In this paper, we present an analytic solution for pulse wave propagation in a flexible arterial model with tapering, physiological boundary conditions and variable wall properties (wall elasticity and thickness). The change of wall properties follows a profile that is proportional to $r^\alpha$, where $r$ represents the lumen radius and $\alpha$ is a material coefficient. The cross-sectionally averaged velocity and pressure are obtained by solving a hyperbolic system derived from the mass and momentum conservations, and they are expressed in Bessel functions of order $(4-\alpha )/(3-\alpha )$ and $1/(3-\alpha )$, respectively. The solution is successfully validated by comparing it with numerical results from three-dimensional (3-D) fluid–structure interaction simulations. Subsequently, the solution is employed to study pulse wave propagation in an arterial model, revealing that the wall properties and the physiological outlet boundary conditions, such as the resistor–capacitor–resistor (RCR) model, play a crucial role in characterizing the input impedance and reflection coefficient. At low-frequency range, the input impedance is found to be insensitive to the wall properties and is primarily determined by the RCR parameters. At high-frequency range, the input impedance oscillates around the local characteristic impedance, and the oscillation amplitude varies non-monotonically with $\alpha$. Expressions for the input impedance at both low-frequency and high-frequency limits are presented. This analytic solution is also successfully applied to model flow inside a patient-specific arterial tree, with the maximum relative errors in pressure and flow rate never exceeding $1.6\,\%$ and $9.0\,\%$ when compared with results from 3-D numerical simulations.

We present a deep probabilistic convolutional neural network (PCNN) model for predicting local values of small-scale mixing properties in stratified turbulent flows, namely the dissipation rates of turbulent kinetic energy and density variance, $\varepsilon$ and $\chi$. Inputs to the PCNN are vertical columns of velocity and density gradients, motivated by data typically available from microstructure profilers in the ocean. The architecture is designed to enable the model to capture several characteristic features of stratified turbulence, in particular the dependence of small-scale isotropy on the buoyancy Reynolds number $Re_b:=\varepsilon /(\nu N^2)$, where $\nu$ is the kinematic viscosity and $N$ is the background buoyancy frequency, the correlation between suitably locally averaged density gradients and turbulence intensity and the importance of capturing the tails of the probability distribution functions of values of dissipation. Empirically modified versions of commonly used isotropic models for $\varepsilon$ and $\chi$ that depend only on vertical derivatives of density and velocity are proposed based on the asymptotic regimes $Re_b\ll 1$ and $Re_b\gg 1$, and serve as an instructive benchmark for comparison with the data-driven approach. When trained and tested on a simulation of stratified decaying turbulence which accesses a range of turbulent regimes (associated with differing values of $Re_b$), the PCNN outperforms assumptions of isotropy significantly as $Re_b$ decreases, and additionally demonstrates improvements over the fitted empirical models. A differential sensitivity analysis of the PCNN facilitates a comparison with the theoretical models and provides a physical interpretation of the features enabling it to make improved predictions.

We study the large-scale dynamics and prediction of hydrodynamic transport in random fracture networks. The flow and transport behaviour is characterized by first passage times and displacement statistics, which show heavy tails and anomalous dispersion with a strong dependence on the injection condition. The origin of these behaviours is investigated in terms of Lagrangian velocities sampled equidistantly along particle trajectories, unlike classical sampling strategies at a constant rate. The velocity series are analysed by their copula density, the joint distribution of the velocity unit scores, which reveals a simple, albeit hidden, correlation structure that can be described by a Gaussian copula. Based on this insight, we derive a Langevin equation for the evolution of equidistant particle speeds. In this framework, particle motion is quantified by a stochastic time-domain random walk, the joint density of particle position, and speed satisfies a Klein–Kramers equation. The upscaled theory quantifies particle motion in terms of the characteristic fracture length scale and the distribution of Eulerian flow velocities. That is, it is predictive in the sense that it does not require the a priori knowledge of transport attributes. The upscaled model captures non-Fickian transport features, and their dependence on the injection conditions in terms of the velocity point statistics and average fracture length. It shows that the first passage times and displacement moments are dominated by extremes occurring at the first step. The presented approach integrates the interaction of flow and structure into a predictive model for large-scale transport in random fracture networks.

