We examine the flow in and around a falling fluid droplet in a vertically oscillating flow. We assume axisymmetric Stokes flow, and for small deformations to the droplet, the governing equations can be linearized leading to an infinite system of linear ordinary differential equations. In this study, we have analytically solved the problem in the small-capillary limit. We note that the solution locally breaks down at the poles of the droplet. The drag and center of the mass were also obtained. In the case when only odd modes are present, the droplet shows three-dimensional axisymmetric heart-shaped solutions oscillating vertically in time. When only even modes are present, the droplet exhibits axisymmetric stretching and squeezing.

In this study we examine the large Péclet number, Pe, limit of a concentration boundary layer in Couette flow. The boundary layer has a thickness of order Pe−1/2. The asymptotic concentration is asymptotically obtained as an integral solution up to order Pe−1/2 using the Fourier sine transform. The asymptotic solution is found to be in good agreement with the full numerical solution for large Péclet numbers. Further, the effective diffusivity obtained from the asymptotic solution is found to be in good agreement with the effective diffusivity obtained from the full numerical solution for large Péclet numbers.

Gas–solid fluidized bed reactors are widely used in the power generation industry. The critical effect of the presence of liquid phase, either as a result of heat, chemical reaction or physical interaction, on the hydrodynamics of the reactor is well recognized by academic researchers and industrial operators. However, theory and simulation frameworks to predict such a condition using the continuum modeling approach are not yet available. This study first shows the significant changes in the flow pattern and distinguishable flow regimes in a slightly wet fluidized bed recorded by an advanced imaging technique. The study then describes the development and implementation of new mathematical formulations for wet particle-particle interactions in a continuum model based on the classic kinetic theory of granular flow (KTGF). Quantitative validation, carried out by comparing the predicted and measured fluidization index (FI) expressed in terms of pressure drop, has shown a good match. The prediction also demonstrates increased bubble splitting, gas channeling, slugging fluidization, and energy dissipation induced by liquid bridges developing from wet particle interactions. These characteristics are similar to those commonly observed in the fluidization of cohesive powders. This model constitutes an important step in extending the continuum theories of dry flow to wet particle-particle interactions. This will allow accurate description and simulation of the fluidized bed in its widest application including power generation systems that involve wet particle fluidization.

Stratified Taylor–Couette flow (STCF) undergoes transient growth. Recent studies have shown that there exists transient amplification in the linear regime of counter-rotating STCF. The kinetic budget of the optimal transient perturbation is analysed numerically to simulate the interaction of the shear production (SP), buoyancy flux (BP), and other energy components that contributes to the total optimal transient kinetic energy. These contributions affect the total energy by influencing the perturbation to extract kinetic energy (KE) from the mean flow. The decay of the amplification factor resulted from the positive amplification of both BP and SP, while the growth is attributed to the negative and positive amplification of BP and SP, respectively. The optimal SP is positively amplified, implying that there is the possibility of constant linear growth. These findings agree with the linear growth rate for increasing values of Grashof number.

The analysis of the Taylor–Couette problem in the small gap limit is extended to the of nearly columnar (NC) solutions. The theoretical results are derived in the small-gap approximation which is not always well approximated in experiments. Despite this studies in the Cartesian frame work when compared with theoretical results and observations yield good agreement. For higher values of the axial wavenumber, β , up to β∼ 3 rather narrow Taylor vortices may be realized for Reynolds number R<80 . These vortices will become unstable to states with columnar components with increasing R . We show that for low R , (R,β)∼(62.2,3.5) , a state with a strong columnar component drifting in stream-wise direction exists for azimuthal wavenumbers α∼ 0.17 with β=3.5 . We examine the bifurcation sequence of these states.

Dynamics of bimolecular A+B & RARR;C reaction-diffusion (RD) fronts have been well characterized in miscible systems. We study here their properties when reactants A and B are initially dissolved in two immiscible solvents put in contact. Neglecting any convective effects, we show that bimolecular RD fronts developing in such immiscible systems upon transfer from the reactants from one phase to the other can change directions up to four times, contrary to miscible systems where only two changes of direction are possible. Further, the conditions for which two local maxima in the reaction rate can exist are identified.& COPY; 2023 Published by Elsevier Ltd.

In this study systematic numerical analyses are outlined searching for additional instabilities in the co-rotating Taylor–Couette system within the fully deterministic sequential approach of bifurcations (SBA) to turbulence. The main idea of the search strategy is the application of a forcing function, rotation, which has a direct physical interpretation, and that was realized in prior experimental work. The forcing induces disturbances that lead to bifurcations of new states. Thus, turbulence can be generated and observed in a rotating fluid without the imposing additional forcing sources. The imposition of thermoconvective forcing in the Taylor–Couette system will be discussed separately. Important findings include the discovery of the interplay of new and already known states, the transition of steady states to oscillatory ones and higher order states in the SBA via vortex merger/separation and re-allocation of symmetries for a more intensified mass transport. The results of the present work enhance the results of [ 1 ]. They will be revisited within an internal length gradient (ILG) framework accounting for weekly nonlocal effects as suggested in the concluding section of the paper.

