When a fluid system is subjected to an acoustic wave (or another periodic actuation), the response of the fluid is not purely periodic, but is rather characterized by the combination of a periodic flow and a steady Stokes drift component, where the former is, in many cases, an acoustic wave and the latter is commonly referred to as acoustic streaming. Classical theories of acoustic streaming have focused on slow acoustic streaming, where the periodic flow is the leading-order flow, and is insensitive to the steady flow component which appears as a small correction and is characterized by a small hydrodynamic Reynolds number. In contrast, Dubrovski et al. (J. Fluid Mech. vol. 975, 2023, A4) tackle the fast acoustic streaming regime – conceived by Zarembo (Acoustic streaming. In High-Intensity Ultrasonic Fields, 1971, pp. 135–199. Springer) approximately fifty years ago – where both the periodic and steady flow components are of a similar order of magnitude such that the periodic flow both supports and is simultaneously impacted by the steady flow. They present a novel theoretical framework that accounts for the convection of momentum both within and between the periodic and steady flow to extend slow-streaming equations to the case of steady flow with arbitrary hydrodynamic Reynolds number. They leverage a scaling analysis of the resulting system of equations and a case study to demonstrate the compatibility of their equations with slow streaming theories and highlight the distinctive features of fast streaming.

The hydrodynamic influence of surface texture on static surfaces ranges from large drag penalties (roughness) to potential performance benefits (shark-like skin). Although it is of wide-ranging research interest, the impact of roughness on flapping systems has received limited attention. In this work, we explore the effect of roughness on the unsteady performance of a harmonically pitching foil through experiments using foils with different surface roughness, at a fixed Strouhal number and within the Reynolds number ($Re$) range of $17\,000\unicode{x2013}33\,000$. The foils’ surface roughness is altered by changing the distribution of spherical-cap-shaped elements over the propulsor area. We find that the addition of surface roughness does not improve the performance compared with a smooth surface over the $Re$ range considered. The analysis of the flow fields shows near-identical wakes regardless of the foil's surface roughness. The performance reduction mainly occurs due to an increase in profile drag. However, we find that the drag penalty due to roughness is reduced from $76\,\%$ for a static foil to $16\,\%$ for a flapping foil at the same mean angle of attack, with the strongest decrease measured at the highest $Re$. Our findings highlight that the effect of roughness on dynamic systems is very different than that on static systems; thereby, it cannot be estimated by only using information obtained from static cases. This also indicates that the performance of unsteady, flapping systems is more robust to the changes in surface roughness.

In two-fluid simulations of gas–solid fluidised beds, the gaseous phase and the particulate phase are modelled as continuous media. The stress exerted by the particulate medium on the container walls should be modelled to predict accurately the bed dynamics. This paper addresses the modelling of sliding particle–wall contacts in two-fluid simulations, based on reference simulations coupling computational fluid dynamics with the discrete element method (CFD-DEM), in which the individual movement of the particles is tracked. The analysis of the CFD-DEM highlights the complex near-wall behaviour of the particles, which is not reproduced by two-fluid models. Nevertheless, the particle–wall shear stress can be expressed based on the total granular pressure within the first cell off the wall. The model is validated for the two-fluid simulation of a bubbling gas–solid fluidised bed of olefin particles in the dense-fluidisation regime.

Flow within submerged canopies influences the transport of nutrients, sediment, pollutants, plant seeds and the settlement of larvae. To improve our understanding of mass transport within canopies, a simple model is proposed to predict the total time-varying velocity within submerged rigid canopies (representing coral reefs) and flexible canopies (representing seagrasses and saltmarshes). The model divides the momentum equations into a canopy layer and free-stream layer. The difference in the time derivative of the velocity between the two layers is balanced by the sum of the shear stress and canopy drag, both of which depend on the in-canopy total velocity. The present model extended the shear stress model developed for steady current to combined current and wave conditions without additional calibrating coefficients. The model agreed well with the in-canopy velocity measured in the present and several previous studies. Importantly, the proposed model significantly improved the accuracy of canopy time-mean velocity prediction, which reduced the root mean square error by more than 50 %, compared with previous models. The model revealed that the addition of waves can significantly decrease the in-canopy time-mean velocity.

