Thin Fluid Layer Research Articles

Asymmetric vertical transport in weakly forced shallow flows

In this paper, we report on an investigation of the vertical transport of tracer particles released within a shallow, continuously-forced flow by means of numerical simulations. The investigation is motivated by the shallow flows encountered in many environmental situations and inspired by the laboratory experiments conducted in electromagnetically forced shallow fluid layers. The flow is confined to a thin fluid layer by stress-free top and no-slip bottom walls. The dynamics and the transport properties of the shallow flow are investigated under various flow conditions characterized by a Reynolds number related to the forcing, ReF, and the aspect ratio of vertical and horizontal length scales δ. The forcing generates an array of vortices that becomes unsteady when ReFδ2≳10. These vortices are accompanied by upwellings in their cores which are surrounded by narrower, stronger downwellings. Hence, upwellings occur where the horizontal flow is vorticity-dominated, while downwellings where it is strain-dominated. The magnitude of the asymmetry in strength and size of the vertical flows and their correlation with horizontal structures depends on the flow conditions and significantly influences the vertical spreading of particles within the fluid volume. Under conditions leading to a large asymmetry, particles within updrafts are transported slowly upwards, while particles within downdrafts rapidly move downwards. In addition, particles are trapped for longer within the updrafts than downdrafts because of their correlation with vorticity-dominated regions. However, when the flow becomes fully three-dimensional and highly unsteady for large ReFδ2 values, this transport asymmetry subsides because the updrafts and downdrafts exhibit similar strength and size in such flow conditions. Consequently, similar amounts of particles are transported upwards and downwards at similar rates.

Curvature-driven transport of thin Bingham fluid layers in airway bifurcations

Curvature-driven transport of thin Bingham fluid layers in airway bifurcations

Vortex lattice state transitions in water surface waves

Surface waves can generate an ordered vortex lattice of fluid particles on the air-water interface. Here, the transitions of the lattice structure are observed through experiments and modelled analytically. A fully-immersed square waveguide system is positioned at the center of a larger bath which directs particles into counter-rotating vortices. Flow interactions with the bath free-surface alter the ordered structure, and a state transition is initiated by the increase of the waveguide immersion depth. It manifests as a change in the orbital momentum and spin properties of the vortices, and develops through the combination of reflected-wave attenuation and mean flow effects from the classical Stokes drift and Eulerian currents. Transitions from a perfect antiferromagnetic state to weakly-ordered ones are achieved, including a monopole vortex reminiscent of spectral condensates in thin fluid layers. Order in the disordered flow is uncovered through velocity field corrections after invoking time-scale separation. The lattice structures are modelled analytically through a superposition of mean flow contributions from the waveguide and bath modes. The results open a new platform for investigating the dynamics of wave and vortex lattice interactions, and devising new techniques to characterize complex flow phenomena on the air-water interface.

Long-wave instabilities of a power-law fluid flowing over a heated, uneven and porous incline: A two-sided model

The stability conditions of a two-dimensional gravity-driven flow of a thin layer of a power-law fluid flowing over a heated, uneven, inclined porous surface are investigated. A two-sided model is employed to account for the bottom filtration in the porous layer. The governing equations are reduced under the long-wave approximation and the cross-stream dependence is eliminated by means of the Integral Boundary Layer technique. Floquet–Bloch theory is used to investigate at linear level how the porous bottom waviness influences the thermocapillarity stability of the flow in a shear-thinning fluid. Differently from the even case, the linear stability analysis suggests that for flow over sufficiently wavy undulations the thermocapillarity may stabilize the equilibrium flow, depending on the values of dimensionless governing numbers and parameters. This stabilizing phenomenon is enhanced by the shear-thinning rheology of the fluid while it is reduced by the permeability of the layer. Numerical simulations, performed solving the reduced nonlinear model through a second order Finite Volume scheme, confirm the results of the linear stability analysis.