The strong coupling interactions of non-equilibrium flow, microscopic particle collisions and radiative transitions within the shock layer of hypersonic atmospheric re-entry vehicles makes accurate prediction of the aerothermodynamics challenging. Therefore, in this study a self-consistent non-equilibrium flow, collisional–radiative reactions and radiative transfer fully coupled model are established to study the non-equilibrium characteristics of the flow field and radiation of vehicle atmospheric re-entry. The comparison of the present calculation results with flight data of FIRE II and previous results in the literature shows a reasonable agreement. The thermal, chemical and excited energy level non-equilibrium phenomena are obtained and analysed for the different FIRE II trajectory points, which form the critical basis for studying the heat transfer and radiation. The non-equilibrium distribution of excited energy levels significantly exists in the post-shock and near-wall regions due to the rapid vibrational dissociation and electronic under-excitation, as well as the wall catalytic reactions. The analysis of stagnation-point heating of FIRE II illustrates that the translational–rotational convection and the dissociation component diffusion play key roles in the aerodynamic heating of the wall region. The spectrally resolved radiative intensity in the entire flow field indicates that the vacuum ultraviolet radiation caused by the high-energy nitrogen atomic spectral lines makes the main contribution to the radiative transfer. Finally, it is found that the non-equilibrium flow–radiation coupling effect can exacerbate the excited energy level non-equilibrium, and further affect the gas radiative properties and radiative transfer. This fully coupled study provides an effective method for reasonable prediction of atmospheric re-entry flow and radiation fields.

In this work, the relationship between the velocity of an elongated bubble and its shape is investigated, in the case where the elongated bubble flows in a viscous liquid initially at rest in a pipe. The velocity, expressed as a Froude number, depends on the angle of the inclined pipe, the Eötvös number and the buoyancy Reynolds number. The diameter of the pipe and the surface tension being fixed, the Eötvös number remains constant; this study focuses on the dependence of the velocity on the pipe inclination angle and the viscosity of the liquid. The velocity of the elongated bubble was measured for different angles between 0 and 15 degrees and for liquid viscosities 10 to 200 times that of water. As the velocity of elongated bubbles depends closely on their shape, shadowgraphy coupled with particle image velocimetry was used. The results show that the velocity of the elongated bubbles is highly sensitive to the inclination angle of the pipe and to the viscosity of the liquid, particularly for low pipe inclinations and large viscosities. In the layer of liquid located downstream of the elongated bubble, laminar flow develops rapidly in the liquid, resulting from a balance between gravity and friction at the wall. The identification of the position of the stagnation point close to the nose of the elongated bubble and the curvature of the interface at this point helps to explain why the velocity of the elongated bubble decreases for low angles and high viscosities.

To explore the impact of surface viscosity on coexisting fluid domains in biomembranes we consider two-phase fluid deformable surfaces as model systems for biomembranes. Such surfaces are modelled by incompressible surface Navier–Stokes–Cahn–Hilliard-like equations with bending forces. We derive this model using the Lagrange–d’Alembert principle considering various dissipation mechanisms. The highly nonlinear model is solved numerically to explore the tight interplay between surface evolution, surface phase composition, surface curvature and surface hydrodynamics. It is demonstrated that hydrodynamics can enhance bulging and furrow formation, which both can further develop to pinch-offs. The numerical approach builds on a Taylor–Hood element for the surface Navier–Stokes part, a semi-implicit approach for the Cahn–Hilliard part, higher-order surface parametrizations, appropriate approximations of the geometric quantities, and mesh redistribution. We demonstrate convergence properties that are known to be optimal for simplified subproblems.

