We examine the dynamic interactions between the large-scale coherent motion and the small-scale turbulence in the passive scalar field of a circular cylinder wake, where the coherent motion exhibits strong periodicity. A combination of four X-wires and four cold wires was used to simultaneously measure the three velocity and temperature fluctuations at nominally the same location. Measurements were taken at $x/d=10$, 20 and 40 in the mean shear plane at Reynolds number 2500, based on the cylinder diameter $d$ and the free-stream velocity. The phase-averaging technique is used to distinguish the large-scale coherent motion from the stochastic motion, enabling the construction of phase-averaged structure functions of the passive scalar in the scale phase plane. The maximum of the coherent scalar $\tilde {\theta }$ closely aligns with the minima of the phase-averaged strain $\langle S \rangle$ and the vortex centre, suggesting that heat is contained within the interior of the vortex. The scale-by-scale distributions of the scalar variance and the streamwise velocity variance exhibit a similar phase dependence associated with the coherent motion. This dependence is perceptible even at the smallest scales. However, as the distance from the cylinder increases, the perceivable scale range decreases and eventually disappears. An expression is formulated to describe the time-averaged second-order structure function of coherent scalar and the time-averaged second-order mixed structure function between the coherent scalar and coherent streamwise velocity at $x/d= 10$ and 20, where the coherent motion is prominent. Furthermore, the scale-by-scale contribution of the coherent scalar variance to the total scalar variance is evaluated. Also, we derive the scale-by-scale scalar variance transport equations that account for the coherent motion in both general and isotropic formulations. Assuming local isotropy, it is found that the equation agrees approximately with the experimental data across all scales at $x/d= 40$. Finally, the differences between the scale-by-scale transport equation for the stochastic scalar variance and that for the stochastic turbulent kinetic energy are discussed.

The flow past a $6:1$ prolate spheroid at a moderate pitch angle $\alpha =10^\circ$ is investigated with a focus on the turbulent wake in a high-fidelity large eddy simulation (LES) study. Two length-based Reynolds numbers, ${\textit{Re}}_L=3\times 10^4$ and $9\times 10^4$, and four Froude numbers, ${\textit{Fr}} = \infty \text{(unstratified)}, 6, 1.9 \text{ and }1$, are selected for the parametric study. Spectral proper orthogonal decomposition (SPOD) analysis of the flow reveals the leading coherent modes in the unsteady separated flow at the tail of the body. At the higher ${\textit{Re}}_L=9\times 10^4$, a high-frequency spanwise flapping of shear layers on either side of the body is observed in the separated boundary layer for all cases. The flapping does not perturb the lateral symmetry of the wake. At ${\textit{Fr}}=\infty$, a low-frequency oscillating laterally asymmetric mode, which is found in addition to the shear-layer mode, leads to a sidewise unsteady lateral load. All temporally averaged wakes at ${\textit{Re}}=9\times 10^4$ are found to be spanwise symmetric in the mean as opposed to the lower ${\textit{Re}}=3\times 10^4$, at which the ${\textit{Fr}}=\infty \text{ and }6$ wakes exhibit asymmetry. The turbulent kinetic energy (TKE) budget is compared among cases. Here, ${\textit{Fr}}=\infty$ exhibits higher production and dissipation compared with ${\textit{Fr}}=6 \text{ and }1.9$. The streamwise vortex pair in the wake induces a significant mean vertical velocity ($U_z$). Therefore, in contrast to straight-on flow, the terms involving gradients of $U_z$ matter to TKE production. Buoyancy reduces $U_z$ and also the Reynolds shear stresses involving $u^{\prime}_z$. Through this indirect mechanism, buoyancy exerts control on the wake TKE budget, albeit being small relative to production and dissipation. Buoyancy, through the baroclinic torque, is found to qualitatively affect the streamwise vorticity. In particular, the primary vortex pair is extinguished in the intermediate wake and two new vortex pairs form with opposite-sense circulation relative to the primary.

