Browsing by Author "Yue, Pengtao"

Now showing 1 - 20 of 32
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    Almost well-posedness of the full water wave equation on the finite stripe domain
    Zhu, Benben (Virginia Tech, 2023-08-18)
    The dissertation gives a rigorous study of surface waves on water of finite depth subjected to gravitational force. As for `water', it is an inviscid and incompressible fluid of constant density and the flow is irrotational. The fluid is bounded above by a free surface separating the fluid from the air above (assumed to be a vacuum) and below by a rigid flat bottom. Then, the governing equations for the motion of the fluid flow are called Euler equations. If the initial fluid flow is prescribed at time zero, i.e., mathematically the initial condition for the Euler equations is given, the long-time existence of a unique solution for the Euler equations is still an open problem, even if the initial condition is small (or initial flow is almost motionless). The dissertation tries to make some progress for proving the long-time existence and show that the time interval of the existence is exponentially long, called almost global well-posedness, if the initial condition is small and satisfies some conditions. The main ideas for the study are from the corresponding almost global well-posedness result for surface waves on water of infinite depth.
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    An arbitrary Lagrangian-Eulerian method for simulating interfacial dynamics between a hydrogel and a fluid
    Li, Lei; Zhang, Jiaqi; Xu, Zelai; Young, Y. -N.; Feng, James J.; Yue, Pengtao (Academic Press/Elsevier, 2022-02-15)
    Hydrogels are crosslinked polymer networks swollen with an aqueous solvent, and play central roles in biomicrofluidic devices. In such applications, the gel is often in contact with a flowing fluid, thus setting up a fluid-hydrogel two-phase system. Using a recently proposed model (Young et al. [41] 2019), we treat the hydrogel as a poroelastic material consisting of a Saint Venant-Kirchhoff polymer network and a Newtonian viscous solvent, and develop a finite-element method for computing flows involving a fluid-hydrogel interface. The interface is tracked by using a fixed-mesh arbitrary Lagrangian-Eulerian method that maps the interface to a reference configuration. The interfacial deformation is coupled with the fluid and solid governing equations into a monolithic algorithm using the finite-element library deal.II. The code is validated against available analytical solutions in several non-trivial flow problems: one-dimensional compression of a gel layer by a uniform flow, two-layer shear flow, and the deformation of a Darcy gel particle in a planar extensional flow. In all cases, the numerical solutions are in excellent agreement with the analytical solutions. Numerical tests show second-order convergence with respect to mesh refinement, and first-order convergence with respect to time-step refinement.
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    Boundary Controllability and Stabilizability of Nonlinear Schrodinger Equation in a Finite Interval
    Cui, Jing (Virginia Tech, 2017-04-24)
    The dissertation focuses on the nonlinear Schrodinger equation iu_t+u_{xx}+kappa|u|^2u =0, for the complex-valued function u=u(x,t) with domain t>=0, 0<=x<= L, where the parameter kappa is any non-zero real number. It is shown that the problem is locally and globally well-posed for appropriate initial data and the solution exponentially decays to zero as t goes to infinity under the boundary conditions u(0,t) = beta u(L,t) and beta u_x(0,t)-u_x(L,t) = ialpha u(0,t), where L>0, and alpha and beta are any real numbers satisfying alpha*beta<0 and beta does not equal 1 or -1. Moreover, the numerical study of controllability problem for the nonlinear Schrodinger equations is given. It is proved that the finite-difference scheme for the linear Schrodinger equation is uniformly boundary controllable and the boundary controls converge as the step sizes approach to zero. It is then shown that the discrete version of the nonlinear case is boundary null-controllable by applying the fixed point method. From the new results, some open questions are presented.
