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In this work we study resonant triad interactions among discrete internal wave modes in a finite-depth, two dimensional uniformly stratified shear flow. The primary wave-field is considered to be a linear superposition of various internal wave modes. The weakly-nonlinear solution of the primary wave-field consists of a superharmonic (2ω) part and a mean-flow part (ω=0). For a given modal interaction, we study the location in the frequency (ω) -Richardson number (Ri) parameter space where the amplitude of the superharmonic part attains a maximum i.e, where two primary internal wave modes of modenumbers 'm' and 'n' resonantly excite a secondary wave mode of modenumber 'q'. Using asymptotic theory we show that, unlike the case of no-shear, the presence of weak-shear, doesn't require the vertical wavenumber condition to be satisfied for resonance. This entails an activation of several new resonances in the presence of arbitrarily weak shear, where only the frequency and the horizontal wavenumber conditions are satisfied. This also leads to the possibility of self-interaction and resonances close to ω = 0. A similar asymptotic theory can be extended to other inhomogeneities (eg: non-uniform stratification) as well. For an exponential background shear flow, we track the location of these resonances in the (ω, Ri) parameter space and present their behaviour.

We study the gravity-induced collisions of charged spheres of dielectric materials dispersed in a gaseous medium. When the gap thickness between the surfaces of two spheres is shorter than the mean free path of the surrounding fluid medium, continuum assumptions for the hydrodynamics interactions are no longer valid, and the non-continuum lubrication interactions result in surface-to-surface contact in finite time. Two like-charged dielectric spheres attract each other at close separations for a wide range of size and charge ratio values. We use trajectory analysis to calculate the collision rate and, thus, explore the role of electrostatic interactions on the collision dynamics of a pair of like-charged dielectric spheres. We present the modifications of pair trajectories due to electrostatic forces and show how collision efficiencies vary with the non-dimensional parameter capturing the relative strength of the electrostatic force to gravity as well as the charge ratio and size ratio.

We study the dynamics of dust particles in various vortical flows which is relevant to geophysical context. The inertial particles are advected by the background vortex flow. The dynamics is tracked using the Maxey-Riley equation. The finite inertia of the particles make their dynamics different from passive fluid parcels, which is interesting. The dust particles may show periodic dynamics or chaotic diffusion depending on parametric variations. The result contradicts the earlier predictions that only density matched inertial particles can have chaotic dynamics, which we justify through our explanation. In addition, the heavy inertial particles in a self rotating vortex patch is observed to be attracted near the vortical region, which is contrary to the physics where they should ideally centrifuged out. The reason behind this phenomena also we explore in detail here.

We study the transportation and rotational dynamics of a finite-sized spheroidal particle in a linear monochromatic surface gravity wave to better understand the transport dynamics of microplastics in oceanic flows. A spheroidal particle, modeled as an anisotropic tracer, attains preferential alignment in a linear wavy flow. We analyze the drift of a finite-size anisotropic particle and find that the horizontal drift of such particles can either increase or decrease depending on the initial orientation and the ratio of the size of the particle to the wavelength of the background wave field. Next, we derive the finite-size modification to the preferred alignment of the spheroidal particle with the flow propagation direction of the wave. In most scenarios, particles in the ocean can have a wide range of densities and are classified into positively and negatively buoyant particles. Negatively buoyant particles settle in a wavy flow with complex trajectories. We study the effect of the orientation and size of such particles on settling and show that the aspect ratio of the particle could alter the trajectory in the wave propagation direction. We also obtain a non-zero vertical Stokes drift. Finally, we consider the effects of fluid and particle inertia in our coupled evolution equations and study the drift and the orientation of an anisotropic particle in a wavy flow field. We demonstrate that considering such an effect could provide a complete picture of the transport and dynamics of microplastics in the upper part of the ocean that can be described more accurately.

Fluid flows over phase-changing boundaries find relevance in the study of many problems in astrophysical, geophysical, and engineering settings. A detailed understanding of the nonlinear coupling between fluid flow and the moving boundary is often critical to the evolution of the system. Growth of sea ice in the Earth's polar regions and the dynamics of ice sheets and the Earth's interior are a few examples that highlight their ubiquity across scales.Here, we consider the stability of a layer of liquid that is bounded by its solid phase at the bottom and a free surface at the top. The liquid has a temperature of maximum density, Tm, and convective motions in the liquid layer ensue when the temperature of the free surface is greater than Tm. We use a combination of linear and asymptotic analyses to explore the effects of the phase-changing boundary, density anomaly, and the free surface on the thermal and thermocapillary instabilities in this system.