Scraping of a thin layer of viscoplastic fluid

We analyze the scraping of a thin layer of viscoplastic fluid residing on a horizontal surface by a translating rigid scraper. This motion generates a mound of fluid upstream of the scraper and a residual layer behind it, both of which are modelled using a shallow layer theory for a Bingham or Herschel-Bulkley fluid. The flows ahead of and behind the scraper are coupled by the motion in the gap under the scraper, which is driven both by the translation of the scraper and by the induced pressure gradient due to the difference in flow thickness upstream and downstream. When the gap between the scraper and the underlying surface is sufficiently small, we find that a steady state emerges after a relatively long transient and that en route to this state, the unsteady dynamics exhibit a variety of regimes that are self-similar to leading order. We construct these solutions explicitly and derive key scalings for the temporal development of the flowing viscoplastic layer, as well as identifying the timescales at which there are transitions between the regimes. These predictions are confirmed by comparison with results from the numerical integration of the full system. Finally, we report results from preliminary laboratory experiments, which are compared with predictions from the shallow-layer theory, obtaining reasonable agreement once a slip boundary condition is included in the model, as motivated by experimental observations. Published by the American Physical Society 2024

Depth-integrated models for three-dimensional flow over topography

Considered in this investigation is the three-dimensional, gravity-driven flow of a thin viscous fluid layer down an incline, and spreading over topography. Three depth-integrated models are presented and contrasted. These include an integral-boundary-layer model, a weighted-residual model and a hybrid model. A numerical solution procedure suited for solving three-dimensional flows is also proposed. Numerous simulations have been conducted using the models for various steady subcritical, and unsteady supercritical flows over several topographies. Good agreement among the three models was found. In addition, the models were also validated using experimental results, and, again, good agreement between the three models and with experiments was obtained.

Fiber reorientation due to obstacles in open channel flow

A model along with experimental validation is presented for fiber re-orientation within the flow of a thin fluid layer down an inclined plane. Unlike the usual approach (Jeffery), the model accommodates the fast-changing velocity gradient defying linearization and the flow constraints due to the thin thickness of the fluid film. Model equations are formulated, and the algorithm for the fiber rotation is developed. The velocity field within the channel is obtained, using COMSOLTM multiphase simulation. The formulated model equations are used to track the orientation of a single fiber. The spatial fiber orientation state is described in terms of the second order tensor by tracking multiple individual fibers with different initial orientations. To validate the model's predictions, an experimental setup was fabricated to record individual fiber kinematics during the flow and to describe the orientation of fibers in the open channel. A novel benchmark for assessing fiber re-orientation models is presented, wherein the dynamics of fiber orientation is observed in response to a semicircular obstacle within the fluid flow. The experimental results are compared to model predictions and are in reasonable agreement. Despite the simplified approach, both the dynamic of single fiber and the development of fiber orientation distribution due to the obstacle can be reasonably predicted.

A novel method to study time fractional coupled systems of shallow water equations arising in ocean engineering

<abstract><p>This study investigates solutions for the time-fractional coupled system of the shallow-water equations. The shallow-water equations are employed for the purpose of elucidating the dynamics of water motion in oceanic or sea environments. Also, the aforementioned system characterizes a thin fluid layer that maintains a hydrostatic equilibrium while exhibiting uniform density. Shallow water flows have a vertical dimension that is considerably smaller in magnitude than the typical horizontal dimension. In the current work, we employ an innovative and effective technique, known as the natural transform decomposition method, to obtain the solutions for these fractional systems. The present methodology entails the utilization of both singular and non-singular kernels for the purpose of handling fractional derivatives. The Banach fixed point theorem is employed to demonstrate the uniqueness and convergence of the obtained solution. The outcomes obtained from the application of the suggested methodology are compared to the exact solution and the results of other numerical methods found in the literature, including the modified homotopy analysis transform method, the residual power series method and the new iterative method. The results obtained from the proposed methodology are presented through the use of tabular and graphical simulations. The current framework effectively captures the behavior exhibited by different fractional orders. The findings illustrate the efficacy of the proposed method.</p></abstract>

Interaction between forced and natural convection in a thin cylindrical fluid layer at low Prandtl number