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    Comparison of four boundary conditions for the fluid-hydrogel interface
    Xu, Zelai; Zhang, Jiaqi; Young, Yuan-Nan; Yue, Pengtao; Feng, James J. (American Physical Society, 2022-09-01)
    In adopting a poroelastic model for a hydrogel, one views its constituent fluid and solid phases as interpenetrating continua, thereby erasing the pore-scale geometry. This gives rise to the need for additional boundary conditions (BCs) at the interface between a hydrogel and a clear fluid to supplement the momentum equations for the fluid and solid phases in the hydrogel. Using a thermodynamic argument on energy dissipation, we propose three sets of BCs for the gel-fluid interface that link the normal and tangential velocity jumps across the interface to the normal and tangential stresses on either side of the interface. Using several flow problems - one-dimensional compression, two-layer Couette and Poiseuille shear flows, and deformation of a gel particle by a planar extension flow - as tests, we compare the predictions of these three BCs with that of a previously proposed BC. Some differences are stark and reveal flaws in certain BCs. Others are subtler and will require quantitative experimental data for validation. Based on these results, we recommend one set of BCs over the other three for computing the flow and deformation of hydrogels in contact with a clear fluid. In addition, we suggest benchmark experiments to validate the BCs and our recommendation.
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    A comparison of viscoelastic stress wakes for two-dimensional and three-dimensional Newtonian drop deformations in a viscoelastic matrix under shear
    Afkhami, Shahriar; Yue, Pengtao; Renardy, Yuriko Y. (American Institute of Physics, 2009-07)
    A recent experimental study of a Newtonian drop suspended in a viscoelastic matrix undergoing simple shear displays a transient overshoot in drop deformation which is qualitatively similar to two-dimensional (2D) numerical simulation results. Despite the similarity, an interpretation in light of the 2D result is misleading because the overshoot is absent in the fully three-dimensional (3D) simulation. This motivates a study of regimes where qualitatively different and interesting features such as overshoots in deformation occur for a 2D drop but not for a 3D drop. The influence of viscoelastic "wakes" that emanate from the drop tips is reported. The viscoelastic wakes are larger and of higher magnitude in 3D than in 2D, and lead to more deformation in 3D. During drop evolution, the less deformed drop is found to be aligned more with the flow direction. As the drop-to-matrix viscosity ratio increases from 1 to past 3, drop rotation is promoted, with accompanying retraction when the capillary number is sufficiently high. Thus, a 3D overshoot in deformation is promoted with increasing viscosity ratio.
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    Compressible Lubrication Theory in Pressurized Gases
    Chien, Ssu-Ying (Virginia Tech, 2019-04-08)
    Lubrication theory plays a fundamental role in all mechanical design as well as applications to biomechanics. All machinery are composed of moving parts which must be protected against wear and damage. Without effective lubrication, maintenance cycles will be shortened to impractical levels resulting in increased costs and decreased reliability. The focus of the work presented here is on the lubrication of rotating machinery found in advanced power systems and designs involving micro-turbines. One of the earliest studies of lubrication is due to Osborne Reynolds in 1886 who recorded what is now regarded as the canonical equation governing all lubrication problems; this equation and its extensions have become known as the Reynolds equation. In the past century, Reynolds equation has been extended to include three-dimensional effects, unsteadiness, turbulence, variable material properties, non-newtonian fluids, multi-phase flows, wall slip, and thermal effects. The bulk of these studies have focused on highly viscous liquids, e.g., oils. In recent years there has been increasing interest in power systems using new working fluids, micro-turbines and non-fossil fuel heat sources. In many cases, the design of these systems employs the use of gases rather than liquids. The advantage of gases over liquids include the reduction of weight, the reduction of adverse effects due to fouling, and compatibility with power system working fluids. Most treatments of gas lubrication are based on the ideal, i.e., low pressure, gas theory and straightforward retro-fitting of the theory of liquid lubrication. However, the 21st Century has seen interest in gas lubrication at high pressures. At pressures and temperatures corresponding to the dense and supercritical gas regime, there is a strong dependence on gas properties and even singular behavior of fundamental transport properties. Simple extrapolations of the intuition and analyses of the ideal gas or liquid phase theory are no longer possible. The goal of this dissertation is to establish the