We study a class of initial value problems (IVPs) involving perturbations on a density stratified, quiescent, viscous liquid layer with a free-surface. The geometry is a two-dimensional, rectangular configuration taking into account surface-tension and gravity. Linearised predictions are obtained by solving the IVP analytically for free-surface and vortical initial perturbations. The viscous spectrum comprises a discrete spectrum of propagating surface and internal modes and damped vorticity modes. Continuous spectra may also exist depending on whether the domain is bounded or unbounded. It is shown that independent of stratification, the vorticity modes in the spectrum play an important role. For an unstratified pool of a highly viscous liquid, the vorticity modes are found to make a large contribution to the temporal evolution of the transient vorticity layer, at the free-surface. It is demonstrated that the temporal evolution of localised vortical perturbations sufficiently deep inside the pool, cannot be captured by surface modes (capillarity-gravity modes) and require contributions from the vorticity modes. The free-surface signature of a deep, localised, vortical perturbation is however found to be negligible. On a stratified pool of viscous liquid, such vortical initial perturbations are shown to have significant projections on vorticity as well as internal gravity modes. In the infinite depth limit of an unstratified pool, the contribution from the continuous spectrum vorticity modes is analysed vis-a-vis that from their finite depth, discrete spectrum counterparts. Several analytical predictions are compared against direct numerical simulations (DNS) obtaining excellent agreement. Our results extend classical ones due to Lamb (1932), Yih (1960), Prosperetti (1976), Prosperetti and Cortelezzi (1982) and receive support from nonlinear simulations., 20 pages, 12 figures, 3 tables

In this paper, we analyse the instability of a thin-film suspension of micro-swimmers subjected to an attractant gradient. The imposed attractant gradient introduces a preferential swimming direction for the swimmers, resulting in an anisotropic orientation field. By modelling the swimmers as force dipoles, the study by Kasyap & Koch (Phys. Rev. Lett., vol. 108, issue 3, 2012, 038101, J. Fluid Mech., vol. 741, 2014, pp. 619–657) found an instability arising from the coupling between the active stress associated with an anisotropic orientation field and perturbations to the swimmer density field. In this work we begin by presenting a stability analysis that calculates the modification of this instability in the presence of an interface. We then detail the presence of a new mode of instability that arises solely from the deformation of the interface mediated by the active stress in the suspension. The resulting hydrodynamic instabilities in the system are observed to be sensitive to the direction of the attractant gradient relative to the interface. Furthermore, we show that the coupling between the two modes involving a jump in interfacial viscous stresses and normal stress differences within the suspension allows for an instability to manifest in a suspension of chemotactic pullers, previously thought to be unconditionally stable. We then use a long-wave theory to write down a set of nonlinear equations governing the film evolution. The numerical solution of the system using the long-wave theory aids in explaining the mechanism associated with the instability in addition to validating the predictions from the linear stability theory.

We present experimental and computational results for the orientation distributions of slender fibers and ramified particles settling in an isotropic turbulent flow. The rotational dynamics of the particles is modeled using a slender-body theory that includes the inertial torque due to sedimentation that tends to rotate the particles toward a broadside orientation. The particles are assumed to rotate due to viscous forces associated with the turbulent velocity gradients occurring on the particle length scale. In the simulations, the turbulence is obtained from a stochastic model of the velocity gradient in a Lagrangian reference frame. In the experiments, the turbulence is generated by active jets in a vertical water tunnel. It is well known that axisymmetric particles rotate according to Jeffery's solution for the rotation of a spheroidal particle if one adopts an appropriate effective aspect ratio. We show that the same result applies to a ramified particle consisting of three coplanar fibers connected with equal angles at a central point which rotates like a thin oblate spheroid. The orientation statistics can be quantified with a single non-dimensional parameter, the settling factor $S_F$, defined as the ratio of rotations due to sedimentation and turbulent shear. For low values of $S_F$, we observe nearly isotropically oriented particles, whereas particles become strongly aligned near the horizontal plane for high values of $S_F$. The variance of the angle away from horizontal scales as $S_F^{-2}$ for $S_F \gg 1$, but the orientation distribution is non-Gaussian due to turbulent intermittency in this limit., Comment: 58 pages, 16 figures, 4 tables