The convection velocity in high-Reynolds-number pipe flow was investigated using two-point correlations obtained from two laser Doppler velocimetry systems. The Reynolds number ranged from ${\textit{Re}}_{{\tau}}=3000$ to 20 800, and profiles were obtained from $y/R=0.002$ up to the pipe centre, where $R$ is the pipe radius. This study examines the scaling behaviour of convection velocity profiles derived from raw velocity signals, and the convection characteristics of very large-scale motions (VLSMs) and large-scale motions extracted via scale-separated or time-resolved velocity signals. The profiles show that convection velocities from raw signals exceed the local mean velocity near the wall and gradually approach it toward the centre. These profiles can be scaled using inner variables, namely $y^+$ and $\Delta x^+$, where $\Delta x^+$ represents the measurement distance. Scale-separated convection velocities for VLSM-scale structures – defined as those larger than $5R$ – were higher than the unfiltered values and remained nearly constant up to $y^+ \leq 2000$ at ${\textit{Re}}_{{\tau}} \approx 20\,000$. In this constant region, the convection velocity of VLSMs scaled well with the bulk velocity $U_{\textit{b}}$, taking values of approximately $0.85U_{\textit{b}}$. Furthermore, analysis of the time-resolved data highlights that, when applying Taylor’s frozen turbulence hypothesis, it is essential to consider both the scale dependence and the temporal fluctuations of the convection velocity, which reflect the underlying spatio-temporal dynamics of the flow structures. The present study provides valuable data for discussions on converting frequency-domain measurements into wavenumber space using Taylor’s hypothesis.

For smectic C* (SmC*) liquid crystals, configured in a bookshelf-type geometry between two horizontal parallel plates, with the bottom plate fixed and the top plate free to move, it is known from experiment that pumping can occur when an electric field is applied, i.e. an upward movement of the top plate through mechanical vibrations when the electric field is suddenly reversed. In this paper we revisit an earlier mathematical model for fast electric field reversal by removing an assumption made there on the velocity field; instead, we arrive at a time-dependent, two-dimensional squeeze-film model, which can ultimately be formulated in terms of a highly nonlinear integro-differential equation. Subsequent analysis leads to an unexpected solvability condition involving the five SmC* viscosity coefficients regarding the existence and uniqueness of solutions. Furthermore, we find that, when solutions do exist, they imply that the plate can move down as well as up, with the final resting position turning out to be dependent on the initial conditions; this is in stark contrast to the results of the earlier model.

This paper investigates the transient characteristics of uniform momentum zones (UMZs) in a rapidly accelerating turbulent pipe flow using direct numerical simulation datasets starting from an initial friction Reynolds number ($Re_{\tau 0}) = 500$ up to a final friction Reynolds number ($Re_{\tau 1}) = 670$. Instantaneous UMZs are identified following the identification methodology proposed by Adrian et al. (2000 J. Fluid Mech. vol. 422, pp. 1–54). The present results reveal that, as the flow rapidly accelerates, the average number of UMZs drops. However, as the flow recovers, it is regained. This result is complemented by the temporal evolution of the average number of internal shear layers. The temporal evolution of UMZs reveals that UMZs sustain their hierarchical flow arrangement with slower zones near the wall and faster zones away from the wall throughout the rapid turbulent flow acceleration. The results show that UMZs speed up during the inertial and pre-transition phases, and progressively slow down during the transition and core-relaxation stages. It is also revealed that UMZs near the wall respond first to flow instability and show earlier signs of recovery based on UMZ kinematic results. Finally, the dominant quadrant behaviour of Reynolds shear stress within UMZs has been investigated. It is found that, prior to the flow excursion, the UMZs nearest to the wall are always $Q2$ dominated, while the rest of the UMZs are always $Q4$ dominated. This behaviour is detected to not change during and after the flow excursion, suggesting that this is a characteristic behaviour of UMZs in accelerating turbulent wall-bounded flows.

Permanent gravity waves propagating in deep water, spanning amplitudes from infinitesimal to their theoretical limiting values, remain a classical yet challenging problem due to its inherent nonlinear complexities. Traditional analytical and numerical methods encounter substantial difficulties near the limiting wave condition due to singularities at sharp wave crests. In this study, we propose a novel hybrid framework combining the homotopy analysis method (HAM) with machine learning (ML) to efficiently compute convergent series solutions of Stokes waves in deep water for arbitrary wave amplitudes from small to theoretical limiting values, which show excellent agreement with established benchmarks. We introduce a neural network trained using only 20 representative cases whose series solution are given by means of HAM, which can rapidly predict series solutions across arbitrary steepness levels, substantially improving computational efficiency. Additionally, we develop a neural network to gain the inverse mapping from the conformal coordinates $(\theta , r)$ to the physical coordinates $(x,y)$, facilitating explicit and intuitive representations of series solutions in physical plane. This HAM–ML hybrid framework represents a powerful and efficient approach to compute convergent series in a whole range of physical parameters for water waves with arbitrary wave height including even limiting waves. In this way we establish a new paradigm to quickly obtain convergent series solutions of complex nonlinear systems for a whole range of physical parameters, thereby significantly broadening the scope of series solutions that can be easily gained by means of HAM even for highly nonlinear problems in science and engineering.