correct form of the Reynolds equation valid for both low and high pressure gases and to explore the dynamics predicted by this new form of the Reynolds equation. The dissertation addresses five problems involving our new Reynolds equation. In the first, we establish the form appropriate for the simple benchmark problem of two-dimensional journal bearings. It is found that the material response is completely determined by a single thermodynamic parameter referred to as the "effective bulk modulus". The validity of our new Reynolds equation has been established using solutions to the full Navier-Stokes-Fourier equations. We have also provided analytical estimates for the range of validity of this Reynolds equation and provided a systematic derivation of the energy equation valid whenever the Reynolds equation holds. The next three problems considered here derive local and global results of interest in high speed lubrication studies. The results are based on a perturbation analysis of our Reynolds and energy equation resulting in simplified formulas and the explicit dependence of pressure, temperature, friction losses, load capacity, and heat transfer on the thermodynamic state and material properties. Our last problem examines high pressure gas lubrication in thrust bearings. We again derive the appropriate form of the Reynolds and energy equations for these intrinsically three-dimensional flows. A finite difference scheme is employed to solve the resultant (elliptic) Reynolds equation for both moderate and high-speed flows. This Reynolds equation is then solved using perturbation methods for high-speed flows. It is found that the flow structure is comprised of five boundary layer regions in addition to the main ``core'' region. The flow in two of these boundary layer regions is governed by a nonlinear heat equation and the flow in three of these boundary layers is governed by nonlinear relaxation equations. Finite difference schemes are employed to obtain detailed solutions in the boundary layers. A composite solution is developed which provides a single solution describing the flow in all six regions to the same accuracy as the individual solutions in their respective regions of validity. Overall, the key contributions are the establishment of the appropriate forms of the Reynolds equation for dense and supercritical flows, analytical solutions for quantities of practical interest, demonstrations of the roles played by various thermodynamic functions, the first detailed discussions of the physics of lubrication in dense and supercritical flows, and the discovery of boundary layer structures in flows associated with thrust bearings.
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    Computational Analysis of Asphalt Binder based on Phase Field Method
    Hou, Yue (Virginia Tech, 2014-04-29)
    The mechanical performance evaluation of asphalt binder has always been a challenging issue for pavement engineers. Recently, the Phase Field Method (PFM) has emerged as a powerful computational tool to simulate the microstructure evolution of asphalt binder. PFM analyzes the structure from the free energy aspect and can provide a view of the whole microstructure evolution process. In this dissertation, asphalt binder performance is analyzed by PFM in three aspects: first, the relationship between asphalt chemistry and performance is investigated. The components of asphalt are simplified to three: asphaltene, resin and oil. Simulation results show that phase separation will occur under certain thermal conditions and result in an uneven distribution of residual thermal stress. Second, asphalt cracking is analyzed by PFM. The traditional approach to analyze crack propagation is Classic Fracture Mechanics first proposed by Griffith, which needs to clearly depict the crack front conditions and may cause complex cracking topologies. PFM describes the microstructure using a phase-field variable which assumes positive one in the intact solid and negative one in the crack void. The fracture toughness is modeled as the surface energy stored in the diffuse interface between the intact solid and crack void. To account for the growth of cracks, a non-conserved Allen-Cahn equation is adopted to evolve the phase-field variable. The energy based formulation of the phase-field method handles the competition between the growth of surface energy and release of elastic energy in a natural way: the crack propagation is a result of the energy minimization in the direction of the steepest descent. Both the linear elasticity and phase-field equation are solved in a unified finite element frame work, which is implemented in the commercial software COMSOL. Different crack mode simulations are performed for validation. It was discovered that the onset of crack propagation agrees very well with the Griffith criterion and experimental results. Third, asphalt self-healing phenomenon is studied based on the Atomic Force Microscopy (AFM) technology. The self-healing mechanism is simulated in two ways: thermodynamic approach and mechanical approach. Cahn-Hilliard dynamics and Allen-Cahn dynamics are adopted, respectively.