We study the collisions in a gaseous medium of a dilute bidisperse suspension of non-Brownian spherical particles sedimenting along the flow axis of a simple shear flow. Continuum lubrication forces prevent particles from coming into contact in a finite time, thus collisions can occur only due to attractive interactions such as the van der Waals force. However, in a low-pressure medium, the lubrication forces are weaker than their continuum counterparts and allow particle pairs to collide, even without any attractive forces. The Knudsen number, defined as the ratio of the mean free path of the medium to the mean radius of the interacting spheres, captures the significance of non-continuum interactions. We use uniformly valid hydrodynamic mobility functions, reflecting non-continuum lubrication at small separations and full continuum hydrodynamic interactions at moderate to larger separations. Due to the nature of the pair trajectory topology, the collision efficiency vanishes at a critical Knudsen number when simple shear flow alone drives the dynamics. Thus we perform collision calculations where particles experience the combined effects of van der Waals attraction and non-continuum hydrodynamics; van der Waals interactions enable collisions below the critical Knudsen number. Next, we calculate the collision efficiency for coupled differential sedimentation and simple shear driven motion in the presence of van der Waals interaction and non-continuum hydrodynamics. Finally, we explore the role of small particle inertia on relative trajectories and collision efficiencies in a non-continuum gas subject to a simple shear flow, ignoring the van der Waals force and gravity.

In this paper, we study the role of shear-induced migration and particle-induced normal stresses in the formation and stability of a particle-laden, gravity-driven shallow flow. We first examine the modification of the base-state Nusselt flow due to the underlying microstructure, how shear-induced migration leads to viscosity stratification. We inspect the development of the base state via the boundary layer formation in the ‘shallow’ limit and find a reduction in entrance length with increasing bulk particle concentration and an increase in entrance length with increasing Péclet number ($Pe_p = \dot {\gamma } a^2 / D_0$, where $\dot{\gamma}$ is the average shear rate, a is the particle size and $D_0$ is the single particle diffusivity). A linear stability analysis is then performed on the fully developed state to identify two modes of instability typically found in gravity-driven falling films – the long-wave surface and the short-wave shear modes. We find that when the associated Péclet number is $Pe_p \ll 1$, increasing bulk particle volume fraction delays the onset of instability for both the surface mode and shear mode. However, with $Pe_p = {O}(1)$, we find an enhancement in both modes of instability. We also find that, beyond a critical Péclet number, for a fixed particle volume fraction, the surface mode is unstable even in the absence of fluid inertia. The enhanced destabilisation is attributed to the combined effects of base-state viscosity stratification and momentum forcing via particle concentration perturbations. We also show that the physics behind the enhancement of instability is independent of the choice of the constitutive model used to describe the dynamics of the particle phase, provided the chosen model has elements of shear-induced migration.

In this paper, we study the role of electrostatic forces on pair trajectories of two uncharged conducting spheres subject to an external electric field. We consider the hydrodynamic interactions between the spheres as they move relative to one another. Previous studies have shown that electric-field-induced forces on a two-sphere system are always attractive, except for the configuration when the line joining the centers is perpendicular to the external electric field. In the current study, we derive the asymptotic form of the interparticle force induced by the electric field in the lubrication limit for arbitrary size ratios. The attractive electric force diverges as the separation approaches zero. Thus, our calculation shows that the electric-field-induced forces can overcome the continuum lubrication resistance and allow finite time contact between the surfaces of two spherical conductors. We calculate the asymptotic variation of interparticle separation using the near-field asymptotic expressions for the electric-field-induced forces, exploring the role of hydrodynamic interactions in interparticle motion parallel and perpendicular to the electric field.

We analyse the instability of a vortex column in a dilute polymer solution at large ${{Re}}$ and ${{De}}$ with ${{El}} = {{De}}/{{Re}}$ , the elasticity number, being finite. Here, ${{Re}} = \varOmega _0 a^2/\nu$ and ${{De}} = \varOmega _0 \tau$ are, respectively, the Reynolds and Deborah numbers based on the core angular velocity ( $\varOmega _0$ ), the radius of the column ( $a$ ), the total (solvent plus polymer) kinematic viscosity ( $\nu = (\mu _s +\mu _p)/\rho$ with $\mu _s$ and $\mu _p$ being the solvent and polymer contributions to the viscosity) and the polymeric relaxation time ( $\tau$ ). The stability of small-amplitude perturbations in this distinguished limit is governed by the elastic Rayleigh equation whose spectrum is parameterized by ${E} = {{El}}(1-\beta )$ , $\beta$ being the ratio of the solvent to the solution viscosity. The neglect of the relaxation terms, in the said limit, implies that the polymer solution supports undamped elastic shear waves propagating relative to the base-state flow. Unlike the neutrally stable inviscid case, an instability of the vortex column arises for finite ${E}$ due to a pair of elastic shear waves being driven into a resonant interaction under the differential convection by the irrotational shearing flow outside the core. An asymptotic analysis for the Rankine profile shows the absence of an elastic threshold for this instability. The growth rate is $O(\varOmega _0)$ for order unity $E$ , although it becomes transcendentally small for ${E} \ll 1$ , being $O(\varOmega _0 {E}^2{\rm e}^{-1/{E}^{{1}/{2}}})$ . An accompanying numerical investigation shows that the instability persists for smooth monotonically decreasing vorticity profiles, provided the radial extent of the transition region (from the rotational core to the irrotational exterior) is less than a certain ${E}$ -dependent threshold.

