Liquid Films Falling Down a Vertical Fiber: Modeling, Simulations and Experiments

We provide a new framework for analyzing the flow of an axisymmetric liquid film flowing down a vertical fiber, applicable to fiber coating flows and those in similar geometries in heat exchangers, water treatment, and desalination processes. The problem considered is that of a viscous liquid film falling under the influence of gravity and surface tension on a solid cylindrical fiber. Our approach is different from existing ones in that we derive our mathematical model by using a control-volume approach to express the conservation of mass and axial momentum in simple and intuitively appealing forms, resulting in a pair of equations that are reminiscent of the Saint-Venant shallow-water equations. Two versions of the model are obtained, one assuming a plug-flow velocity profile with a linear drag force expression, and the other using the fully-developed laminar velocity profile for a locally uniform film to approximate the drag. These can, respectively, model high- and low-Reynolds number regimes of flow. Linear stability analyses and fully nonlinear numerical simulations are presented that show the emergence of traveling wave solutions representing chains of identical droplets falling down the fiber. Physical experiments with safflower oil on a fishing line are also undertaken and match the theoretical predictions from the laminar flow model well when machine learning methods are used to estimate the parameters.

Simulation Tests of a Cow Milking Machine—Analysis of Design Parameters

The objective of this paper was to create a mathematical model of vacuum drops in a form that enables the testing of the impact of design parameters of a milking cluster on the values of vacuum drops in the claw. Simulation tests of the milking cluster were conducted, with the use of a simplified model of vacuum drops in the form of a fourth-degree polynomial. Sensitivity analysis and a simulation of a model with a simplified structure of vacuum drops in the claw were carried out. As a result, the impact of the milking machine’s design parameters on the milking process could be analysed. The results showed that a change in the local loss and linear drag coefficient in the long milk duct will have a lower impact on vacuum drops if a smaller flux of inlet air, a higher head of the air/liquid mix, and a higher diameter of the long milk tube are used.

Global dissipative solutions of the defocusing isothermal Euler–Langevin–Korteweg equations

We construct global dissipative solutions on the torus of dimension at most three of the defocusing isothermal Euler–Langevin–Korteweg system, which corresponds to the Euler–Korteweg system of compressible quantum fluids with an isothermal pressure law and a linear drag term with respect to the velocity. In particular, the isothermal feature prevents the energy and the BD-entropy from being positive. Adapting standard approximation arguments we first show the existence of global weak solutions to the defocusing isothermal Navier–Stokes–Langevin–Korteweg system. Introducing a relative entropy function satisfying a Gronwall-type inequality we then perform the inviscid limit to obtain the existence of dissipative solutions of the Euler–Langevin–Korteweg system.

Parameterizing non-propagating form drag over rough bathymetry

Slowly-evolving stratified flow over rough topography is subject to substantial drag due to internal motions, but often numerical simulations are carried out at resolutions where this “wave” drag must be parameterized. Here we highlight the importance of internal drag from topography with scales that cannot radiate internal waves, but may be highly non-linear, and we propose a simple parameterization of this drag that has a minimum of fit parameters compared to existing schemes. The parameterization smoothly transitions from a quadratic drag law () for low- (linear wave dynamics) to a linear drag law () for high- flows (non-linear blocking and hydraulic dynamics), where N is the stratification, h is the height of the topography, and u0 is the near-bottom velocity; the parameterization does not have a dependence on Coriolis frequency. Simulations carried out in a channel with synthetic bathymetry and steady body forcing indicate that this parameterization accurately predicts drag across a broad range of forcing parameters when the effect of reduced near-bottom mixing is taken into account by reducing the effective height of the topography. The parameterization is also tested in simulations of wind-driven channel flows that generate mesoscale eddy fields, a setup where the downstream transport is sensitive to the bottom drag parameterization and its effect on the eddies. In these simulations, the parameterization replicates the effect of rough bathymetry on the eddies. If extrapolated globally, the sub-inertial topographic scales can account for 2.7 TW of work done on the low-frequency circulation, an important sink that is redistributed to mixing in the open ocean.

Equatorial retrograde flow in WASP-43b elicited by deep wind jets?

ABSTRACT We present WASP-43b climate simulations with deep wind jets (down to 700 bar) that are linked to retrograde (westward) flow at the equatorial day side for p < 0.1 bar. Retrograde flow inhibits efficient eastward heat transport and naturally explains the small hotspot shift and large day-night-side gradient of WASP-43b (Porb = Prot = 0.8135 d) observed with Spitzer. We find that deep wind jets are mainly associated with very fast rotations (Prot = Porb ≤ 1.5 d) which correspond to the Rhines length smaller than 2 planetary radii. We also diagnose wave activity that likely gives rise to deviations from superrotation. Further, we show that we can achieve full steady state in our climate simulations by imposing a deep forcing regime for p > 10 bar: convergence time-scale τconv = 106–108 s to a common adiabat, as well as linear drag at depth (p ≥ 200 bar), which mimics to first-order magnetic drag. Lower boundary stability and the deep forcing assumptions were also tested with climate simulations for HD 209458b (Porb = Prot = 3.5 d). HD 209458b simulations always show shallow wind jets (never deeper than 100 bar) and unperturbed superrotation. If we impose a fast rotation (Porb = Prot = 0.8135 d), also the HD 209458b-like simulation shows equatorial retrograde flow at the day side. We conclude that the placement of the lower boundary at p = 200 bar is justified for slow rotators like HD 209458b, but we suggest that it has to be placed deeper for fast-rotating, dense hot Jupiters (Porb ≤ 1.5 d) like WASP-43b. Our study highlights that the deep atmosphere may have a strong influence on the observable atmospheric flow in some hot Jupiters.

The Control of Surface Friction on the Scales of Baroclinic Eddies in a Homogeneous Quasigeostrophic Two-Layer Model

In idealized models of the extratropical troposphere, both β and surface friction can control the equilibrated scales of baroclinic eddies by stopping the inverse cascade. A scaling theory on how surface friction alone sets these scales was proposed by Held in 1999 in the case of a quadratic drag law. However, the theory breaks down when friction is modeled by linear damping, and there are other reasons to suspect that it is oversimplified. An ideal system to test the theory is the homogeneous two-layer quasigeostrophic model in the β = 0 limit with quadratic damping. This study investigates some numerical simulations of the model to analyze two causes of the theory’s breakdown. They are 1) the asymmetry between two layers due to confinement of friction to the lower layer and 2) deviation from a spectrally local inverse energy cascade due to the spread of wavenumbers over which energy is input into the barotropic mode. The former is studied by comparing the simulations with drag appearing asymmetrically or symmetrically between the two layers. The latter is addressed with a heuristic modification of the theory. A regime where eddies equilibrate without an inverse cascade is also examined. A comparison is then made between quadratic and linear drag simulations. The connection to a competing theory based on the dynamics of equivalent barotropic vortices with thermal signatures is further discussed. Finally, we present an example of an inhomogeneous statistically steady state to argue that the diffusivity obtained from the homogeneous model has relevance to more realistic configurations.