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    Condensation Frosting: From Ice Bridges to Dry Zones
    Nath, Saurabh (Virginia Tech, 2017-09-18)
    The most ubiquitous mode of frost formation on substrates is condensation frosting, where dew drops condense on a supercooled surface and subsequently freeze, and has been known since the time of Aristotle. The physics of frost incipience at a microscopic scale has, nevertheless, eluded researchers because of an unjustified ansatz regarding the primary mechanism of condensation frosting. It was widely assumed that during condensation frosting each supercooled droplet in the condensate population freezes in isolation by heterogeneous nucleation at the solid-liquid interface, quite analogous to the mechanism of icing. This assumption has very recently been invalidated with strong experimental evidence which shows that only a single droplet has to freeze by heterogeneous nucleation (typically by edge effects) in order to initiate condensation frosting in a supercooled condensate population. Once a droplet has frozen, it subsequently grows an ice bridge towards its nearest neighboring liquid droplet, freezing it in the process. Thus ensues a chain reaction of ice bridging where the newly frozen droplets grow ice bridges toward their nearest neighbor liquid droplets forming a percolating network of interconnected frozen droplets. Not always are these ice bridges successful in connecting to their adjacent liquid droplets. Sometimes the liquid droplet can completely evaporate before the ice bridges can connect, thus forming a local dry region in the vicinity of the ice bridge. In this work, we first formulate a thermodynamic framework in order to understand the localized vapor pressure gradients that emerge in mixed-mode phase-change systems and govern condensation and frost phenomena. Following this, we study droplet pair interactions between a frozen droplet and a liquid droplet to understand the physics behind the local ice bridge connections. We discuss the emergent scaling laws in ice bridging dynamics, their relative size dependencies, and growth rates. Thereafter, we show how with spatial control of interdroplet distances in a supercooled condensate and temporal control of the first freezing event, we can tune global frost propagation on a substrate and even cause a global failure of all ice bridges to create a dry zone. Subsequently, we perform a systematic study of dry zones and derive a scaling law for dry zones that collapses all of our experimental data spanning a wide parameter space. We then show that almost always the underlying mechanism behind the formation of dry zones around any hygroscopic droplet is inhibition of growth and not inhibition of nucleation. We end with a discussion and preliminary results of our proposed anti-frosting surface that uses ice itself to prevent frost.
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    Design, Analysis, and Application of Immersed Finite Element Methods
    Guo, Ruchi (Virginia Tech, 2019-06-19)
    This dissertation consists of three studies of immersed finite element (IFE) methods for inter- face problems related to partial differential equations (PDEs) with discontinuous coefficients. These three topics together form a continuation of the research in IFE method including the extension to elasticity systems, new breakthroughs to higher degree IFE methods, and its application to inverse problems. First, we extend the current construction and analysis approach of IFE methods in the literature for scalar elliptic equations to elasticity systems in the vector format. In particular, we construct a group of low-degree IFE functions formed by linear, bilinear, and rotated Q1 polynomials to weakly satisfy the jump conditions of elasticity interface problems. Then we analyze the trace inequalities of these IFE functions and the approximation capabilities of the resulted IFE spaces. Based on these preparations, we develop a partially penalized IFE (PPIFE) scheme and prove its optimal convergence rates. Secondly, we discuss the limitations of the current approaches of IFE methods when we try to extend them to higher degree IFE methods. Then we develop a new framework to construct and analyze arbitrary p-th degree IFE methods. In this framework, each IFE function is the extension of a p-th degree polynomial from one subelement to the whole interface element by solving a local Cauchy problem on interface elements in which the jump conditions across the interface are employed as the boundary conditions. All the components in the analysis, including existence of IFE functions, the optimal approximation capabilities and the trace inequalities, are all reduced to key properties of the related discrete extension operator. We employ these results to show the optimal convergence of a discontinuous Galerkin IFE (DGIFE) method. In the last part, we apply the linear IFE methods in the literature together with the shape optimization technique to solve a group of interface inverse problems. In this algorithm, both the governing PDEs and the objective functional for interface inverse problems are discretized optimally by the IFE method regardless of the location of the interface in a chosen mesh. We derive the formulas for the gradients of the objective function in the optimization problem which can be implemented efficiently in the IFE framework through a discrete adjoint method. We demonstrate the properties of the proposed algorithm by applying it to three representative applications.