Cavitation inception originates from nuclei in a liquid. This paper proposes a Gibbs free energy approach that provides a smooth transition from homogeneous to heterogeneous nucleation when gas is present. The impact of gas content on nucleation is explored. It is found that the gas content stabilises nuclei, a phenomenon not present in pure liquid–vapour systems. This reduces the energy barrier over that required to nucleate a vapour bubble. Different gas saturation levels are studied. Gas content can significantly reduce the energy barrier required for nucleation, and under certain circumstances eliminate it. An analytic solution for the critical radius and activation energy is obtained that accounts for gas content. The classical Blake radius is recovered as a limiting case. The hysteresis between incipience and desinence is explained using the asymmetry observed in the critical radii. The solution is used to obtain the initial bubble radius, given a critical pressure condition in cavitation susceptibility meter experiments. The relationship between initial bubble diameter and critical pressure is described by an analytic solution that accounts for gas content. A model for the derivative of the cumulative nuclei histogram with respect to bubble diameter is proposed. An analytic expression is obtained that shows good agreement with decades worth of experimental data compiled by Khoo et al. (Exp. Fluids, vol. 61, issue 2, 2020, pp. 1–20) from ocean to water tunnels. The expression recovers the $-4$ power law that is observed experimentally.

The numerical investigation focuses on the flow patterns around a rectangular cylinder with three aspect ratios ($L/D=5$, $10$, $15$) at a Reynolds number of $1000$. The study delves into the dynamics of vortices, their associated frequencies, the evolution of the boundary layer and the decay of the wake. Kelvin–Helmholtz (KH) vortices originate from the leading edge (LE) shear layer and transform into hairpin vortices. Specifically, at $L/D=5$, three KH vortices merge into a single LE vortex. However, at $L/D=10$ and $15$, two KH vortices combine to form a LE vortex, with the rapid formation of hairpin vortex packets. A fractional harmonic arises due to feedback from the split LE shear layer moving upstream, triggering interaction with the reverse flow. Trailing edge (TE) vortices shed, creating a Kármán-like street in the wake. The intensity of wake oscillation at $L/D=5$ surpasses that in the other two cases. Boundary layer transition occurs after the saturation of disturbance energy for $L/D=10$ and $15$, but not for $L/D=5$. The low-frequency disturbances are selected to generate streaks inside the boundary layer. The TE vortex shedding induces the formation of a favourable pressure gradient, accelerating the flow and fostering boundary layer relaminarization. The self-similarity of the velocity defect is observed in all three wakes, accompanied by the decay of disturbance energy. Importantly, the decrease in the shedding frequency of LE (TE) vortices significantly contributes to the overall decay of disturbance energy. This comprehensive exploration provides insights into complex flow phenomena and their underlying dynamics.

We present the results from a series of experiments investigating the dynamics of gravity currents which form when a dense saline or particle-laden plume issuing from a moving source interacts with a horizontal surface. We define the dimensionless parameter $P$ as the ratio of the source speed, $u_a$, to the buoyancy speed, $(B_0/z_0)^{1/3}$, where $B_0$ and $z_0$ are the source buoyancy flux and height above the horizontal surface, respectively. Using our experimental data, we determine that the limiting case in which $P=P_c$ the gravity current only spreads downstream of the initial impact point occurs when $P_c=0.83\pm 0.02$. For $P< P_c$, from our experiments we observe that the plume forms a gravity current that spreads out in all directions from the point of impact and the propagation of the gravity current is analogous to a classical constant-flux gravity current. For $P>P_c$, we observe that the descending plume is bent over and develops a pair of counter-rotating line vortices along the axis of the plume. The ensuing gravity current spreads out downstream of the source, normal to the motion of the source. Analogous processes occur with particle-laden plumes, but there is a second dimensionless parameter $S$, the ratio of the particle fall speed, $v_s$, to the vertical speed of a plume in a crossflow, $(B_0/u_a z_0)^{1/2}$. For $S\ll 1$, particles remain well mixed in the plume and a particle-driven gravity current develops. For $S\gg 1$, particles separate from the plume prior to impacting the boundary which leads to a fall deposit and no gravity current. We discuss these results in the context of deep-sea mining.