Non-linear solutions and their stability are presented for homogeneously heated channel flows with a simple geometry under the influence of a constant pressure gradient or when the vanishing of the mass flux across any lateral cross-section of the channel is imposed. The critical Grashof number is determined by linear stability analysis for various values of the Prandtl number. In our numerical study the angle of inclination of the channel is taken into account. We found that in each case studied, with the exception of a horizontal layer of fluid and when the applied constant pressure gradient is zero, the basic flow looses stability through a Hopf bifurcation. Following the linear stability analysis our numerical studies are focused on the emerging secondary flows and their stability, in order to identify possible bifurcation points for tertiary flows. We conclude with a few comments on revisiting the present results within an internal length gradient (ILG) framework accounting for higher order velocity and temperature gradients.

This study focuses on the investigation of stratified Taylor–Couette flow (STCF) using non-modal analysis, which has received relatively limited attention compared to other shear flows. The dynamics of perturbations under different temperature conditions are explored, and their patterns of amplification are analyzed. The study highlights the correlation between flow configurations, emphasizing the similarity in transient dynamics despite different speed ratios. The subcritical effects of thermal stratification on disturbance dynamics are examined, considering the interplay between viscous and buoyancy effects counteracted by strong centrifugal forces. It is found that increasing the wall temperature beyond a critical value leads to buoyancy forces dominating, resulting in a linear increase in the amplification factor. The research reveals significant deviations from previous results, indicating the significant role of temperature stratification.

Non-linear solutions and their stability are presented for homogeneously heated fluids bounded by rigid conducting and insulating plates. In particular, we sought roll-type solutions emerging from the neutral stability curve for fluids with Prandtl numbers of 0.025, 0.25, 0.705, and 7. We determined the stability boundaries for the roll states in order to identify possible bifurcation points for the secondary flow in the form of regions that are equivalent to the Busse balloon. We also compared the stability exchange between ‘‘up’’ and ‘‘down’’ hexagons for a Prandtl number of $$0.25$$ obtained from weakly non-linear analysis in relation to the fully non-linear analysis, consistent with earlier studies. Our numerical analysis showed that there are potential bistable regions for both hexagons and rolls, a result that requires further investigations with a fully non-linear analysis.

Chaos is a relatively new subject in mathematical science. It has rapidly, however, found applications in a diverse range of areas in pure science and engineering. This work considers the application of chaos in viscous fluid mixing, fluids that are, for example, found in the food or polymer industry. It has been shown by many authors that under steady state conditions laminar fluids flow in a streamline motion. It has also been shown that it is possible to increase the degree of disorder in such a system by introducing time dependant moving boundaries. It is well established however, that chaotic motion is not dependent on the variation in streamlines alone. The system is required to be operated within a specific range of boundary conditions for chaos to occur. This work presents further investigations of chaotic flow domains concentrating on a property that relates to the divergence and separation of particles originating from small generation zones within the flow domain. The property of divergence of particles is noted to yield conclusions consistent with previously published work in two dimensions. It is then applied to investigate the degree of disorder in three three-dimensional continuous chaotic systems, operating with a range of time and spatially periodic boundary conditions. In order to highlight the benefits of increased levels of mixing when operating the systems under specific conditions characteristic plots, for incremental variations of operating conditions, are presented. It is hoped that by the development of the characteristic plots and concepts presented here that engineers will be able to predict and characterise the degree of disorder in future industrial chaotic mixing equipment.

We employ a homotopy method, rather than conventional stability theory, in order to resolve the degeneracy due to resonance, which exists in fluid motion associated with a channel of infinite extent in ventilated double glazing. The introduction of a symmetry breaking perturbation, in the form of a Poiseuille flow component, alters substantially the resonant bifurcation tree of the original flow. Previously unknown resonant higher order nonlinear solutions, i.e. after the removal of the perturbative Poiseuille flow component, are discovered. A possible extension of the methodology to consider non-Newtonian gradient enhanced incompressible viscous fluids is also briefly discussed.

