Finite element solution of viscous jet flows with surface tension

A finite element method is used to study the effect of Reynolds number and surface tension on the expansion and contraction of jets of Newtonian liquids. For values of Reynolds numbers (based on tube diameter), below 14 the jets expand, and when Re > 14 the jets contract. For higher Reynolds numbers the jet diameter approaches a limiting value. It is also found that the surface tension has a considerable effect on low Reynolds number jet flows, becoming negligible at higher Reynolds numbers. As an example, if the surface tension parameter $\text{σ}\text{η}\text{u}$ is equal to unity, the creeping flow jet expansion is reduced by 4% relative to the case with no surface tension but when Re is equal to 20 and 50 the final jet diameters increase by only 0.2%. The calculated jet shapes are compared with available experimental results.     

Stabilization of a class of nonlinear uncertain ordinary differential equation by parabolic partial differential equation controller

This article is concerned with stabilization for a class of uncertain nonlinear ordinary differential equation (ODE) with dynamic controller governed by linear 1?d heat partial differential equation (PDE). The control input acts at the one boundary of the heat's controller domain and the second boundary injects a Dirichlet term in ODE plant. The main contribution of this article is the use of the recent infinite‐dimensional backstepping design for state feedback stabilization design of coupled PDE‐ODE systems, to stabilize exponentially the nonlinear uncertain systems, under the restrictions that (a) the right‐hand side of the ODE equation has the classical particular form: linear controllable part with an additive nonlinear uncertain function satisfying lower triangular linear growth condition, and (b) the length of the PDE domain has to be restricted. We solve the stabilization problem despite the fact that all known backstepping transformation in the literature cannot decouple the PDE and the ODE subsystems. Such difficulty is due to the presence of a nonlinear uncertain term in the ODE system. This is done by introducing a new globally exponentially stable target system for which the PDE and ODE subsystems are strongly coupled. Finally, an example is given to illustrate the design procedure of the proposed method.     

L2 disturbance attenuation for highly dissipative nonlinear spatially distributed processes via HJI approach

For many practical industrial spatially distributed processes (SDPs), their dynamics are usually described by highly dissipative nonlinear partial differential equations (PDEs). In this paper, we address the L₂ disturbance attenuation problem of nonlinear SDPs using the Hamilton–Jacobi–Isaacs (HJI) approach. Firstly, by collecting an ensemble of PDE states, Karhunen–Loève decomposition (KLD) is employed to compute empirical eigenfunctions (EEFs) of the SDP based on the method of snapshots. Subsequently, these EEFs together with singular perturbation (SP) technique are used to obtain a finite-dimensional slow subsystem of ordinary differential equation (ODE) that accurately describes the dominant dynamics of the PDE system. Secondly, based on the slow subsystem, the L₂ disturbance attenuation problem is reformulated and a finite-dimensional H_∞ controller is synthesized in terms of the HJI equation. Moreover, the stability and L₂-gain performance of the closed-loop PDE system are analyzed. Thirdly, since the HJI equation is a nonlinear PDE that has proven to be impossible to solve analytically, we combine the method of weighted residuals (MWR) and simultaneous policy update algorithm (SPUA) to obtain its approximate solution. Finally, the simulation studies are conducted on a nonlinear diffusion-reaction process and a temperature cooling fin of high-speed aerospace vehicle, and the achieved results demonstrate the effectiveness of the developed control method.     

L1 adaptive control for general partial differential equation (PDE) systems

This paper addresses the L₁ adaptive control problem for general Partial Differential Equation (PDE) systems. Since direct computation and analysis on PDE systems are difficult and time-consuming, it is preferred to transform the PDE systems into Ordinary Differential Equation (ODE) systems. In this paper, a polynomial interpolation approximation method is utilized to formulate the infinite dimensional PDE as a high-order ODE first. To further reduce its dimension, an eigenvalue-based technique is employed to derive a system of low-order ODEs, which is incorporated with unmodeled dynamics described as bounded-input, bounded-output (BIBO) stable. To establish the equivalence with original PDE, the reduced-order ODE system is augmented with nonlinear time-varying uncertainties. On the basis of the reduced-order ODE system, a dynamic state predictor consisting of a linear system plus adaptive estimated parameters is developed. An adaptive law will update uncertainty estimates such that the estimation error between predicted state and real state is driven to zero at each time-step. And a control law is designed for uncertainty handling and good tracking delivery. Simulation results demonstrate the effectiveness of the proposed modeling and control framework.     