This study examines the effects of an evolving attractant field, due to the phenomenon of autochemotaxis, on the stability of a dilute suspension of active swimmers in a channel. Motile swimmers typically exhibit an orientation bias that is dependent upon the local gradient in the attractant field. By modelling the autochemotactic behaviour of the swimmers through a non-dimensional number $Da$ that characterizes the rate of attractant secretion, we are able to show that the active stress-driven instability predicted by Kasyap & Koch (Phys. Rev. Lett., vol. 108, issue 3, 2012, p. 038101) for a suspension of pushers is stabilized with increasing $Da$. We also show that the autochemotactic behaviour results in an active stress-driven flow instability for a suspension of pullers. Furthermore, we show analytically and numerically that in the absence of convective transport of the attractant, pushers and pullers undergo a simultaneous switch of stabilities at a critical $Da$.

We revisit the somewhat classical problem of the linear stability of a rigidly rotating liquid column in this communication. Although literature pertaining to this problem dates back to 1959, the relation between inviscid and viscous stability criteria has not yet been clarified. While the viscous criterion for stability, given by $We = n^2+k^2-1$, is both necessary and sufficient, this relation has only been shown to be sufficient in the inviscid case. Here, $We = \rho \Omega^2 a^3/\gamma$ is the Weber number and measures the relative magnitudes of the centrifugal and surface tension forces, with $\Omega$ being the angular velocity of the rigidly rotating column, $a$ the column radius, $\rho$ the density of the fluid, and $\gamma$ the surface tension coefficient; $k$ and $n$ denote the axial and azimuthal wavenumbers of the imposed perturbation. We show that the subtle difference between the inviscid and viscous criteria arises from the surprisingly complicated picture of inviscid stability in the $We-k$ plane. For all $n >1$, the viscously unstable region, corresponding to $We > n^2+k^2-1$, contains an infinite hierarchy of inviscidly stable islands ending in cusps, with a dominant leading island. Only the dominant island, now infinite in extent along the $We$ axis, persists for $n= 1$. This picture may be understood, based on the underlying eigenspectrum, as arising from the cascade of coalescences between a retrograde mode, that is the continuation of the cograde surface-tension-driven mode across the zero Doppler frequency point, and successive retrograde Coriolis modes constituting an infinite hierarchy., Comment: 22 pages, 13 figures

We investigate the effect of inertial particles dispersed in a circular patch of finite radius on the stability of a two-dimensional Rankine vortex in semi-dilute dusty flows. Unlike the particle-free case where no unstable modes exist, we show that the feedback force from the particles triggers a novel instability. The mechanisms driving the instability are characterized using linear stability analysis for weakly inertial particles and further validated against Eulerian–Lagrangian simulations. We show that the particle-laden vortex is always unstable if the mass loading $M>0$ . Surprisingly, even non-inertial particles destabilize the vortex by a mechanism analogous to the centrifugal Rayleigh–Taylor instability in radially stratified vortex with density jump. We identify a critical mass loading above which an eigenmode $m$ becomes unstable. This critical mass loading drops to zero as $m$ increases. When particles are inertial, modes that fall below the critical mass loading become unstable, whereas modes above it remain unstable but with lower growth rates compared with the non-inertial case. Comparison with Eulerian–Lagrangian simulations shows that growth rates computed from simulations match well the theoretical predictions. Past the linear stage, we observe the emergence of high-wavenumber modes that turn into spiralling arms of concentrated particles emanating out of the core, while regions of particle-free flow are sucked inward. The vorticity field displays a similar pattern which leads to the breakdown of the initial Rankine structure. This novel instability for a dusty vortex highlights how the feedback force from the disperse phase can induce the breakdown of an otherwise resilient vortical structure.

We study the role of phase change and thermal noise in particle transport in turbulent flows. We employ a toy model to extract the main physics: condensing droplets are modelled as heavy particles which grow in size, the ambient flow is modelled as a two-dimensional Taylor-Green (TG) flow consisting of an array of vortices delineated by separatrices, and thermal noise are modelled as uncorrelated Gaussian white noise. In general, heavy inertial particles are centrifuged out of regions of high vorticity and into regions of high strain. In cellular flows, we find, in agreement with earlier results, that droplets with Stokes numbers smaller than a critical value, $St < St_{\rm{cr}}$, remain trapped in the vortices in which they are initialised, while larger droplets move ballistically away from their initial positions by crossing separatrices. We independently vary the P��clet number $Pe$ characterising the amplitude of thermal noise and the condensation rate $��$ to study their effects on the critical Stokes number for droplet trapping, as well as on the final states of motion of the droplets. We find that the imposition of thermal noise, or of a finite condensation rate, allows droplets of $St < St_{\rm{cr}}$ to leave their initial vortices. We find that the effects of thermal noise become negligible for growing droplets, and that growing droplets achieve ballistic motion when their Stokes numbers become $\mathcal{O}(1)$. We also find an intermediate regime prior to attaining the ballistic state, in which droplets move diffusively away from their initial vortices in the presence of thermal noise., 14 pages, 11 figures