Granular flow down an inclined plane is ubiquitous in geophysical and industrial applications. On rough inclines, the flow exhibits Bagnold’s velocity profile and follows the so-called $\mu (I)$ local rheology. On insufficiently rough or smooth inclines, however, velocity slip occurs at the bottom and a basal layer with strong agitation emerges below the bulk, which is not predicted by the local rheology. Here, we use discrete element method simulations to study detailed dynamics of the basal layer in granular flows down both smooth and rough inclines. We control the roughness via a dimensionless parameter, $R_a$, varied systematically from 0 (flat, frictional plane) to near 1 (very rough plane). Three flow regimes are identified: a slip regime ($R_a \lesssim 0.45$) where a dilated basal layer appears, a no-slip regime ($R_a \gtrsim 0.6$) and an intermediate transition regime. In the slip regime the kinematics profiles (velocity, shear rate and granular temperature) of the basal layer strongly deviate from Bagnold’s profiles. General basal slip laws are developed that express the slip velocity as a function of the local shear rate (or granular temperature), base roughness and slope angle. Moreover, the basal layer thickness is insensitive to flow conditions but depends somewhat on the interparticle coefficient of restitution. Finally, we show that the rheological properties of the basal layer do not follow the $\mu (I)$ rheology, but are captured by Bagnold’s stress scaling and an extended kinetic theory for granular flows. Our findings can help develop more predictive granular flow models in the future.

We examine the linear stability of a shear flow driven by wind stress at the free surface and rotation at the lower boundary, mimicking oceanic flows influenced by surface winds and the Earth’s rotation. The linearised eigenvalue problem is solved using the Chebyshev spectral collocation method and a long-wave asymptotic analysis. Our results reveal new long-wave instability modes that emerge for non-zero rotational Reynolds numbers. It is observed that the most unstable mode, characterised by the lowest critical parameters, corresponds to long-wave spanwise disturbances with vanishing streamwise wavenumber. The asymptotic analysis, which shows excellent agreement with numerical results, analytically confirms the existence of this instability. Thus, the present study demonstrates the hitherto unreported combined influence of wind stress and the Earth’s rotation on ocean dynamics.

In the paper, we consider a two-dimensional free-surface flow past a single point vortex in fluid of infinite depth. The flow moves from left to right with uniform speed $c$ far upstream and is subject to the downward acceleration $g$ of gravity. A point vortex of circulation $\varGamma$ is located at depth $H$. The positive direction of circulation is counterclockwise. The flow is characterised by two dimensionless parameters which are the dimensionless vortex circulation $\gamma =\varGamma /(\textit{cH}\,)$ and the Froude number $ \textit{Fr}=c/\sqrt {gH}$. The goal of the paper is to find the solutions of the solitary wave type with one or several crests on the free surface. These solutions are waveless far downstream and have a vertical line of symmetry. We have established that for a fixed Froude number $ \textit{Fr}\le 0.8$, there exists a finite set of positive $\gamma$ for which the solutions of the solitary wave type occur.

Secondary flows induced by spanwise heterogeneous surface roughness play a crucial role in determining engineering-relevant metrics such as surface drag, convective heat transfer and the transport of airborne scalars. While much of the existing literature has focused on idealized configurations with regularly spaced roughness elements, real-world surfaces often feature irregularities, clustering and topographic complexity for which the secondary flow response remains poorly understood. Motivated by this gap, we investigate multicolumn roughness configurations that serve as a regularized analogue of roughness clustering. Using large-eddy simulations, we systematically examine secondary flows across a controlled set of configurations in which cluster density and local arrangement are varied in an idealized manner, and observe that these variations give rise to distinct secondary flow polarities. Through a focused parameter study, we identify the spanwise gap between the edge-most roughness elements of adjacent columns, normalized by the channel half-height ($s_a/H$), as a key geometric factor governing this polarity. In addition to analysing the time-averaged structure, we investigate how variations in polarity affect the instantaneous dynamics of secondary flows. Here, we find that the regions of high- and low-momentum fluid created by the secondary flows alternate in a chaotic, non-periodic manner over time. Further analysis of the vertical velocity signal shows that variability in vertical momentum transport is a persistent and intrinsic feature of secondary flow dynamics. Taken together, these findings provide a comprehensive picture of how the geometric arrangement of roughness elements governs both the mean structure and temporal behaviour of secondary flows.