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    Dynamical Phase-Change Phenomena
    Ahmadi, Seyedfarzad (Virginia Tech, 2019-06-28)
    Matter on earth exists mostly in three different phases of solid, liquid, and gas. With extreme amounts of energy, temperature, or pressure, a matter can be changed between the phases. Six different types of phase-change phenomena are possible: freezing (the substance changes from a liquid to a solid), melting (solid to liquid), condensation (gas to liquid), vaporization (liquid to gas), sublimation (solid to gas), and desublimation (gas to solid). Another form of phase change which will be discussed here is the wetting or dewetting transitions of a superhydrophobic surface, in which the phase residing within the surface structure switches between vapor and liquid. Phase transition phenomena frequently occur in our daily life; examples include: a ``liquid'' to ``solid'' transition when cars decrease their distance at a traffic light, solidification of liquids droplets during winter months, and the dancing of droplets on a non-sticking pan. In this dissertation we will address seven different phase-change problems occurring in nature. We unveil completely new forms of phase-change phenomena that exhibit rich physical behavior. For example, during traffic flow, drivers keep a large distance from the vehicle in front of them to ensure safe driving. When vehicles come to a stop, for example at a red light, drivers voluntarily induce a ``phase transition'' from this ``liquid phase'' to a close-packed ``solid phase''. This phase transition is motivated by the intuition that traveling as far as possible before stopping will minimize the overall travel time. However, we are going to investigate this phase-change process and show that this long standing intuition is wrong. Phase-change of solidification will be discussed for different problems. Moreover, the complex physics of oil as it wicks up sheets of frost and freezing of bubble unveil completely new forms of multiphase flows that exhibit rich physical behavior. Finally, the ``Cassie'' to ``Wenzel'' transition will be investigated for layered nano-textured surfaces. These phenomena will be modeled using thermodynamics and fluid mechanics equations.
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    Editorial for Special Issue “Drop, Bubble and Particle Dynamics in Complex Fluids”
    Afkhami, Shahriar; Yue, Pengtao (MDPI, 2020-01-02)
    The presence of drops, bubbles, and particles affects the behavior and response of complex multiphase fluids [...]
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    Finite-element simulations of interfacial flows with moving contact lines
    Zhang, Jiaqi (Virginia Tech, 2020-06-19)
    In this work, we develop an interface-preserving level-set method in the finite-element framework for interfacial flows with moving contact lines. In our method, the contact line is advected naturally by the flow field. Contact angle hysteresis can be easily implemented without explicit calculation of the contact angle or the contact line velocity, and meshindependent results can be obtained following a simple computational strategy. We have implemented the method in three dimensions and provide numerical studies that compare well with analytical solutions to verify our algorithm. We first develop a high-order numerical method for interface-preserving level-set reinitialization. Within the interface cells, the gradient of the level set function is determined by a weighted local projection scheme and the missing additive constant is determined such that the position of the zero level set is preserved. For the non-interface cells, we compute the gradient of the level set function by solving a Hamilton-Jacobi equation as a conservation law system using the discontinuous Galerkin method. This follows the work by Hu and Shu [SIAM J. Sci. Comput. 