Experimental measurements of the force and torque on freely settling fibres are compared with predictions of the slender-body theory of Khayat & Cox (J. Fluid Mech., vol. 209, 1989, pp. 435–462). Although the flow is viscous dominated at the scale of the fibre diameter, fluid inertia is important on the scale of the fibre length, leading to inertial torques which tend to rotate symmetric fibres toward horizontal orientations. Experimentally, the torque on symmetric fibres is inferred from the measured rate of rotation of the fibres using a quasi-steady torque balance. It is shown theoretically that fibres with an asymmetric radius or mass density distribution undergo a supercritical pitch-fork bifurcation from vertical to oblique settling with increasing Archimedes number, increasing Reynolds number or decreasing asymmetry. This transition is observed in experiments with asymmetric mass density and we find good agreement with the predicted symmetry breaking transition. In these experiments, the steady orientation of the oblique settling fibres provides a means to measure the inertial torque in the absence of transient effects since it is balanced by the known gravitational torque.

This work aims to implement the asymptotic homogenization method (AHM) to predict the effective thermal/electrical conductivity for suspensions with aligned inclusions. Exploiting the substantial separation of length scales between the macroscopic and microscopic structures, multiscale modeling using the AHM capitalizes on the perturbations of the potential field caused due to the presence of an inclusion under a macroscopic loading used to predict the effective property. The analytical formulation for the thermal/electrical conductivity problem is derived, and subsequently, the finite element formulation required to solve the unit cell problem is described. The results obtained for a cylindrical inclusion are validated against known analytical solutions for both the dilute [Mori–Tanaka (MT)] and concentrated volume fractions ([Formula: see text]) of the inclusion. This study revealed that MT estimate and AHM agree well at [Formula: see text] less than 0.4. However, in near-maximum packing fractions, the AHM results fared significantly better than MT when compared with known asymptotic forms [J. Keller, “Conductivity of a medium containing a dense array of perfectly conducting spheres or cylinders or nonconducting cylinders,” J. Appl. Phys. 34, 991 (1963)]. The proposed AHM method is then implemented in structures with aligned spheroidal inclusions of various aspect ratios and conductivity ratios, thus providing a more generalized approach to predict the effective thermal/electrical conductivity. The results obtained are systematically benchmarked and validated against known analytical expressions.

The collisions in a dilute polydisperse suspension of sub-Kolmogorov spheres with negligible inertia settling in a turbulent flow and interacting through hydrodynamics including continuum breakdown on close approach are studied. A statistically significant decrease in ideal collision rate without gravity is resolved via a Lagrangian stochastic velocity-gradient model at Taylor microscale Reynolds number larger than those accessible by current direct numerical simulation capabilities. This arises from the difference between the mean inward velocity and the root-mean-square particle relative velocity. Differential sedimentation, comparable to the turbulent shear relative velocity, but minimally influencing the sampling of the velocity gradient, diminishes the Reynolds number dependence and enhances the ideal collision rate i.e. the rate without interactions. The collision rate is retarded by hydrodynamic interactions between sphere pairs and is governed by non-continuum lubrication as well as full continuum hydrodynamic interactions at larger separations. The collision efficiency (ratio of actual to ideal collision rate) depends on the relative strength of differential sedimentation and turbulent shear, the size ratio of the interacting spheres and the Knudsen number (defined as the ratio of the mean-free path of the gas to the mean radius of the interacting spheres). We develop an analytical approximation to concisely report computed results across the parameter space. This accurate closed form expression could be a critical component in computing the evolution of the size distribution in applications such as water droplets in clouds or commercially valuable products in industrial aggregators.