The effect of magnetic as well as electromagnetic fields on the stability of an electrically conducting viscous liquid film flowing down an inclined plane has been investigated for the full range of inclination angles $\theta$ ($0 < \theta \le 90^{\circ }$) in association with a given value of the Reynolds number $Re$ ($0 < Re \le 100$), and vice versa. A nonlinear evolution equation is derived by using the momentum-integral method, which is valid for both small and large values of $Re$. Use of the normal mode approach on the linearized surface evolution equation gives the stability criterion and the critical value of the wavenumber $k_c$ (for which the imaginary part of the complex frequency $\omega _i^+$ is zero) which conceive the electric parameter $E$, magnetic parameter $M$, Reynolds number $Re$, Weber number $We$ and inclination angle $\theta$. The nonlinear stability analysis based on the second Landau constant $J_2$ helps to demarcate all four possible distinct flow zones (explosive, supercritical, unconditional and subcritical) of this problem. A novel result of this analysis is a simple relationship between the critical values of $k_c$ and $k_j$ (for which $J_2$ is zero) that basically gives the necessary conditions for the existence of the range of $k$ for an explosive unstable zone, which is either one or two accordingly as $k_j >k_c$ or $k_j< k_c$, and the non-existence of an unconditional stable zone is $k_j \le k_c$ depending upon the values of $M$. The analysis confirms the existence of two critical values of $M$, namely, $M_c$ (for which $k_c$ is zero) and $M_j$ (for which $k_j$ is zero). Here, $M_j > M_c$ except for $\theta = 90^{\circ }$; and we have found the existence of all four or two (unconditional and subcritical) or one (subcritical) zone(s) of this flow problem accordingly, as $0 \le M < M_c$ or $M_c \le M < M_j$ and $M > M_j$ or $M = M_j$.

We derive reduced models for extrusion problems where it is necessary to account for fluid compressibility. We consider the two-dimensional extensional flow of a compressible viscous fluid and discuss two specific applications: weakly compressible fluids and bubbly liquid–gas mixtures that behave as a single compressible fluid. The mathematical model we present consists of equations for conservation of mass, conservation of momentum and a closure relationship between the pressure and density. The most substantial differences between compressible extrusion problems is in the closure relationship. By integrating the conservation equations across the fluid cross-section and exploiting a slender aspect ratio, we derive reduced equations for conservation of mass and conservation of momentum in the direction of flow. The reduced system of equations relating cross-sectionally averaged quantities is closed by a relationship between the averaged pressure and density, which will differ substantially depending on the application. We demonstrate the utility of a reduced model for both the weakly compressible fluid and bubbly mixture applications; namely, in providing valuable quantitative insights without needing to solve a complicated free-boundary problem.

Applying a focused ultrasonic field on a free liquid surface results in its growth eventually leading to the so-called acoustic fountain. In this work, a numerical approach is presented to further increase the understanding of the acoustic fountain phenomenon. The developed simulation method enables the prediction of the free surface motion and the dynamic acoustic field in the moving liquid. The dynamic system is a balance between inertia, surface tension and the acoustic radiation force, and its nonlinearity is demonstrated by studying the relation between the ultrasonic excitation amplitude and corresponding liquid deformation. We show that dynamic resonance is the main mechanism causing the specific acoustic fountain shapes, and the analysis of the dynamic acoustic pressure allows us to predict Faraday-instability atomisation. We show that strong resonance peaks cause atomisation bursts and strong transient deformations corresponding to previously reported experimental observations. The quantitative prediction of the dynamic acoustic pressure enables us to assess the potential of cavitation generation in acoustic fountains. The observed local high acoustic pressures above both the cavitation and the atomisation threshold hint at the coexistence of these two phenomena in acoustic fountains.