21 (1999) 660-690]. The missing constant for these cells is recovered using the continuity of the level set function while taking into account the characteristics. To treat highly distorted initial conditions, we develop a hybrid numerical flux that combines the Lax-Friedrichs flux and a penalty flux. Our method is accurate for non-trivial test cases and handles singularities away from the interface very well. When derivative singularities are present on the interface, a second-derivative limiter is designed to suppress the oscillations. At least (N + 1)th order accuracy in the interface cells and Nth order accuracy in the whole domain are observed for smooth solutions when Nth degree polynomials are used. Two dimensional test cases are presented to demonstrate superior properties such as accuracy, long-term stability, interface-preserving capability, and easy treatment of contact lines. We then develop a level-set method in the finite-element framework. The contact line singularity is removed by the slip boundary condition proposed by Ren and E [Phys. Fluids, vol. 19, p. 022101, 2007], which has two friction coefficients: βN that controls the slip between the bulk fluids and the solid wall and βCL that controls the deviation of the microscopic dynamic contact angle from the static one. The predicted contact line dynamics from our method matches the Cox theory very well. We further find that the same slip length in the Cox theory can be reproduced by different combinations of (βN; βCL). This combination leads to a computational strategy for mesh-independent results that can match the experiments. There is no need to impose the contact angle condition geometrically, and the dynamic contact angle automatically emerges as part of the numerical solution. With a little modification, our method can also be used to compute contact angle hysteresis, where the tendency of contact line motion is readily available from the level-set function. Different test cases, including code validation and mesh-convergence study, are provided to demonstrate the efficiency and capability of our method. Lastly, we extend our method to three-dimensional simulations, where an extension equation is solved on the wall boundary to obtain the boundary condition for level-set reinitializaiton with contact lines. Reinitialization of ellipsoidal interfaces is presented to show the accuracy and stability of our method. In addition, simulations of a drop on an inclined wall are presented that are in agreement with theoretical results.
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    Higher Order Immersed Finite Element Methods for Interface Problems
    Meghaichi, Haroun (Virginia Tech, 2024-05-17)
    In this dissertation, we provide a unified framework for analyzing immersed finite element methods in one spatial dimension, and we design a new geometry conforming IFE space in two dimensions with optimal approximation capabilities, alongside with applications to the elliptic interface problem and the hyperbolic interface problem. In the first part, we discuss a general m-th degree IFE space for one dimensional interface problems with many polynomial-like properties, then we develop a general framework for obtaining error estimates for the IFE spaces developed for solving a variety of interface problems, including but not limited to, the elliptic interface problem, the Euler-Bernoulli beam interface problem, the parabolic interface problem, the transport interface problem, and the acoustic interface problem. In the second part, we develop a new m-th degree finite element space based on the differential geometry of the interface to solve interface problems in two spatial dimensions. The proposed IFE space has optimal approximation capabilities, easy to construct, and the IFE functions satisfy the interface conditions exactly. We provide several numerical examples to demonstrate that the IFE space yields optimally converging solutions when applied to the elliptic interface problem and the hyperbolic interface problem with a symmetric interior penalty discontinuous Galerkin formulation.
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    How soap bubbles freeze
    Ahmadi, S. Farzad; Nath, Saurabh; Kingett, Christian M.; Yue, Pengtao; Boreyko, Jonathan B. (Springer Nature, 2019-06-18)
    Droplets or puddles tend to freeze from the propagation of a single freeze front. In contrast, videographers have shown that as soap bubbles freeze, a plethora of growing ice crystals can swirl around in a beautiful effect visually reminiscent of a snow globe. However, the underlying physics of how bubbles freeze has not been studied. Here, we characterize the physics of soap bubbles freezing on an icy substrate and reveal two distinct modes of freezing. The first mode, occurring for isothermally supercooled bubbles, generates a strong Marangoni flow that entrains ice crystals to produce the aforementioned snow globe effect. The second mode occurs when using a cold stage in a warm ambient, resulting in a bottom-up freeze front that eventually halts due to poor conduction along the bubble. Blending experiments, scaling analysis, and numerical methods, the dynamics of the freeze fronts and Marangoni flows are characterized.