Collisions in a dilute polydisperse suspension of spheres of negligible inertia interacting through non-continuum hydrodynamics and settling in a slow uniaxial compressional flow are studied. The ideal collision rate is evaluated as a function of the relative strength of gravity and uniaxial compressional flow and it deviates significantly from a linear superposition of these driving terms. This non-trivial behaviour is exacerbated by interparticle interactions based on uniformly valid non-continuum hydrodynamics, that capture non-continuum lubrication at small separations and full continuum hydrodynamic interactions at larger separations, retarding collisions driven purely by sedimentation significantly more than those driven purely by the linear flow. While the ideal collision rate is weakly dependent on the orientation of gravity with the axis of compression, the rate including hydrodynamic interactions varies by more than with orientation. This dramatic shift can be attributed to complex trajectories driven by interparticle interactions that prevent particle pairs from colliding or enable a circuitous path to collision. These and other important features of the collision process are studied in detail using trajectory analysis at near unity and significantly smaller than unity size ratios of the interacting spheres. For each case analysis is carried for a large range of relative strengths and orientations of gravity to the uniaxial compressional flow, and Knudsen numbers (ratio of mean free path of the media to mean radius).

A detailed measurement of rainfall is an important input to various climate models. Accuracy of rainfall measurements also has a socio-economic impact since it forms an important constituent in the region-wise and seasonal water budget. Radar measurements are most promising in this direction since they provide the detailed spatio-temporal data than any conventional method. However, it also has the disadvantage that rainfall estimation using radar is an indirect measurement and often involves the use of a number of correlations. The amount of scattered radiation is directly related to the raindrop size and shape distribution. Therefore, this project is aimed at solving Maxwell’s equations of electromagnetism using the lattice Boltzmann method with the intention of accurately relating the size and shape distribution of falling raindrops to scattered intensity and thus improving the rainfall measurement. Towards this objective, a D3Q7 code for a pseudo vector discrete Boltzmann distribution function has been developed following the work of Hauser and Verhey (2017) [1]. Standard validations involving scattering of electromagnetic waves by a dielectric interface and a circular scatterer have been performed. Error in the solutions was found to be minimal on comparing with analytical solutions. The applicability of the proposed numerical method towards analyzing scattering of electromagnetic waves by a three-dimensional scatterer, modelled on realistic hydrometeors, will be discussed..

In this paper, the linear stability characteristics of a two-layered liquid–liquid flow in an inclined channel with slippery walls are investigated. Previous studies on two-layered inclined channel flows have observed the presence of multiple base state flow profiles, two for countercurrent flow and up to three base states for co-current flow. The role of wall slip on the multiple base states associated with each holdup solution is analyzed here. Subsequently, a linear stability analysis, using a combination of a long-wave asymptotic analysis and finite wavenumber numerical calculation, is carried out with the slip boundary condition. Neutral stability boundaries are presented for each base state, with comparisons made with the previous results obtained for the no-slip boundary condition. It was found that the wall slip could have both stabilizing and destabilizing effects depending on the flow rates and the value of holdup—the location of an interface.

We study the role of wall slip on the stability of a two-layered plane Poiseuille flow. The equations of motion for the base flow state are derived, and a linear stability analysis is carried out to arrive at the fourth-order Orr–Sommerfeld equations for the fluid layers. An asymptotic analysis is conducted for wavenumbers in the long wavelength limit. The Orr–Sommerfeld equations are solved numerically using a multidomain Chebyshev Collocation Method to arrive at the spectrum of eigenvalues and their associated eigenfunctions. The effect of wall slip on the stability characteristics of the flow system is examined in greater detail. It is observed that slip brings about a stabilizing, as well as a destabilizing effect on the flow system.

In this paper, we give a modal interpretation of the lift-up effect, one of two well-known mechanisms that lead to an algebraic instability in parallel shearing flows, the other being the Orr mechanism. To this end, we first obtain the two families of continuous spectrum modes that make up the complete spectrum for a non-inflectional velocity profile. One of these families consists of modified versions of the vortex-sheet eigenmodes originally found by Case (Phys. Fluids, vol. 3, 1960, pp. 143–148) for plane Couette flow, while the second family consists of singular jet modes first found by Sazonov (Izv. Acad. Nauk SSSR Atmos. Ocean. Phys., vol. 32, 1996, pp. 21–28), again for Couette flow. The two families are used to construct the modal superposition for an arbitrary three-dimensional distribution of vorticity at the initial instant. The so-called non-modal growth that underlies the lift-up effect is associated with an initial condition consisting of rolls, aligned with the streamwise direction, and with a spanwise modulation (that is, a modulation along the vorticity direction of the base-state shearing flow). This growth is shown to arise from an appropriate superposition of the aforementioned continuous spectrum mode families. The modal superposition is then generalized to an inflectional velocity profile by including additional discrete modes associated with the inflection points. Finally, the non-trivial connection between an inviscid eigenmode and the viscous eigenmodes for large but finite Reynolds number, and the relation between the corresponding modal superpositions, is highlighted.