Structured on the paradigmatic Navier-Stokes flow model, we study a stochastically forced Taylor-Couette system in the narrow gap limit, in order to analyze the simultaneous impact of a non-conserved (Gaussian) force and a nonlinear perturbation, in determining linear stability. Our analysis identifies key parametric windows within which the model remarkably retains its linear stability, even against jointly varying stochastic forcing and nonlinear fluctuations. We identify this feature as a {\it latent universality}, that we then utilize to answer the elusive question as to how a recent groundbreaking accretion flow model retains its stability, even in presence of nonlinear perturbations and non-conserved stochastic forcing. Our analytical outline goes beyond the immediate model studied and lays a generic foundation of analyzing stability for nonlinear sheared models acted on by external forces and subjected to boundary layer instability. presence of nonlinear perturbations and non-conserved stochastic forcing. Our analytical outline goes beyond the immediate model studied and lays a generic foundation of analyzing stability for nonlinear sheared models acted on by external forces and subjected to boundary layer instability.

The analysis of the Taylor-Couette problem in the small gap limit is extended to the cases of tertiary and quaternary solutions. The theoretical results are compared with experimental observations. Although in the latter the small-gap approximation is not always well approximated, the comparison of theoretical results and observations yields reasonable agreements. The absence of the wavy twist mode in the observed patterns is explained by the presence of no-slip boundary conditions in the axial direction of the experimental apparatus, which differ from the periodic conditions imposed in the theoretical analysis. Quaternary solutions bifurcating from the tertiary ones through subharmonic instabilities are presented and compared with experimental observations. Reasonable agreement has been found.

Brine disposal is a major challenge facing the desalination industry. Discharged brines pollute the oceans and aquifers. Here is it proposed to reduce the volume of brines by means of evaporative coolers in seawater greenhouses, thus enabling the cultivation of high-value crops and production of sea salt. Unlike in typical greenhouses, only natural wind is used for ventilation, without electric fans. We present a model to predict the water evaporation, salt production, internal temperature and humidity according to ambient conditions. Predictions are presented for three case studies: (a) the Horn of Africa (Berbera) where a seawater desalination plant will be coupled to salt production; (b) Iran (Ahwaz) for management of hypersaline water from the Gotvand dam; (c) Gujarat (Ahmedabad) where natural seawater is fed to the cooling process, enhancing salt production in solar salt works. Water evaporation per face area of evaporator pad is predicted in the range 33 to 83m3/m2·yr, and salt production up to 5.8tonnes/m2·yr. Temperature is lowest close to the evaporator pad, increasing downwind, such that the cooling effect mostly dissipates within 15m of the cooling pad. Depending on location, peak temperatures reduce by 8–16°C at the hottest time of year.

The transition of laterally heated flows in a vertical layer and in the presence of a streamwise pressure gradient is examined numerically for the case of different values Prandtl number. The stability analysis of the basic flow for the pure hydrodynamic case (Pr=0) was reported in [1]. We find that in the absence of transverse pumping the previously known critical parameters are recovered [2], while as the strength of the Poiseuille flow component is increased the convective motion is delayed considerably. Following the linear stability analysis for the vertical channel flow our attention is focused on a study of the finite amplitude secondary travelling-wave (TW) solutions that develop from the perturbations of the transverse roll type imposed on the basic flow and temperature profiles. The linear stability of the secondary TWs against three-dimensional perturbations is also examined and it is shown that the bifurcating tertiary flows are phase-locked to the secondary TWs.

We propose a simple computational procedure in order to resolve the degeneracy, which invariably exists on the background of fluid motion associated with a channel of infinite extent. The procedure is applied to elucidate the bifurcation structure for the particular case of laterally heated flow with the addition of a perturbative Poiseuille flow component. The introduction of a symmetry breaking perturbation as the simplest imperfection alters the bifurcation tree of the original shear flow. As a result, the previously unknown higher order nonlinear solutions for the unperturbed flow are discovered, without implementing classical stability theory.

Finite element simulations have been performed along side normal mode analysis on the linear stability that examined the development of volumetrically heated flow patterns in a horizontal layer controlled by the Prandtl number, Pr, and the Grashof number, Gr. The fluid was bounded by an isothermal plane above an adiabatic plane. In the simulations performed here, a number of convective polygonal planforms occurred, as Gr increased above the critical Grashof number, Grc at Pr = 7, while roll structures were observed for Pr < 1 at 2Grc.

An outline of the state space of planar Couette flow at high Reynolds numbers (Re < 10^4) is investigated via a variety of efficient numerical techniques. It is verified from nonlinear analysis that the lower branch of Hairpin Vortex State (HVS) asymptotically approaches the primary (laminar) state with increasing Re. It is also predicted that the lower branch of HVS at high Re belongs to the stability boundary that initiates transition to turbulence. These facts suggest HVS may provide a criterion to estimate a minimum perturbation arising transition to turbulent states at the infinite Re limit.