Motivated by nuclear safety issues, we study the heat transfers in a thin cylindrical fluid layer with imposed fluxes at the bottom and top surfaces (not necessarily equal) and a fixed temperature on the sides. We combine direct numerical simulations and a theoretical approach to derive scaling laws for the mean temperature and for the temperature difference between the top and bottom of the system. We find two asymptotic scaling laws depending on the flux ratio between the upper and lower boundaries. The first one is controlled by heat transfer to the side, for which we recover scaling laws characteristic of natural convection (Batchelor, Q. Appl. Maths, vol. 12, 1954, pp. 209–233). The second one is driven by vertical heat transfers analogous to Rayleigh–Bénard convection (Grossmann & Lohse, J. Fluid Mech., vol. 407, 2000, pp. 27–56). We show that the system is inherently inhomogeneous, and that the heat transfer results from a superposition of both asymptotic regimes. Keeping in mind nuclear safety models, we also derive a one-dimensional model of the radial temperature profile based on a detailed analysis of the flow structure, hence providing a way to relate this profile to the imposed boundary conditions.

Exact Solutions of the Oberbeck–Boussinesq Equations for the Description of Shear Thermal Diffusion of Newtonian Fluid Flows

We present a new exact solution of the thermal diffusion equations for steady-state shear flows of a binary fluid. Shear fluid flows are used in modeling and simulating large-scale currents of the world ocean, motions in thin layers of fluid, fluid flows in processes, and apparatuses of chemical technology. To describe the steady shear flows of an incompressible fluid, the system of Navier–Stokes equations in the Boussinesq approximation is redefined, so the construction of exact and numerical solutions to the equations of hydrodynamics is a very difficult and urgent task. A non-trivial exact solution is constructed in the Lin-Sidorov-Aristov class. For this class of exact solutions, the hydrodynamic fields (velocity field, pressure field, temperature field, and solute concentration field) were considered as linear forms in the x and y coordinates. The coefficients of linear forms depend on the third coordinate z. Thus, when considering a shear flow, the two-dimensional velocity field depends on three coordinates. It is worth noting that the solvability condition given in the article imposes a condition (relation) only between the velocity gradients. A theorem on the uniqueness of the exact solution in the Lin–Sidorov–Aristov class is formulated. The remaining coefficients of linear forms for hydrodynamic fields have functional arbitrariness. To illustrate the exact solution of the overdetermined system of Oberbeck–Boussinesq equations, a boundary value problem was solved to describe the complex convection of a vertical swirling fluid without its preliminary rotation. It was shown that the velocity field is highly stratified. Complex countercurrents are recorded in the fluid.

From lab to technical scale: Ultraviolet C treatment for the reduction of Lactococcus lactis bacteriophage P008 in whey

UV–C irradiation is a powerful non-thermal principle to inactivate lactococcal bacteriophages, which pose a high risk for fermentation failures in dairies. However, milk products limit ray penetration with their turbidity and absorption properties. In reactor design, attention must be paid to uniform and effective dose distribution, which can be achieved by thin fluid layers and/or turbulence. Preliminary tests performed in a lab-scale batch mode irradiation chamber showed filtered whey to be the most suitable application site. In a subsequent larger-scale feasibility study, two continuous flow-through reactors (lab-scale and technical-scale) were investigated. A bacteriophage reduction of 5 log units required a volume-related UV–C dose of 10.6 J mL−1 in the irradiation chamber, but only 1.8 J mL−1 in the continuous lab-scale reactor and 2.4 J mL−1 in the continuous technical-scale reactor. These data show that upscaling is feasible and that an effective bacteriophage inactivation with gentle product treatment is allowed.

Instability evolution of a shock-accelerated thin heavy fluid layer in cylindrical geometry

Instability evolutions of shock-accelerated thin cylindrical SF $_6$ layers surrounded by air with initial perturbations imposed only at the outer interface (i.e. the ‘Outer’ case) or at the inner interface (i.e. the ‘Inner’ case) are numerically and theoretically investigated. It is found that the instability evolution of a thin cylindrical heavy fluid layer not only involves the effects of Richtmyer–Meshkov instability, Rayleigh–Taylor stability/instability and compressibility coupled with the Bell–Plesset effect, which determine the instability evolution of the single cylindrical interface, but also strongly depends on the waves reverberated inside the layer, thin-shell correction and interface coupling effect. Specifically, the rarefaction wave inside the thin fluid layer accelerates the outer interface inward and induces the decompression effect for both the Outer and Inner cases, and the compression wave inside the fluid layer accelerates the inner interface inward and causes the decompression effect for the Outer case and compression effect for the Inner case. It is noted that the compressible Bell model excluding the compression/decompression effect of waves, thin-shell correction and interface coupling effect deviates significantly from the perturbation growth. To this end, an improved compressible Bell model is proposed, including three new terms to quantify the compression/decompression effect of waves, thin-shell correction and interface coupling effect, respectively. This improved model is verified by numerical results and successfully characterizes various effects that contribute to the perturbation growth of a shock-accelerated thin heavy fluid layer.