In 1880 Lord Kelvin analysed the linearized inviscid oscillations of a Rankine vortex as part of a theory of vortex atoms. These eponymously named neutrally stable modes are, however, exceptional regular oscillations that make up the discrete spectrum of the Rankine vortex. In this paper, we examine the singular oscillations that make up the continuous spectrum (CS) and span the entire base state range of frequencies. In two dimensions, the CS eigenfunctions have a twin-vortex-sheet structure similar to that known from earlier investigations of parallel flows with piecewise linear velocity profiles. The vortex sheets are cylindrical, being threaded by axial lines, with one sheet at the edge of the core and the other at the critical radius in the irrotational exterior; the latter refers to the radial location at which the fluid co-rotates with the eigenmode. In three dimensions, the CS eigenfunctions have core vorticity and may be classified into two families based on the singularity at the critical radius. For the first family, the singularity is a cylindrical vortex sheet threaded by helical vortex lines, while for the second family it has a localized dipole structure with radial vorticity. The presence of perturbation vorticity in the otherwise irrotational exterior implies that the CS modes, unlike the Kelvin modes, offer a modal interpretation for the (linearized) interaction of the Rankine vortex with an external vortical disturbance. It is shown that an arbitrary initial distribution of perturbation vorticity, both in two and three dimensions, may be evolved as a superposition over the discrete and CS modes; this modal representation being equivalent to a solution of the corresponding initial value problem. For the restricted case of an initial axial vorticity distribution in two dimensions, the modal representation may be generalized to a smooth vortex. Finally, for the three-dimensional case, the analogy between rotational flows and stratified shear flows, and the known analytical solution for stratified Couette flow, are used to clarify the singular manner in which the modal superposition for a smooth vortex approaches the Rankine limit.

We consider the linear stability and nonlinear evolution of two-dimensional shear flows that take the form of an unstratified plane Couette flow that is seeded with a localized ‘defect’ containing sharp density and vorticity variations. For such flows, matched asymptotic expansions furnish a reduced model that allows a straightforward and computationally efficient exploration of flows at sufficiently high Reynolds and Péclet numbers that sharp density and vorticity gradients persist throughout the onset, growth and saturation of instability. We are thereby able to study the linear and nonlinear dynamics of three canonical variants of stratified shear instability: Kelvin–Helmholtz instability, the Holmboe instability, and the lesser-considered Taylor instability, all of which are often interpreted in terms of the interactions of waves riding on sharp interfaces of density and vorticity. The dynamics near onset is catalogued; if the interfaces are sufficiently sharp, the onset of instability is subcritical, with a nonlinear state existing below the linear instability threshold. Beyond onset, both Holmboe and Taylor instabilities are susceptible to inherently two-dimensional secondary instabilities that lead to wave mergers and wavelength coarsening. Additional two-dimensional secondary instabilities are also found to appear for higher Prandtl numbers that take the form of parasitic Holmboe-like waves.

We demonstrate that upstream swimming of sperm emerges via an orientation disorder-order transition. The order parameter, the average orientation of the sperm head against the flow, follows a 0.5 power law with the deviation from the critical flow shear rate (γ − γc). This transition is successfully explained by a hydrodynamic bifurcation theory, which extends the sperm upstream swimming to a broad class of near surface micro-swimmers that possess front-back asymmetry and circular motion.

The recently developed pushover analysis procedure has led a new dimension to performance-based design in structural engineering practices. With the increase in the magnitude of monotonic loading, weak links and failure modes in the multi-storey RC frames are usually formed. The force distribution and storey displacements are evaluated using static pushover analysis based on the assumption that the response is controlled by fundamental mode and no mode shift takes place. Himalayan-Nagalushai region, Indo-Gangetic plain, Western India, Kutch and Kathiawar regions are geologically unstable parts of India and some devastating earthquakes of remarkable intensity have occurred here. In view of the intensive construction activity in India, where even a medium intensity tremor can cause a calamity, the authors feel that a completely up-to-date, versatile method of aseismic analysis and design of structures are essential. A detailed dynamic analysis of a 10-storey RC frame building is therefore performed using response spectrum method based on Indian Standard Codal Provisions and base shear, storey shear and storey drifts are computed. A modal pushover analysis (MPA) is also carried out to determine the structural response of the same model for the same acceleration spectra used in the earlier case. The major focus of study is to bring out the superiority of pushover analysis method over the conventional dynamic analysis method recommended by the code. The results obtained from the numerical studies show that the response spectrum method underestimates the response of the model in comparison with modal pushover analysis. It is also seen that modal participation of higher modes contributes to better results of the response distribution along the height of the building. Also pushover curves are plotted to illustrate the displacement as a function of base shear.