On the die-swell of an axisymmetric Newtonian jet

This paper investigates the behaviour of an axisymmetric Newtonian jet of incompressible viscous fluid. Numerical solutions are obtained via the Galerkin formulation of the finite element method for values of Reynolds number between 0.0001 and 1000, and for various values of the surface tension parameter. Attention is focussed on the shape of the free surface profile and the die-swell ratio. The numerical solutions are subjected to tests for validity based upon the momentum balance and mass balance equations. The satisfaction of stress boundary condition is also tested. The numerical are compared with experimental results.     

Simulation of liquid micro-jet in free expanding high-speed co-flowing gas streams

We present the development of an experimentally validated computational fluid dynamics model for liquid micro jets. Such jets are produced by focusing hydrodynamic momentum from a co-flowing sheath of gas on a liquid stream in a nozzle. The numerical model based on laminar two-phase, Newtonian, compressible Navier–Stokes equations is solved with finite volume method, where the phase interface is treated by the volume of fluid approach. A mixture model of the two-phase system is solved in axisymmetry using?~?300,000 finite volumes, while ensuring mesh independence with the finite volumes of the size 0.25 µm in the vicinity of the jet and drops. The numerical model is evaluated by comparing jet diameters and jet lengths obtained experimentally and from scaling analysis. They are not affected by the strong temperature and viscosity changes in the focusing gas while expanding at nozzle outlet. A range of gas and liquid-operating parameters is investigated numerically to understand their influence on the jet performance. The study is performed for gas and liquid Reynolds numbers in the range 17–1222 and 110–215, and Weber numbers in the range 3–320, respectively. A reasonably good agreement between experimental and scaling results is found for the range of operating parameters never tackled before. This study provides a basis for further computational designs as well as adjustments of the operating conditions for specific liquids and gases.     

The FEM based liquid transfer model in gravure offset printing using phase field method

The velocity control of a roller is crucial in gravure offset printing for determining the quality of the printed images such as width and thickness of an electric circuit. The velocity control also affects mass printability, especially when using micro-scale liquid of high conductivity ink. In this work, a liquid transfer model for gravure offset printing is developed using the phase field method to investigate interfacial dynamics. As a numerical scheme, the finite element method is used for discretization of the partial differential equation. The interfacial layer governed by the phase field variable is embodied by the Cahn–Hilliard equation for a convection–diffusion problem. The numerical results are compared with those from the literatures for their validation. The results were found to be in good agreement with both analytical and experimental results in the literatures. After the validation, the effects of several key factors in gravure offset printing, such as velocity, gravity, surface tension and viscosity on liquid transfer are studied with respect to the contact angle of the upper plate. The ranges of the velocity and contact angle are varied from 0.01 to 0.25?m/s and from 30° to 70°, respectively. Also, the values of the surface tension and viscosity are changed from 0.5 to 1.5?N/m and from 0.05 to 0.15 N?s/m², respectively. The simulation result showed that at α?=?β?=?60° regardless of gravity, the liquid transfer rate (R _%) is increased as the velocity of the upper plate is increased at velocities below 0.01?m/s for liquid with low density, whereas the liquid transfer rate is decreased as the velocity is increased for liquid with high density. Also, the liquid transfer rate is increased as the surface tension is increased until the contact angle (α?≤?β?=?60°) approached 60°. Whereas the liquid transfer rate is decreased as the surface tension is increased until the contact angle (α?≤?β?=?60°) is increased to 60°.     

A novel technique for producing metallic microjets and microdrops

A new technique for producing steady metallic jets is proposed. It allows the production of supercritical jets with Weber numbers well below unity, which entails important technological advantages over existing techniques. The metallic liquid is injected through a micrometer converging nozzle located inside a gas stream. Both the liquid jet and the coflowing gas current cross an orifice located in front of the nozzle. The gas stream stabilizes the jet by sweeping away the capillary waves growing on the free surface. In this way, one can steadily produce microjets with a kinetic energy much lower than the interfacial energy, a possibility that has been predicted theoretically (Gañán-Calvo in Phys Rev E 78:026304, 2008). Experiments were conducted with mercury to assess the performance of the new technique. The experimental results agreed remarkably well with the predictions calculated from the convective/absolute instability transition of the jet. The jet breakup mechanism did not correspond to classical Rayleigh instability, but to the growth of surface waves over a capillary column which ends at a fixed location. The results were compared with those obtained with the well-established flow focusing method to show that the new technique considerably favors the jet’s stability.     

Optimal fault detection for linear discrete time-varying systems

This paper deals with the problem of observer-based fault detection for linear discrete time-varying (LDTV) systems. A problem formulation is first proposed to address the optimization of the fault detection filter (FDF) design, which is expressed in terms of maximizing a finite horizon H_∞/H_∞ or H₋/H_∞ performance index. This formulation can be applied to FDF design of LDTV systems subject to l₂-norm bounded unknown inputs or stochastic noise sequences. It is shown that a unified optimal solution to the FDF can be obtained by solving the discrete time Riccati equation and the optimal FDF is not unique. A numerical example is given to illustrate the proposed method.     