Seeing the invisible: convection cells revealed with thermal imaging

Fluid instabilities are ubiquitous phenomena of great theoretical and applied importance. In particular, an intriguing example is the thermocapillary or Bénard-Marangoni instability which occurs when a thin horizontal fluid layer, whose top surface is free, is heated from below. In this phenomenon, after passing a certain temperature difference threshold, the fluid develops a regular pattern, usually hexagonal, of convection cells known as Bénard convection. In general, this pattern is not visible to the naked eye unless specific tracers are incorporated into the fluid. The use of thermal imaging is a simple alternative not only for directly observing this phenomenon but also for obtaining valuable quantitative information, such as the relationship between the critical wavelength and the thickness of the fluid layer. Here, we propose an experiment specially suited for laboratory courses in fluid mechanics or nonlinear physics that involves the use of thermal cameras, or appropriate smartphone accessories, to study Bénard convection.

Hydrodynamics of an odd active surfer in a chiral fluid

We theoretically and computationally study the low-Reynolds-number hydrodynamics of a linear active microswimmer surfing on a compressible thin fluid layer characterized by an odd viscosity. Since the underlying three-dimensional fluid is assumed to be very thin compared to any lateral size of the fluid layer, the model is effectively two-dimensional. In the limit of small odd viscosity compared to the even viscosities of the fluid layer, we obtain analytical expressions for the self-induced flow field, which includes non-reciprocal components due to the odd viscosity. On this basis, we fully analyze the behavior of a single linear swimmer, finding that it follows a circular path, the radius of which is, to leading order, inversely proportional to the magnitude of the odd viscosity. In addition, we show that a pair of swimmers exhibits a wealth of two-body dynamics that depends on the initial relative orientation angles as well as on the propulsion mechanism adopted by each swimmer. In particular, the pusher–pusher and pusher–puller-type swimmer pairs exhibit a generic spiral motion, while the puller–puller pair is found to either co-rotate in the steady state along a circular trajectory or exhibit a more complex chaotic behavior resulting from the interplay between hydrodynamic and steric interactions. Our theoretical predictions may pave the way toward a better understanding of active transport in active chiral fluids with odd viscosity, and may find potential applications in the quantitative microrheological characterization of odd-viscous fluids.

A new thin layer model for viscous flow between two nearby non‐static surfaces

AbstractWe propose a two‐dimensional flow model of a viscous fluid between two close moving surfaces. We show, using a formal asymptotic expansion of the solution, that its asymptotic behavior, when the distance between the two surfaces tends to zero, is the same as that of the the Navier‐Stokes equations.The leading term of the formal asymptotic expansions of the solutions to the new model and Navier‐Stokes equations are solution of the same limit problem, and the type of the limit problem depends on the boundary conditions. If slip velocity boundary conditions are imposed on the upper and lower bound surfaces, the limit is a solution of a lubrication model, but if the tractions and friction forces are known on both bound surfaces, the limit is a solution of a thin fluid layer model.The model proposed has been obtained to be a valuable tool for computing viscous fluid flow between two nearby moving surfaces, without the need to decide a priori whether the flow is typical of a lubrication or a thin fluid layer problem, and without the enormous computational effort that would be required to solve the Navier‐Stokes equations in such a thin domain.

A new IBL model for quasi-unidirectional gravity-driven flow over topography

Discussed in this investigation is the three-dimensional gravity-driven flow of a thin fluid layer down an incline and over topography. A new three-dimensional integral-boundary-layer (IBL) model is proposed to describe the flow. Numerical simulations are presented and comparisons with lubrication and two-dimensional IBL models are discussed for both steady and unsteady flows. Subcritical and supercritical cases are considered, along with symmetrical and asymmetrical bottom topographies. Also, good agreement was found with experiments.