Existing theoretical analyses of Faraday waves in Hele-Shaw cells rely on the Darcy approximation and assume a parabolic flow profile in the narrow direction. However, Darcy's model is known to be inaccurate when convective or unsteady inertial effects are important. In this work, we propose a gap-averaged Floquet theory accounting for inertial effects induced by the unsteady terms in the Navier–Stokes equations, a scenario that corresponds to a pulsatile flow where the fluid motion reduces to a two-dimensional oscillating Poiseuille flow, similarly to the Womersley flow in arteries. When gap-averaging the linearised Navier–Stokes equation, this results in a modified damping coefficient, which is a function of the ratio between the Stokes boundary layer thickness and the cell's gap, and whose complex value depends on the frequency of the wave response specific to each unstable parametric region. We first revisit the standard case of horizontally infinite rectangular Hele-Shaw cells by also accounting for a dynamic contact angle model. A comparison with existing experiments shows the predictive improvement brought by the present theory and points out how the standard gap-averaged model often underestimates the Faraday threshold. The analysis is then extended to the less conventional case of thin annuli. A series of dedicated experiments for this configuration highlights how Darcy's thin-gap approximation overlooks a frequency detuning that is essential to correctly predict the locations of the Faraday tongues in the frequency–amplitude parameter plane. These findings are well rationalised and captured by the present model.

The inverse problem of steady two-dimensional open channel free-surface flow is considered, with the focus on determining two types of disturbances: a surface pressure distribution and solid channel bottom topography. A closed-form expression for the inverse surface pressure is derived, and a linear Fredholm equation of the first kind is shown to describe the inverse topography problem, which then needs to be descretised and solved numerically. However, the equation for the channel bottom is prone to instability, so the truncated singular value decomposition (TSVD) method is proposed as a way to stabilise the associated discrete solution. The effectiveness of the TSVD method is demonstrated through several numerical examples, and its performance in the presence of error-contaminated input data is also examined. The results show that the TSVD method can recover the topography accurately from the forward free-surface problem, and provide good approximations even with noisy input data.

In this work, smoothed particle hydrodynamics (SPH) is employed to investigate the segregation evolution in granular flows. We first provide the Lagrangian description-based governing equations, including the linear momentum conservation and the segregation–diffusion equation. Then the hybrid continuum surface reaction scheme is introduced to formulate the concentration-related inhomogeneous Neumann boundary condition on the free and wall surfaces. We follow a two-stage strategy to advance boundary particle searching and normal direction identification. Moreover, $C^1$ consistency is considered based on the Taylor series to obtain accurate segregation flux gradient along the boundary. Our SPH model is validated with a shear box experiment. The model is then applied to investigate the segregation mechanism in bidisperse-sized granular flows in a rotating drum.

The spectral behaviour of random sawtooth waves propagating in the inner surf zone is investigated in this study. We show that the elevation energy spectrum exhibits a universal shape with an $\omega ^{-2}$ tendency in the inertial subrange and an exponential decay in the diffusive subrange ($\omega$ being the angular frequency). A theoretical spectrum is derived based on the similarities between sawtooth waves in the inner surf zone and Burgers wave solutions. Very good agreement is shown between this theoretical spectrum and laboratory experiments covering a large range of incident random wave conditions. Additionally, an equation describing the universal shape of the dissipation spectrum is derived. It highlights that the dissipation spectrum is nearly constant in the inertial subrange, consistent with prior laboratory observations. The findings presented in this study can be useful to improve broken wave dissipation parametrizations in stochastic spectral wave models.

We exploit the similarity between the mean momentum equation and the mean energy equation and derive transformations for mean temperature profiles in compressible wall-bounded flows. In contrast to prior studies that rely on the strong Reynolds analogy and the presumed similarity between the instantaneous and mean velocity and temperature signals, the discussion in this paper involves the Farve-averaged equations only. We establish that the compressible momentum and energy equations can be made identical to their incompressible counterparts under appropriate normalizations and coordinate transformations. Two types of transformations are explored for illustration purposes: Van Driest (VD)-type transformations and semi-local-type or Trettel–Larsson (TL)-type transformations. In our derivations, it becomes clear that VD-type velocity and temperature transformations hold exclusively within the logarithmic layer. On the other hand, TL-type transformations extend their applicability to incorporate wall-damping effects, at least in principle. Each type of transformation serves its distinct purpose and has its applicable range. However, it is noteworthy that while VD-type transformations can be assessed using measurements obtained from laboratory experiments, TL-type transformations necessitate viscosity and density information typically accessible only through numerical simulations. Finally, we justify the omission of the turbulent kinetic energy transfer term, a term that is unclosed, in the energy equation. This omission leads to closed-form temperature transformations that are valid for both adiabatic and isothermal walls.