The dynamics and nonlinear wave forcing of a flexible floating structure are investigated experimentally and numerically. The floater was designed to match sub-harmonic rigid-body natural frequencies of typical floating wind turbine substructures, with the addition of a flexible bending mode. Experiments were carried out for three sea states with phase-shifted input signals to allow harmonic separation of the measured response. We find for the weakest sea states that sub-harmonic rigid-body motion is driven by even-harmonic difference frequency forcing, and by linear forcing for the strongest sea state. The flexible mode was tested in a soft, linearly forced layout, and a stiff layout, forced by second-, third- and fourth-harmonic frequency content, for increasing severity of the sea state. Further insight is gained by analysis of the amplitude scaling of the resonant response. A new simplified approach is proposed and compared with the recent method of Orszaghova et al. (J. Fluid Mech., vol. 929, 2021, A32). We find that resonant surge and pitch motions are dominated by even-harmonic potential-flow forcing and that odd-harmonic response is mainly potential-flow driven in surge and mainly drag driven in pitch. The measured responses are reproduced numerically with second-order forcing and quadratic drag loads, using a recent and computationally efficient calculation method, extended here for the heave, pitch and flexible motions. We are able to reproduce the response statistics and power spectra for the measurements, including the subharmonic pitch and heave modes and the flexible mode. Deeper analysis reveals that inaccuracies in the even-harmonic forcing content can be compensated by the odd-harmonic loads.

We experimentally investigate the forced synchronization of a self-excited chaotic thermoacoustic oscillator with two natural frequencies, $f_1$ and $f_2$. On increasing the forcing amplitude, $\epsilon _f$, at a fixed forcing frequency, $f_f$, we find two different types of synchronization: (i) $f_f/f_1 = 1:1$ or $2:1$ chaos-destroying synchronization (CDS), and (ii) phase synchronization of chaos (PSC). En route to $1:1$ CDS, the system transitions from an unforced chaotic state (${\rm {CH}}_{1,2}$) to a forced chaotic state (${\rm {CH}}_{1,2,f}$), then to a two-frequency quasiperiodic state where chaos is destroyed ($\mathbb {T}^2_{2,f}$), and finally to a phase-locked period-1 state (${\rm {P1}}_f$). The route to $2:1$ CDS is similar, but the quasiperiodic state hosts a doubled torus $(2\mathbb {T}^2_{2,f})$ that transforms into a phase-locked period-2 orbit $({\rm {P2}}_f)$ when CDS occurs. En route to PSC, the system transitions to a forced chaotic state (${\rm {CH}}_{1,2,f}$) followed by a phase-locked chaotic state, where $f_1$, $f_2$ and $f_f$ still coexist but their phase difference remains bounded. We find that the maximum reduction in thermoacoustic amplitude occurs near the onset of CDS, and that the critical $\epsilon _f$ required for the onset of CDS does not vary significantly with $f_f$. We then use two unidirectionally coupled Anishchenko–Astakhov oscillators to phenomenologically model the experimental synchronization dynamics, including (i) the route to $1:1$ CDS, (ii) various phase dynamics, such as phase drifting, slipping and locking, and (iii) the thermoacoustic amplitude variations in the $f_f/f_1$–$\epsilon _f$ plane. This study extends the applicability of open-loop control further to a chaotic thermoacoustic system, demonstrating (i) the feasibility of using an existing actuation strategy to weaken aperiodic thermoacoustic oscillations, and (ii) the possibility of developing new active suppression strategies based on both established and emerging methods of chaos control.