Evolution of an elastic blister in the presence of sloping topography

The propagation of a finite blister of fluid beneath an elastic sheet is controlled by the local dynamics at the peeling periphery of the current. Previous works have described constant volume elastic blisters by considering the peeling due to curvature at these edges. Here, we show that an along-slope component of gravity fundamentally changes the dynamics by removing the role of curvature at the trailing, upslope edge. The local dynamics of this trailing edge is instead controlled by shear stress in the sheet, as in the elastic Landau–Levich problem, and thereby allows for a receding edge, in contrast to propagation by peeling for which only an advancing contact line is possible. Using an asymptotic analysis, we show that this receding edge condition allows for a new, nearly translating regime in which the body of the blister moves at an approximately constant speed, leaving behind a thin layer of fluid. This prediction is verified by detailed numerical modelling of the two-dimensional downslope spreading. We conclude by discussing the applicability of these results in the rapid spreading of subglacial meltwater.

Why Sonochemistry in a Thin Layer? Constructive Interference.

Sonochemistry in a thin fluid layer has advantages of no visible cavitation, no turbulence, negligible temperature changes (≲1 °C), low power transducers, and transmissibility (sound pressure amplification) of ≳106. Unlike sonochemistry in semi-infinite fluids, resonance and so constructive interference of sound pressure can be established in thin layers. Constructive interference enables substantial amplification of sound pressure at solid fluid interfaces. Fluid properties of sound velocity and attenuation, oscillator input frequency, and thin fluid layer thickness couple to established resonance in underdamped conditions. In thin layer sonochemistry (TLS), thin layers are established where ultrasonic wavelength and oscillator-interface separation are comparable, about a centimeter in water. Solution of a one dimensional wave equation identifies explicit relationships between the system parameters required to establish resonance and constructive interference in a thin layer.

An OpenFOAM framework to model thermal bubble-driven micro-pumps

Thermal bubble-driven micro-pumps (also known as inertial pumps) are an upcoming micro-pump technology that can be integrated directly into micro/mesofluidic channels to displace fluid without moving parts. These micro-pumps are high-power resistors that locally vaporize a thin layer of fluid above the resistor surface to form a high-pressure vapor bubble which performs mechanical work. Despite their geometric simplicity, thermal bubble-driven micro-pumps are complex to model due to the multiphysics couplings of Joule heating, thermal bubble nucleation, phase change, and multiphase flow. As such, most simulation approaches simplify the physics by neglecting Joule heating, nucleation, and phase change effects as done in this study. To date, there are no readily available, reduced physics open-source modeling tools that can resolve both pre-collapse (defined as when the bubble is expanding and collapsing) and post-collapse (defined as when the bubble has re-dissolved back into the subcooled fluid) bubble and flow dynamics. In this study, an OpenFOAM framework for modeling thermal bubble-driven micro-pumps is presented, validated, and applied. The developed OpenFOAM model agrees with both experimental data and commercial computational fluid dynamics (CFD) software, FLOW-3D. Additionally, we assess the shape of the transient velocity profile during a pump cycle for the first time and find that it varies substantially from theoretical Poiseuille flow during pre-collapse but is within 25% of the theoretical flow profile during post-collapse. We find that this deviation is due to flow never becoming fully developed during each pump cycle. We envision the developed OpenFOAM framework as an open-source CFD toolkit for microfluidic designers to simulate devices with thermal bubble-driven micro-pumps.

A gauge theory for shallow water

The shallow water equations describe the horizontal flow of a thin layer of fluid with varying height. We show that the equations can be rewritten as a d=2+1d=2+1 dimensional Abelian gauge theory. The magnetic field corresponds to the conserved height of the fluid, while the electric charge corresponds to the conserved vorticity. In a certain linearised approximation, the shallow water equations reduce to relativistic Maxwell-Chern-Simons theory. This describes Poincaré waves. The chiral edge modes of the theory are identified as coastal Kelvin waves.