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    Large-Eddy Simulations of Hydrocyclones
    Bukhari, Mustafa Mohammedamin T. (Virginia Tech, 2023-01-20)
    This dissertation investigates the flow physics, turbulence structure, and particle classification process in hydrocyclones using large-eddy simulations of turbulent multiphase flow. Two types of hydrocyclones are considered. The first is a classifying hydrocyclone, and the second is a mineral flotation hydrocyclone, also known as an air-sparged hydrocyclone (ASH). Large-eddy simulations (LES) are conducted for multi-phase flow (air, water, and sand particles) so that the complex anisotropic turbulence of a swirling flow is computed correctly. The effects of mesh refinements on the mean flow and turbulence stresses are investigated, and (LES) results are validated by comparisons with experimental data for classifying hydrocyclone. The two-phase flow in air-sparged hydrocyclone has not been analyzed before. ANSYS CFX software V17.2 has been used to conduct the simulations. Firstly, large-eddy simulations have been conducted for two-phase flow (water and air) in a conventional hydrocyclone using the Eulerian two-fluid (Eulerian-Eulerian) and Volume-of- Fluid (VOF) models. Subgrid stresses are modeled using a dynamic eddy–viscosity model, and results are compared to those using the Smagorinsky model. The effects of grid resolutions on the mean flow and turbulence statistics have been thoroughly investigated. Five block-structured grids of 0.72, 1.47, 2.4, 3.81, and 7.38 million elements have been used for the simulations of a typical conventional hydrocyclone designed and tested by Hsieh (75 mm hydrocyclone) [1]. Mean velocity profiles and normal Reynolds stresses have been compared with experimental data. The results of the Eulerian two-fluid model agree with those of the VOF model. A fine mesh in the axial and radial directions is necessary for capturing the turbulent vortical structures. Turbulence structures in the hydrocyclone are dominated by helical vortices around the air core. Energy spectra are analyzed at different points in the hydrocyclone, and regions of low turbulent kinetic energy are identified and attributed to stabilizing effects of the swirling velocity component. Turbulent energy spectra in the different regions of the hydrocyclone have been analyzed. The energy spectra are calculated at two points near the air-water interface. They show a short inertial subrange where energy decays as f−5/3, followed by viscous damping where energy drops as f−7, where f is frequency. However, for the points located near the boundary where high turbulent kinetic energy is found, the energy spectra exhibit f^(−4) decay. Secondly, the two-fluid (Eulerian two-fluid) model and large-eddy simulation are used to compute the turbulent two-phase flow of air and water in a cyclonic flotation device known as an Air-Sparged Hydrocyclone (ASH). In the operation of ASH, the air is injected through a porous cylindrical wall. The study considers a 48-mm diameter hydrocyclone and uses a block-structured fine mesh of 10.5 million hexahedral elements. The air-to-water injection ratio is 4, and a uniform air bubble diameter of 0.5 mm has been specified. The flow field in ASH has been investigated for the inlet flow rate of water of 30.6 L/min at different values of underflow exit pressure. The present simulations show that the value of static pressure imposed at the underflow section strongly affects the distribution of air volume fraction, water axial velocity, tangential velocity, and swirling layer thickness in ASH. The loci of zero-axial velocity surfaces have been determined for different exit pressures. The water split ratio through the overflow opening varies with underflow exit pressure as 6%, 8%, 16%, and 26% for 3, 4, 5, and 6 kPa, respectively. These results indicate that regulating the pressure at the underflow exit can be used to optimize ASH's performance. Turbulent energy spectra in different regions of the hydrocyclone have been analyzed. Small-scale turbulence spectra at near-wall points exhibit f^(−4) law, where f is frequency. Whereas for points at the air-column interface, the energy spectra show an inertial subrange f^(−5/3) followed by a dissipative range of f^(−7) law. Thirdly, large-eddy simulation (LES) has been used to investigate the flow separation in multi-phase flow (gas, liquid, and solid) in a classifying hydrocyclone using the multi-fluid (Eulerian multi-fluid) model. The results of the CFD simulation are compared with the Hsieh [1] experimental data. The water phase is considered a continuous phase, while air and solid particles are considered dispersed phases. Drag between water-air and water-sand is the only considered interfacial force. The Schiller-Naumann and Wen-Yu models are used to model the drag, and the Gidaspow model is used to calculate the solid pressure term. Various particle sizes are tested in the hydrocyclone to investigate the underflow recovery percentages. The results agree with the experimental data for the particles of a diameter smaller than 20 μm, while the results vary based on the model for the large particles. Therefore, using the Wen Yu and Schiller-Naumann model for the drag model and the Gidaspow model for the solid pressure in the three-fluid model could give acceptable results for the small particles underflow recovery and volume fraction distribution. However, the models failed for large particles. Finally, the large particle size separation needs more investigation.