We investigate the Faraday instabilities of a three-layer fluid system in a cylindrical container containing low-viscosity liquid metal, sodium hydroxide solution and air by establishing the Mathieu equations with considering the viscous model derived by Labrador et al. (J. Phys.: Conf. Ser., vol. 2090, 2021, 012088). The Floquet analysis, asymptotic analysis, direct numerical simulation and experimental method are adopted in the present study. We obtain the dispersion relations and critical oscillation amplitudes of zigzag and varicose modes from the analysis of the Mathieu equations, which agree well with the experimental result. Furthermore, considering the coupling strength of two interfaces, besides zigzag and varicose modes, we find a beating instability mode that contains two primary frequencies, with its average frequency equalling half of the external excitation frequency in the strongly coupled system. In the weakly coupled system, the $A$-interface instability, $B$-interface instability and $A$&$B$-interface instability are defined. Finally, we obtain a critical wavenumber $k_c$ that can determine the transition from zigzag or varicose modes to the corresponding $A$-interface or $B$-interface instability.

The relationships between characteristic large-scale wake structures appearing behind a free-stream-aligned circular cylinder are investigated and discussed from the velocity field obtained by wind tunnel tests. The tests were conducted under a supportless condition using a magnetic suspension and balance system and stereo PIV measurements at a Reynolds number of $3.46\times 10^4$. The velocity fields were analysed with a modal decomposition combining azimuthal Fourier decomposition and proper orthogonal decomposition. The wake behind the free-stream-aligned circular cylinder with three different fineness ratios of 1.0, 1.5 and 2.0 was investigated, and the wake structures in a non-reattaching flow formed by the cylinder at a fineness ratio of 1.0 are mainly discussed in the present study. Four characteristic large-scale wake structures of the recirculation bubble pumping, azimuthal shear mode, large-scale vortex shedding and streaks are identified and mainly focused on in the present study. The state of the vortex shedding is classified into three: anticlockwise/clockwise circular and flapping patterns. Each state has a relationship with the azimuthal shear mode and it tends to appear when the state is circular. Furthermore, from the analysis of the relationship between modes, the recirculation bubble pumping is found to be related to the vortex shedding position in the radial direction and the strength of the streaks. Particularly, analysis of causality shows that the recirculation bubble pumping is affected by them in the low-frequency range.

Over the last decade, substantial progress has been made in understanding the topology of quasi-two-dimensional (2-D) non-equilibrium fluid flows driven by ATP-powered microtubules and microorganisms. By contrast, the topology of three-dimensional (3-D) active fluid flows still poses interesting open questions. Here, we study the topology of a spherically confined active flow using 3-D direct numerical simulations of generalized Navier–Stokes (GNS) equations at the scale of typical microfluidic experiments. Consistent with earlier results for unbounded periodic domains, our simulations confirm the formation of Beltrami-like bulk flows with spontaneously broken chiral symmetry in this model. Furthermore, by leveraging fast methods to compute linking numbers, we explicitly connect this chiral symmetry breaking to the entanglement statistics of vortex lines. We observe that the mean of linking number distribution converges to the global helicity, consistent with the asymptotic result by Arnold [In Vladimir I. Arnold – Collected Works (ed. A.B. Givental, B.A. Khesin, A.N. Varchenko, V.A. Vassiliev & O.Y. Viro), pp. 357–375. Springer]. Additionally, we characterize the rate of convergence of this measure with respect to the number and length of observed vortex lines, and examine higher moments of the distribution. We find that the full distribution is well described by a k-Gamma distribution, in agreement with an entropic argument. Beyond active suspensions, the tools for the topological characterization of 3-D vector fields developed here are applicable to any solenoidal field whose curl is tangent to or cancels at the boundaries of a simply connected domain.