A thin, evaporating sessile droplet with a pinned contact line containing inert particles is considered. In the limit in which the liquid flow decouples from the particle transport, we discuss the interplay between particle advection, diffusion and adsorption onto the solid substrate on which the droplet sits. We perform an asymptotic analysis in the physically relevant regime in which the Péclet number is large, i.e. ${\textit{Pe}}\gg 1$, so that advection dominates diffusion in the droplet except in a boundary layer near the contact line, and in which the ratio of the particle velocities due to substrate adsorption and diffusion is at most of order unity as ${\textit{Pe}}\rightarrow \infty$. We use the asymptotic model alongside numerical simulations to demonstrate that substrate adsorption leads to a different leading-order distribution of particle mass compared with cases with negligible substrate adsorption, with a significant reduction of the mass in the suspension – the nascent coffee ring reported in Moore et al. (J. Fluid Mech., vol. 920, 2021, A54). The redistribution leads to an extension of the validity of the dilute suspension assumption, albeit at the cost of breakdown due to the growth of the deposited layer, which are important considerations for future models that seek to accurately model the porous deposit regions.

We propose a novel multiple-scale spatial marching method for flows with slow streamwise variation. The key idea is to couple the boundary region equations, which govern large-scale flow evolution, with local exact coherent structures that capture the small-scale dynamics. This framework is consistent with high-Reynolds-number asymptotic theory and offers a promising approach to constructing time-periodic finite-amplitude solutions in a broad class of spatially developing shear flows. As a first application, we consider a non-uniformly curved channel flow, assuming that a finite-amplitude travelling-wave solution of plane Poiseuille flow is sustained at the inlet. The method allows for the estimation of momentum transport and highlights the impact of the inlet condition on both the transport properties and the overall flow structure. We then consider a case with gradually decreasing curvature, starting with Dean vortices at the inlet. In this setting, small external oscillatory disturbances can give rise to subcritical self-sustained states that persist even after the curvature vanishes.

Spectral analysis of the transport process of turbulence kinetic energy (TKE) in a channel roughened with spanwise-aligned circular-arc ribs is conducted based on direct numerical simulations (DNS). Test cases of varying pitch-to-height ratios ($P/H=3.0$, 5.0 and 7.5) and bulk Reynolds numbers (${\textit{Re}}_b=5600$ and 14 600) are compared. It is observed that the characteristic spanwise wavelength of the energy-containing eddies in the internal shear layer (ISL) increases as the value of $P/H$ increases, but decreases as the Reynolds number increases. In the ISL, the energy transport processes are dominated by turbulent production as the lead source term, but by turbulent diffusion and dissipation as the lead sink terms. It is found that regions with high production and dissipation rates of TKE in the ISL are associated with moderate and small wavelengths, respectively. The TKE production for sustaining moderate- and large-scale motions enhances gradually with an increasing value of $P/H$, while that for sustaining small-scale motions augments as the Reynolds number increases. It is interesting to observe that the interscale-transport term plays a critical role in draining TKE at moderate wavelengths as a sink and carries the drained TKE to small-scale eddies as a source. It is discovered that a higher pitch-to-height ratio leads to shortening of the characteristic spanwise wavelength of the dissipation process but prolongation of those of the production, interscale-transport and turbulent-diffusion processes in the ISL. By contrast, a higher Reynolds number results in reductions in the characteristic spanwise wavelengths of all spectral transport terms.

