., the critical layer lies on the axis of the duct: (case II C) negative, i.e., two critical layers exist in the duct. In all cases (I, II A, II B, II C) it is possible to obtain the exact acoustic field over the whole flow region by expanding around the singularities and matching solutions. Since the acoustic wave equation in a shear flow does not lead to a Sturm–Liouville problem, the eigenfunctions need not be orthogonal or complete. There is a single set of natural frequencies and normal modes in cases I, II A, and II B, but not in case II C; in the latter case, where two critical layers lie in the flow, they may separate three sets of eigenvalues/eigenfunctions, in the regions between by the critical layers and the walls.

Transient Motion of a Neutrally Buoyant Disk in Plane Parabolic Shear Flow

Transient Motion of a Neutrally Buoyant Disk in Plane Parabolic Shear Flow

Flow separation in a viscous parabolic shear past a cylinder

Low Reynolds number parabolic shear flow past a circular cylinder, placed asymetrically with its centre at a distance k from the shear axis, is investigated employing matching technique. The analysis of flow separation indicates coalescion of wakes with increasing k. The drag is governed by a parameter λ which changes sign depending on the magnitude of k, while the torque is proportional to k. The case of a rotating cylinder including free rotation is also discussed.

Flow separation in a viscous parabolic shear past a sphere

Axisymmetric parabolic shear flow past a spinning sphere in an unbounded viscous medium is considered using the matching technique and taking the non-uniform shear as the dominant feature, where the Reynolds number based on parabolic shearRe≪1 and the rotational Reynolds number based on the angular rotation of the sphereR0≪1 are such thatR02/Re=O(1). Forces on the body are calculated and flow separation from the boundary is studied. Coalescion of two wakes or detachment of a wake into two are observed, depending on the increase in magnitudeU′ of the free stream for a fixedR02/Re or increase inR02/Re for a fixedU′ respectively.

Forces on a neutrally buoyant disk in plane parabolic shear flow

A two-dimensional calculation of the force acting on a neutrally buoyant circular disk in the plane of an infinite parabolic shear flow profile is made in the creeping flow approximation.

The exterior Dirichlet problem for a quasilinear elliptic boundary value problem suggested by plane shear flow

We study the existence of a classical solution of the exterior Dirichlet problem for a class of quasilinear elliptic boundary value problems that are suggested by plane shear flow. In this connection only bounded solutions which tend to zero at infinity are of interest. A priori bounds on solutions and constructive existence proofs are given. Finally, we prove the existence of a unique bounded solution of the shear flow and we show, under certain hypotheses about the asymptotic behavior of the nonlinearity, that this solution tends to zero at infinity. As an example, we consider the case of the parabolic shear flow.

The Forces on a Thin Aerofoil in Slightly Parabolic Shear Flow

AbstractThe aerodynamic characteristics of a thin cambered aerofoil are determined, when the aerofoil is set at incidence in a shear flow of incompressible inviscid fluid. First order effects are deduced assuming that the deviation from uniform flow is small, the velocity profile being parabolic in form.