Wall-based spanwise forcing has been experimentally used with success by Auteri et al. (Phys. Fluids, vol. 22, 2010, 115103) to obtain large reductions of turbulent skin-friction drag and considerable energy savings in a pipe flow. The spatial distribution of the azimuthal wall velocity used in the experiment was not continuous, but piecewise constant. The present study is a numerical replica of the experiment, based on a set of direct numerical simulations (DNS); its goal is the identification of the effects of spatially discrete forcing, as opposed to the idealized sinusoidal forcing considered in the majority of numerical studies. Regardless of the discretization, with DNS the maximum drag reduction is found to be larger: the flow easily reaches complete relaminarization, whereas the experiment was capped at 33 % drag reduction. However, the key result stems from the observation that, for the piecewise-constant forcing, the apparent irregularities of the experimental data appear in the simulation data too. They derive from the rich harmonic content of the discontinuous travelling wave, which alters the drag reduction of the sinusoidal forcing. A detailed understanding of the contribution of each harmonic reveals that, whenever for example technological limitations constrain one to work far from the optimal forcing parameters, a discrete forcing may perform very differently from the corresponding ideal sinusoid, and in principle can outperform it. However, care should be exercised in comparison, as discrete and continuous forcing have different energy requirements.

To investigate the influence of inertia and slip on the instability of a liquid film on a fibre, a theoretical framework based on the axisymmetric Navier–Stokes equations is proposed via linear instability analysis. The model reveals that slip significantly enhances perturbation growth in viscous film flows, whereas it exerts minimal influence on flows dominated by inertia. Moreover, under no-slip boundary conditions, the dominant instability mode of thin films remains unaltered by inertia, closely aligning with predictions from a no-slip lubrication model. Conversely, when slip is introduced, the dominant wavenumber experiences a noticeable reduction as inertia decreases. This trend is captured by an introduced lubrication model with giant slip. Direct numerical simulations of the Navier–Stokes equations are then performed to further confirm the theoretical findings at the linear stage. For the nonlinear dynamics, no-slip simulations show complex vortical structures within films, driven by fluid inertia near surfaces. Additionally, in scenarios with weak inertia, a reduction in the volume of satellite droplets is observed due to slip, following a power-law relationship.

We develop a theoretical and experimental framework for generating slip underneath thin-film flows of viscous fluids in the laboratory, with the ability to control slip as desired. Such a framework is useful for large-scale fluid-mechanical experiments in which basal sliding is important. In particular, we consider the flow of a thin film of viscous fluid spreading over a structured, slippery substrate, involving a sequence of two-dimensional cavities that are prewetted with a fluid of smaller viscosity. By averaging over small-scale inhomogeneities, we demonstrate that such a substrate gives rise to a macroscopic linear sliding law, or Navier slip condition, that is effectively homogeneous on the large scale. The slip length, determining the slipperiness of the substrate, is proportional to the viscosity ratio and width of each cavity. As such, the slipperiness of the substrate can be controlled by altering the viscosity ratio, as desired. Two asymptotic regimes arise, describing flow over very slippery substrates and flow over no-slip substrates. The former regime is valid for early times, when the depth of the overlying fluid is much less than the slip length, and the latter is valid for late times, when the depth is much greater than the slip length. Solutions to the full model approach similarity solutions describing the two regimes for early and late times. We confirm our theoretical predictions by conducting a series of analogue laboratory experiments.