The triadic interactions and nonlinear energy transfer are investigated in a subsonic turbulent jet at $Re = 450\,000$. The primary focus is on the role of these interactions in the formation and attenuation of streaky structures. To this end, we employ bispectral mode decomposition, a technique that extracts coherent structures associated with dominant triadic interactions. A strong triadic correlation is identified between Kelvin–Helmholtz (KH) wavepackets and streaks: interactions between counter-rotating KH waves generates streamwise vortices, which subsequently give rise to streaks through the lift-up mechanism. The most energetic streaks occur at azimuthal wavenumber $m = 2$, with the dominant contributing triad being $[m_1, m_2, m_3] = [1, 1, 2]$. The spectral energy budget reveals that the net effect of nonlinear triadic interactions is an energy loss from the streaks. As these streaks convect downstream, they engage in further nonlinear interactions with other frequencies, which drain their energy and ultimately lead to their attenuation. Further analysis identifies the dominant scales and direction of energy transfer across different spatial regions of the jet. While the turbulent jet exhibits a forward energy cascade in a global sense, the direction of energy transfer varies locally: in the shear layer near the nozzle exit, triadic interactions among smaller scales dominate, resulting in an inverse energy cascade, whereas farther downstream, beyond the end of the potential core, interactions among larger scales prevail, leading to a forward cascade.

Sea surface films significantly influence air–sea interaction. While their damping effect on gravity–capillary waves is well recognised, the detailed mechanisms by which surface films alter small-scale wave dynamics – particularly energy dissipation and near-surface flow patterns – remain insufficiently understood. This paper presents experimental observations focusing on small-scale wave profiles and surface-flow dynamics in the presence of surfactants, providing direct experimental evidence of underlying mechanisms such as Marangoni effects. The experiments demonstrate enhanced energy dissipation and significant alterations in near-surface flow caused by surfactants, including the transformation of typical circular motion into elliptical-like trajectories and the emergence of reverse surface drift.

Ionic surfactants are commonly employed to modify the rheological properties of fluids, particularly in terms of surface viscoelasticity. Concurrently, external electric fields can significantly impact the dynamics of liquid threads. A key question is how ionic surfactants affect the dynamic behaviour of threads in the presence of an electric field? To investigate this, a one-dimensional model of a liquid thread coated with surfactants within a radial electric field is established, employing the long-wave approximation. We systematically investigate the effects of dimensionless parameters associated with the surfactants, including surfactant concentration, dilatational Boussinesq number ${\textit{Bo}}_{\kappa \infty }$ and shear Boussinesq number ${\textit{Bo}}_{\mu \infty }$. The results indicate that increasing the surfactant concentration and the two Boussinesq numbers reduces both the maximum growth rate and the dominant wavenumber. In addition, both the electric field and surfactants mitigate the breakup of the liquid thread and the formation of satellite droplets. At low applied electric potentials, the surface viscosity induced by surfactants predominantly governs this suppression. Surface viscosity suppresses the formation of satellite droplets by maintaining the neck point at the centre of the liquid thread within a single disturbance wavelength. When the applied potential is high, the electric stress has two main effects: the external electric field exerts a normal pressure on the liquid thread surface, suppressing satellite droplet formation, while the internal electric field inhibits liquid drainage. Surface viscosity further stabilizes the system by suppressing flow dynamics during this process.

Slip effects on solid boundaries are common in complex fluids. Boundary depletion layers in polymer solutions can create apparent slip effects, which can in turn significantly impact the dynamics of moving bodies. Motivated by microswimmer locomotion in such environments, we derive a series of slip slender-body theories for filamentous bodies experiencing slip-like boundary conditions. Using Navier’s slip model, we derive three slip slender-body theories, linking the body’s velocity to the distribution of hydrodynamic forces. The models are shown to be consistent with each other and with existing numerical computations. As the slip length increases, we show that the drag parallel to the body decreases towards zero while the perpendicular drag remains finite, in a manner which we quantify. This reduction in drag ratio is shown to be inversely related to microswimmer mobility in two simple swimmer models. This increase could help rationalise empirically observed enhanced swimming in complex fluids.