A rigid object moving in a viscous fluid and in close proximity to an elastic wall experiences self-generated elastohydrodynamic interactions. This has been the subject of intense research activity, with recent and growing attention given to the particular case of elastomeric and gel-like substrates. Here, we address the situation where the elastic wall is replaced by a capillary surface. Specifically, we analyse the lubrication flow generated by the prescribed normal motion of a rigid infinite cylinder near the deformable interface separating two immiscible and incompressible viscous fluids. Using a combination of analytical and numerical treatments, we compute the emergent capillary-lubrication force at leading order in capillary compliance, and characterize its dependencies with the interfacial tension, viscosities of the fluids, and length scales of the problem. Interestingly, we identify two main contributions: (i) a velocity-dependent adhesive-like force; (ii) an acceleration-dependant inertia-like force. Our results may have implications for the mobility of colloids near complex interfaces and for the motility of confined microbiological entities.

In this study, mean velocity and temperature profiles for turbulent vertical convection (VC) confined in an infinite channel are investigated theoretically. The analysis starts from the governing equations of the thermal flow, with Reynolds shear stress and turbulent heat flux closed by the mixing length theory. Employing a three-sublayer description of the mean fields, the mean velocity and temperature profiles are found to be linear laws near the channel wall (viscosity-dominated sublayer), and they follow power laws close to the channel centre (turbulence-dominated sublayer). The characteristic scales of velocity, temperature and length in the present profiles arise naturally from the system normalisation, rather than from scaling analyses, thus ensuring a sound mathematical description. The derived profiles are verified fully via various literature data available in the classical regime; further, they are compared with the reported profiles, and the results indicate that the present profiles are the only ones with the ability to interpret data accurately from different sources, demonstrating much better versatility. Meanwhile, we provide analytical arguments showing that in the ultimate regime, the mean profiles in VC may remain in power laws, rather than the log laws inferred by analogy with Rayleigh–Bénard convection (RBC) systems. The power profiles recognised in this study are induced by the effect of buoyancy, which is in parallel with the mean flow in VC and contributes to the streamwise momentum transport, whereas in RBC systems, buoyancy is perpendicular to the mean flow, and does not influence the streamwise momentum transport, resulting in log profiles, being similar to the case of wall shear flows.

The unsteady wake interference of unequal-height tandem finite wall-mounted cylinders (FWMCs) fully submerged in a turbulent boundary layer (TBL) was investigated using time-resolved particle image velocimetry. The aspect ratios of the cylinders were fixed at $h/d = 5.3$ for the upstream cylinder (UC) and $H/d = 7.0$ for the downstream cylinder (DC) to achieve a height ratio of $h/H = 0.75$, where d is the diameter of the cylinders. The Reynolds number based on the cylinder diameter was $Re = 5540$ and the submergence ratio was $\delta /H = 1.2$, where $\delta $ is the TBL thickness. Three main flow regimes of tandem FWMCs were examined by varying the centre-to-centre spacing ($s$) between the cylinders: extended-body ($s/d = 2$), reattachment ($s/d = 4$) and co-shedding ($s/d = 6$) regimes. These test cases denoted as SR2, SR4 and SR6, respectively, were compared with a reference isolated cylinder (SC) with an aspect ratio similar to that of the DC. Spatio-temporal analysis of the flow field showed that the gap region of SR2 is characterized by a strong downwash of alternating low- and high-momentum fluid induced by the approach flow that is deflected from the unsheltered portion of the DC. In contrast, the gap region of SR4 and SR6 exhibited both downwash and upwash flow with a saddle point that moves closer to the mid-height of the UC as the spacing ratio increases. The upwash and downwash shear layers were associated with small-scale vortices with Strouhal numbers larger than that of the Kármán vortex shedding in the spanwise shear layers. The wake structure behind the DC was significantly altered compared with the SC due to sheltering effects, and the spacing ratio had a significant impact on the spatio-temporal evolution of the vortices.