Prediction of penetration depth in a plunging water jet using soft computing approaches

The flow characteristics of the plunging water jets can be defined as volumetric air entrainment rate, bubble penetration depth, and oxygen transfer efficiency. In this study, the bubble penetration depth is evaluated based on four major parameters that describe air entrainment at the plunge point: the nozzle diameter (D _N), jet length (L _j), jet velocity (V _N), and jet impact angle (θ). This study presents artificial neural network (ANN) and genetic expression programming (GEP) model, which is an extension to genetic programming, as an alternative approach to modeling of the bubble penetration depth by plunging water jets. A new formulation for prediction of penetration depth in a plunging water jets is developed using GEP. The GEP-based formulation and ANN approach are compared with experimental results, multiple linear/nonlinear regressions, and other equations. The results have shown that the both ANN and GEP are found to be able to learn the relation between the bubble penetration depth and basic water jet properties. Additionally, sensitivity analysis is performed for ANN, and it is found that D _N is the most effective parameter on the bubble penetration depth.     

Pinch-off mechanism for Taylor bubble formation in a microfluidic flow-focusing device

The present work aims at studying the nonlinear breakup mechanism for Taylor bubble formation in a microfluidic flow-focusing device by using a high-speed digital camera. Experiments were carried out in a square microchannel with cross section of 600 × 600 μm. During the nonlinear collapse process, the variation of the minimum radius of bubble neck (r ₀) with the remaining time until pinch-off (τ) can be scaled by a power–law relationship: \(r_{0} \propto \tau^{\alpha } .\) Due to the interface rearrangement around the neck, the nonlinear collapse process can be divided into two distinct stages: liquid squeezing collapse stage and free pinch-off stage. In the liquid squeezing collapse stage, the neck collapses under the constriction of the liquid flow and the exponent α approaches to 0.33 with the increase in the liquid flow rate Q _l. In the free pinch-off stage, the value of α is close to the theoretical value of 0.50 derived from the Rayleigh–Plesset equation and is independent of Q _l.     

Nonlocal vibration and instability analysis of carbon nanotubes conveying fluid considering the influences of nanoflow and non-uniform velocity profile

In this article, the influences of non-uniform velocity profile attributable to slip boundary condition and viscosity of fluid on the dynamic instability of carbon nanotubes (CNTs) conveying fluid are investigated. The nonlocal elasticity theory and the Euler–Bernoulli beam theory are employed to derive partial differential equation of nanotubes conveying fluid. Furthermore, a dimensionless momentum correction factor (MCF) is obtained as a function of Knudsen number (Kn) so as to insert the effects of non-uniform velocity profile into the equation of motion. In continuation, complex eigen-frequencies of the system are attained with respect to different boundary conditions, the momentum correction factor, slip boundary condition and nonlocal parameter. The results delineate that considering the effects of non-uniform velocity profile could diminish predicted critical velocity of flow. Therefore, the divergence instability occurs in the lower values of flow velocity. In addition, the MCF decreases through enhancement of Kn; hence, the effects of non-uniform velocity profile are more noticeable for liquid fluid than gas fluid.     

Max-plus eigenvector representations for solution of nonlinear H/sub /spl infin// problems: basic concepts

The H/sub /spl infin// problem for a nonlinear system is considered. The corresponding dynamic programming equation is a fully nonlinear, first-order, steady-state partial differential equation (PDE), possessing a term which is quadratic in the gradient. The solutions are typically nonsmooth, and further, there is nonuniqueness among the class of viscosity solutions. In the case where one tests a feedback control to see if it yields an H/sub /spl infin// controller, the PDE is a Hamilton-Jacobi-Bellman equation. In the case where the "optimal" feedback control is being determined as well, the problem takes the form of a differential game, and the PDE is, in general, an Isaacs equation. The computation of the solution of a nonlinear, steady-state, first-order PDE is typically quite difficult. In this paper, we develop an entirely new class of methods for obtaining the "correct" solution of such PDEs. These methods are based on the linearity of the associated semigroup over the max-plus (or, in some cases, min-plus) algebra. In particular, solution of the PDE is reduced to solution of a max-plus (or min-plus) eigenvector problem for known unique eigenvalue 0 (the max-plus multiplicative identity). It is demonstrated that the eigenvector is unique, and that the power method converges to it. An example is included.     