The present work focuses on the effect of rough horizontal boundaries on the heat transfer in rotating Rayleigh–Bénard convection. We measure the non-dimensional heat transfer, the Nusselt number $Nu$, for various strengths of the buoyancy forcing characterized by the Rayleigh number $Ra$ (${10^5}\mathrm{\ \mathbin{\lower.3ex\hbox{$\buildrel< \over {\smash{\scriptstyle\sim}\vphantom{_x}}$}}\ }Ra\mathrm{\ \mathbin{\lower.3ex\hbox{$\buildrel< \over {\smash{\scriptstyle\sim}\vphantom{_x}}$}}\ }5 \times {10^8}$), and rotation rates characterized by the Ekman number E ($1.4 \times {10^{ - 5}}\mathrm{\ \mathbin{\lower.3ex\hbox{$\buildrel< \over {\smash{\scriptstyle\sim}\vphantom{_x}}$}}\ }E\mathrm{\ \mathbin{\lower.3ex\hbox{$\buildrel< \over {\smash{\scriptstyle\sim}\vphantom{_x}}$}}\ }7.6 \times {10^{ - 4}}$) for aspect ratios $\varGamma \approx 1$, $2.8$ and $6.7$. Similar to rotating convection with smooth horizontal boundaries, the so-called rotationally constrained (RC), rotation-affected (RA) and rotation-unaffected (RuA) regimes of heat transfer seem to persist for rough horizontal boundaries. However, the transition from the RC regime to RA regime occurs at a lower Rayleigh number for rough boundaries. For all experiments with rough boundaries in this study, the thermal and Ekman boundary layers are in a perturbed state, leading to a significant enhancement in the heat transfer as compared with that for smooth walls. However, the enhancement in heat transfer due to wall roughness is observed to attain a maximum in the RC regime. We perform companion direct numerical simulations of rotating convection over smooth walls to suggest a phenomenology explaining this observation. We propose that the heat transfer enhancement due to wall roughness reaches a maximum when the strength and coherence of the columnar structures are both significant, which enables efficient vertical transport of the additional thermal anomalies generated by the roughness at the top and bottom walls.

Subgrid-scale (SGS) modelling is formulated using a local transport of spectral kinetic energy estimated by a wavelet multiresolution analysis. Using a spectrally and spatially local decomposition by wavelet, the unresolved inter-scale energy transfer and modelled SGS dissipation are evaluated to enforce explicitly and optimally their balance a priori over a range of large-eddy simulation (LES) filter widths. The formulation determines SGS model constants that optimally describe the spectral energy balance between the resolved and unresolved scales at a given cutoff scale. The formulation is tested for incompressible homogeneous isotropic turbulence (HIT). One-parameter Smagorinsky- and Vreman-type eddy-viscosity closures are optimised for their model constants. The algorithm discovers the theoretical prediction of Lilly (The representation of small-scale turbulence in numerical simulation experiments. In Proceedings of the IBM Scientific Computing Symposium on Environmental Sciences, pp. 195–210) at a filter cutoff scale in the inertial subrange, whereas the discovered constants deviate from the theoretical value at other cutoff scales so that the spectral optimum is achieved. The dynamic Smagorinsky model used a posteriori shows a suboptimal behaviour at filter scales larger than those in the inertial subrange. A two-parameter Clark-type closure model is optimised. The optimised constants provide evidence that the nonlinear gradient model of Clark et al. (J. Fluid Mech., vol. 91, issue 1, 1979, pp. 1–16) is prone to numerical instability due to its model form, and combining the pure gradient model with a dissipative model such as the classic Smagorinsky model enhances numerical stability but the standard mixed model is not optimal in terms of spectral energy transfer. A posteriori analysis shows that the optimised SGS models produce accurate LES results.