In this paper, we numerically investigate the orbit dynamics of three-dimensional symmetric Janus drops in shear flow using an improved ternary-fluids phase field method, focusing on how drop deformation and initial orientation affect the orbit drift of two configurations of Janus drops: dumbbell-shaped and near-spherical. We find that the motion of dumbbell-shaped drops eventually evolves into tumbling, while near-spherical drops attain stable spinning. We attribute this bifurcation in orbit drift to contrasting deformation dynamics and shape-dependent hydrodynamics of the two configurations. Specifically, the drift bifurcation is closely related to the aspect ratio of Janus drops at equilibrium, giving rise to two distinct mechanisms: (1) coupling between outer interface deformation and the surrounding flow field; and (2) interplay between inner interface deformation and vortices enclosed within the drop. In addition, we observe that for the dumbbell-shaped Janus drops with different aspect ratios, their tumbling dynamics resembles ellipsoids in shear flow. Moreover, the trajectories of the dumbbell-shaped Janus drops during orbit drift collapse onto a universal curve, independent of their initial orientations, and significant deformation and inertia accelerate the orbit transition. To quantitatively evaluate the effect of drop deformation on the orbit drift of the dumbbell-shaped Janus drops, we propose an effective aspect ratio model based on the drop shapes at equilibrium and at the maximum elongation. By incorporating the effective aspect ratio into Jeffery’s theory for solid particles, we accurately predict the rotation period and angular velocity of Janus drops in the tumbling regime and during the orbit drift, especially for drops with linear deformation. Moreover, the orbit parameter $C$ is found to vary exponentially with time for drops with linear deformation, while the time variation of $C$ transits from one exponential function to another for drops with nonlinear deformation.

A fully resolved numerical study was performed to investigate interfacial heat and mass transfer enhanced by the fully developed Rayleigh–Bénard–Marangoni instability in a relatively deep domain. The instability was triggered by evaporative cooling modelled by a constant surface heat flux. The latter allowed for temperature-induced variations in surface tension giving rise to Marangoni forces reinforcing the Rayleigh instability. Simulations were performed at a fixed Rayleigh number (${\textit{Ra}}_h$) and a variety of Marangoni numbers (${\textit{Ma}}_h$). In each simulation, scalar transport equations for heat and mass concentration at various Schmidt numbers (${\textit{Sc}}=16{-}200$) were solved simultaneously. Due to the fixed (warm) temperature prescribed at the bottom of the computational domain, large buoyant plumes emerged quasi-periodically both at the top and bottom. With increasing Marangoni number a decrease in the average convection cell size at the surface was observed, with a simultaneous improvement in near-surface mixing. The presence of high aspect ratio rectangular convection cell footprints was found to be characteristic for Marangoni-dominated flows. Due to the promotion of interfacial mass transfer by Marangoni forces, the power in the scaling of the mass transfer velocity, $K_{\!L}\!\propto\! \textit{Sc}^{-n}$, was found to decrease from $n=0.50$ at ${\textit{Ma}}_h=0$ to $\approx 0.438$ at ${\textit{Ma}}_h=13.21\times 10^5$. Finally, the existence of a buoyancy-dominated and a Marangoni-dominated regime was investigated in the context of the interfacial heat and mass transfer scaling as a function of ${\textit{Ma}}_h+\varepsilon {\textit{Ra}}_h$, where $\varepsilon$ is a small number determined empirically.

As a step towards realising a skin-friction drag reduction technique that scales favourably with Reynolds number, the impact of a synthetic jet on a turbulent boundary layer was explored through a study combining wind-tunnel measurements and large eddy simulations. The jet was ejected in the wall-normal direction through a rectangular slot whose spanwise dimension matched that of dominant large-scale structures in the logarithmic region to target structures of that size and smaller simultaneously. Local skin-friction reduction was observed at both $x/\delta =2$ and $x/\delta =5$ downstream of the orifice centreline, where $\delta$ is the boundary-layer thickness. At $x/\delta =2$, the skin-friction reduction was observed to be due to the synthetic-jet velocity deficit intersecting the wall. At $x/\delta =5$, evidence from the simulations and wind-tunnel measurements suggests that a weakening of wall-coherent velocity scales is primarily responsible for the skin-friction reduction. Local skin-friction reduction which scales favourably with Reynolds number may be achievable with the synthetic jet employed in this study. However, there are many technical hurdles to overcome to achieve net skin-friction drag reduction over the entire region of influence. For instance, regions of skin-friction increase were observed close to the orifice ($x/\delta \lt 2$) and downstream of the orifice edge due to the induced motion of synthetic-jet vortical structures. Additionally, a recirculation region was seen to form during expulsion, which has implications for pressure drag on non-planar surfaces.