Robust/H∞ filtering for nonlinear systems

A robust (or H_∞) approach to filtering for nonlinear systems is considered. A bound on the estimate error as a function of the disturbance energy is obtained. The corresponding dynamic programming equation is a first-order PDE. This has computational ramifications. The case where the measurements are discrete time is considered also. A numerical method is discussed.     

Numerical investigations in Rayleigh breakup of round liquid jets with VOF methods

Simulations of the growth of a capillary instability and of the breakup of a jet were carried out using a one-fluid model to describe the two-phase flow motion and a VOF approach to capture the interface. The model considered each phase as fictitious sub-domains and accounted implicitly for jump conditions at the interface through a unique set of equations for which a source term of surface tensions appeared in momentum equations. The predominance of capillary effects in the breakup mechanism required to accurately describe the surface tension contribution. The Brackbill surface model was chosen because of its simplicity to represent tension forces, although it was known to generate parasitic currents susceptible to limit its precision. The flow incompressibility was ensured with an augmented Lagrangian method in case of sequential calculations and by a predictor/corrector approach for 3D simulations that required parallel computations. As a first step, the numerical methods were validated by simulating the growth of a capillary instability and comparing results to those predicted by the Rayleigh theory for capillary instabilities. The consistency of the Brackbill surface tension model and the accuracy of the methods were evaluated via a convergence study. As a second step, the simulation of a jet breakup was carried out using water as injected liquid and compressed carbon dioxide as surrounding medium. It was shown that the simulation predicted accurately the breakup length and the droplet size evidenced experimentally in literature.     

CFD prediction of the trajectory of a liquid jet in a non-uniform air crossflow

This paper describes a predictive numerical modelling methodology for calculation of deflection and deformation of liquid jets in air crossflows. The methodology combines Jet Embedding (JE) with Volume-of-Fluid (VOF) in a computational fluid dynamics (CFD) based approach. The combined JE/VOF methodology applies the JE concept of modelling the air and liquid phases in separate but linked models, with a representation of the liquid column embedded in the crossflow model. The crossflow is modelled in a CFD calculation using CFX4.2 and the separate jet model is written in FORTRAN 77. A multi-fluid implementation of the VOF technique, in CFX4.2, is used to model deformation of the liquid column cross-section in a series of two-dimensional models along the column trajectory. The crossflow, jet and deformation calculations are linked by an iterative procedure which advances from the nozzle exit in a series of time-steps. The JE/VOF methodology is used to make a prediction of a time-average trajectory of a deflected liquid column, with a progressively deformed cross-section, in an air crossflow. The prediction demonstrates that the JE/VOF methodology is capable of producing a physically realistic result.     

A time domain approach to robust fault detection of linear time-varying systems

This paper gives the optimal solutions to several robust fault detection problems such as ?₋/?_∞, ?₂/?_∞ and ?_∞/?_∞ problems for linear time-varying systems in a time domain, which extends the previous results on linear time-invariant systems in a frequency domain. It is shown that all three problems have the same optimal detection filter and the filter is a simple observer obtained by solving a standard differential Riccati equation. Finally, an example is given to illustrate our results.     

Lattice Boltzmann simulation of the dispersion of aggregated particles under shear flows

The lattice Boltzmann method (LBM) for multicomponent immiscible fluids is applied to simulations of the deformation and breakup of a particle-cluster aggregate in shear flows. In the simulations, the solid particle is modeled by a droplet with strong interfacial tension and large viscosity. The van der Waals attraction force is taken into account for the interaction between the particles. The ratio of the hydrodynamic drag force to cohesive force, I, is introduced, and the effect of I on the aggregate deformation and breakup in shear flows is investigated. It is found that the aggregate is easier to deform and to be dispersed when I is over 100.     

Delay compensated control of the Stefan problem and robustness to delay mismatch

This paper presents a control design for the one‐phase Stefan problem under actuator delay via a backstepping method. The Stefan problem represents a liquid‐solid phase change phenomenon which describes the time evolution of a material's temperature profile and the interface position. The actuator delay is modeled by a first‐order hyperbolic partial differential equation (PDE), resulting in a cascaded transport‐diffusion PDE system defined on a time‐varying spatial domain described by an ordinary differential equation (ODE). Two nonlinear backstepping transformations are utilized for the control design. The setpoint restriction is given to guarantee a physical constraint on the proposed controller for the melting process. This constraint ensures the exponential convergence of the moving interface to a setpoint and the exponential stability of the temperature equilibrium profile and the delayed controller in the norm. Furthermore, robustness analysis with respect to the delay mismatch between the plant and the controller is studied, which provides analogous results to the exact compensation by restricting the control gain.