Transient growth of perturbations by linear non-modal evolution is studied here in a stably stratified bounded Couette flow. The density stratification is linear. Classical inviscid stability theory states that a parallel shear flow is stable to exponential disturbances if the Richardson number (Ri) is greater than 1/4 everywhere in the flow. Experiments and numerical simulations at higher Ri show however that algebraically growing disturbances can lead to transient amplification. The complexity of a stably stratified shear flow stems from its ability to combine this transient amplification with propagating internal gravity waves (IGWs). The optimal perturbations associated with maximum energy amplification are numerically obtained at intermediate Reynolds numbers. It is shown that in this wall-bounded flow, the optimal perturbations are oblique, unlike in unstratified flow, with the structure realigning as time progresses, such that the initial and final obliqueness are different. A partitioning of energy into kinetic and potential helps in understanding the exchange of energies and how it modifies the transient growth. We show that the apportionment between potential and kinetic energy depends in an interesting manner on the Richardson number, and on time, as the transient growth proceeds from an optimal perturbation. The oft-quoted stabilizing role of stratification is also probed in the non-diffusive limit in the context of disturbance energy amplification.

Exclusive and kinematically complete high-statistics measurements of quasifree polarized (n) over right arrowp scattering have been performed in the energy region of the narrow resonancelike structure d* with I(J(P)) = 0(3(+)),M approximate to 2380 MeV, and Gamma approximate to 70 MeV observed recently in the double-pionic fusion channels pn -> d pi(0)pi(0) and pn -> d pi(+)pi(-). The experiment was carried out with the WASA detector setup at COSY having a polarized deuteron beam impinged on the hydrogen pellet target and utilizing the quasifree process (d) over right arrowp -> np + p(spectator). This allowed the np analyzing power, A(y), to be measured over a broad angular range. The obtained A(y) angular distributions deviate systematically from the current SAID SP07 NN partial-wave solution. Incorporating the new A(y) data into the SAID analysis produces a pole in the D-3(3) - (3)G(3) waves in support of the d* resonance hypothesis.

In this chapter, our objective is twofold: (1) to describe common physical mechanisms which cause flows to become unstable, and (2) to introduce recent viewpoints on the subject. In the former, we present some well-known instabilities, and also discuss how surface tension and viscosity can act as both stabilisers and destabilisers. The field has gone through a somewhat large upheaval over the last two decades, with the understanding of algebraic growth of disturbances, and of absolute instability. In the latter part we touch upon these aspects.

A novel form of an explicit numeric-analytic technique is developed for solving strongly nonlinear oscillators of engineering interest. The analytic part of this technique makes use of Adomian Decomposition Method (ADM), but unlike other analytical solutions it does not rely on the functional form of the solution over the whole domain of the independent variable. Instead it discretizes the domain and solves multiple IVPs recursively. ADM uses a rearranged Taylor series expansion about a function and finds a series of functions which add up to generate the required solution. The present method discretizes the axis of the independent variable and only collects lower powers of the chosen step size in series solution. Each function constituting the series solution is found analytically. It is next shown that the modified ADM can be used to obtain the analytical solution,in a piecewise form. For nonlinear oscillators such a piecewise solution is valid only within a chosen time step. An attempt has been made to address few issues like the order of local error and convergence of the method. Emphasis has been on the application of the present method to a number of well known oscillators. The method has the advantage of giving a functional form of the solution within each time interval thus one has access to finer details of the solution over the interval. This is not possible in purely numerical techniques like the Runge–Kutta method, which provides solution only at the two ends of a given time interval, provided that the interval is chosen small enough for convergence. It is shown that the present technique successfully overcomes many limitations of the conventional form of ADM. The present method has the versatility and advantages of numerical methods for being applied directly to highly nonlinear problems and also have the elegance and other benefits of analytical techniques.

We investigate analytically the short-time response of disturbances in a density-varying Couette flow without viscous and diffusive effects. The complete inviscid problem is also solved as an initial value problem with a density perturbation. We show that the kinetic energy of the disturbances grows algebraically at early times, contrary to the well-known algebraic decay at time tending to infinity. This growth can persist for arbitrarily long times in response to sharp enough initial perturbations. The simplest in our three-stage study is a model problem forced by a buoyancy perturbation in the absence of background stratification. A linear growth with time is obtained in the vertical velocity component. This model provides an analogy between the transient mechanism of kinetic energy growth in a two-dimensional density-varying flow and the lift-up mechanism of the three-dimensional constant density flow. Next we consider weak stable background stratification. Interestingly, the lowest order solution here is the same as that of the model flow. Our final study shows that a strong background stratification results in a sub-linear growth with time of the perturbation. A framework is thus presented where two-dimensional streamwise disturbances can lead to large transient amplification, unlike in constant density flow where three dimensions are required.