Under adverse pressure gradient (APG) conditions, the outer regions of turbulent boundary layers (TBLs) are characterized by an increased velocity defect $U_{e}-U$, an outwards shift of the peak value of the Reynolds shear stress $-\langle uv\rangle$ and an appearance of the outer peak value of the Reynolds normal stress $\langle uu\rangle$. Here $U_{e}$ is the TBL edge velocity. Scaling APG TBLs is challenging due to the non-equilibrium effects caused by changes in the APG. To address this, the response distance of TBLs to non-equilibrium conditions is utilized to extend the Zagarola–Smits scaling $U_{zs} = U_{e}({\delta ^{*} }/{\delta })$ and ensure that the original properties of the Zagarola–Smits scaling are maintained as $Re \to \infty$. Here $\delta ^{*}$ is the displacement thickness and $\delta$ is the boundary layer thickness. Based on the established correlation between $U_{e}-U$ and $-\langle uv\rangle$, the scaling is extended to $-\langle uv\rangle$. Furthermore, considering the coupling relationship between Reynolds stress components, the scaling is extended to encompass each Reynolds stress component. The proposed consistent scaling is verified using five non-equilibrium databases and five near-equilibrium databases, successfully collapsing the data of the TBL outer region. The pressure gradient parameter $\beta =({\delta ^{*} }/{\rho u_{\tau }^{2} }) ({\mathrm {d} P_{e} }/{\mathrm {d}\kern0.7pt x})$ of these databases spans two orders of magnitude. Here $P_{e}$ is the boundary layer edge pressure, $u_{\tau }$ is the friction velocity and $\rho$ is the density. Finally, the influence of the APG on the inner and outer regions of TBLs is analysed using the mean momentum balance equation. The analysis suggests that the shift of the $-\langle uv\rangle$ peak to the outer region under APG conditions is due to an insufficient inertia term near the inner region to balance the APG. It is observed that the APG promotes interaction between the inner and outer regions of TBLs, but the inner and outer regions still retain distinctive properties.

We simulate the head-on collision between vortex rings with circulation Reynolds numbers of 4000 using an adaptive, multiresolution solver based on the lattice Green's function. The simulation fidelity is established with integral metrics representing symmetries and discretization errors. Using the velocity gradient tensor and structural features of local streamlines, we characterize the evolution of the flow with a particular focus on its transition and turbulent decay. Transition is excited by the development of the elliptic instability, which grows during the mutual interaction of the rings as they expand radially at the collision plane. The development of antiparallel secondary vortex filaments along the circumference mediates the proliferation of small-scale turbulence. During turbulent decay, the partitioning of the velocity gradients approaches an equilibrium that is dominated by shearing and agrees well with previous results for forced isotropic turbulence. We also introduce new phase spaces for the velocity gradients that reflect the interplay between shearing and rigid rotation and highlight geometric features of local streamlines. In conjunction with our other analyses, these phase spaces suggest that, while the elliptic instability is the predominant mechanism driving the initial transition, its interplay with other mechanisms, e.g. the Crow instability, becomes more important during turbulent decay. Our analysis also suggests that the geometry-based phase space may be promising for identifying the effects of the elliptic instability and other mechanisms using the structure of local streamlines. Moving forward, characterizing the organization of these mechanisms within vortices and universal features of velocity gradients may aid in modelling turbulent flows.

We examined the influences of slip parameters on the velocity and thermal characteristics of a ferrofluid film of fixed thickness. The flow is generated on a rough and inclined whirling surface that is positioned in an external magnetic (dipole) field. The similarity transformation reduces the model equations (continuity, momentum, energy and concentration), and the solution of the normalized coupled ordinary differential equations is carried out through the finite element process. The influences of slip effects, Brownian motion, thermophoresis and a heat source on the velocity (radial, tangential and axial), gravity (drainage, induced), temperature profile and concentration profile are determined. The tangential flow and temperature are both decreased by an increase in the velocity slip parameter, whereas drainage, induced, radial and axial flows are increased. Enlarging the thermal slip parameter decreases the temperature. Improving slip parameters (velocity and thermal) also improves the concentration profile. Both Nusselt and Sherwood numbers are found to improve on improving the velocity slip parameter, while they decrease on decreasing the thermal slip parameter. The results and insights from this work could be applied to a wide range of medicinal fields, such as targeted medication therapy and delivery, tissue engineering, etc. as well as different industrial processes including coating, lubrication, heat transfer, etc.