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    A level-set method for moving contact lines with contact angle hysteresis
    Zhang, Jiaqi; Yue, Pengtao (Academic Press/Elsevier, 2020-10-01)
    We develop a level-set method in the finite-element framework. The contact line singularity is removed by the slip boundary condition proposed by Ren and E (2007) [6], which has two friction coefficients: βN that controls the slip between the bulk fluids and the solid wall and βCL that controls the deviation of the microscopic dynamic contact angle from the static one. The predicted contact line dynamics from our method matches the Cox theory very well. We further find that the same slip length in the Cox theory can be reproduced by different combinations of (βN,βCL), based on which we come up with a computational strategy for mesh-independent results that can match the experiments. There is no need to impose the contact angle condition geometrically, and the dynamic contact angle automatically emerges as part of the numerical solution. With a little modification, our method can also be used to compute contact angle hysteresis, where the tendency of contact line motion is readily available from the level-set function. Different test cases, including code validation and mesh-convergence study, are provided to demonstrate the efficiency and capability of our method.
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    Modeling Superparamagnetic Particles in Blood Flow for Applications in Magnetic Drug Targeting
    Rukshin, Iris; Mohrenweiser, Josef; Yue, Pengtao; Afkhami, Shahriar (MDPI, 2017-06-04)
    Magnetic drug targeting is a technique that involves the binding of medicine to magnetizable particles to allow for more specific transport to the target location. This has recently come to light as a method of drug delivery that reduces the disadvantages of conventional, systemic treatments. This study developed a mathematical model for tracking individual superparamagnetic nanoparticles in blood flow in the presence of an externally applied magnetic field. The model considers the magnetic attraction between the particles and the external magnet, influence of power law flow, diffusive interaction between the particles and blood, and random collisions with red blood cells. A stochastic system of differential equations is presented and solved numerically to simulate the paths taken by particles in a blood vessel. This study specifically focused on localized cancer treatment, in which a surface tumor is accessed through smaller blood vessels, which are more conducive to this delivery method due to slower flow velocities and smaller diameters. The probability of the particles reaching the tumor location is found to be directly dependent on ambient factors; thus, diffusion through Brownian motion and red blood cell collisions, different magnetic field and force models, blood viscosities, and release points are considered.
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    Multiphase Interfacial Phenomena for Liquid Manipulation and Defrosting
    Lolla, Venkata Yashasvi (Virginia Tech, 2024-10-07)
    Interfacial phenomena are prevalent in various natural and engineered systems. A thorough understanding of these phenomena is essential for a complete understanding of processes such as phase transitions and interaction of liquid droplets with different surfaces. The insights gained from understanding interfacial behavior are pivotal in fields such as pharmaceuticals, microfluidics, material sciences, and environmental engineering. This dissertation aims to advance our understanding of interfacial behaviors, thereby facilitating the development of innovative technologies for applications in health, defrosting, and omniphobic surfaces. In Chapters 1 and 2, relevant background information and goals are provided to contextualize the research being presented in this dissertation. Chapter 3 introduces a novel metal-free alternative to conventional antiperspirants (containing aluminum salts and zirconium salts). We leverage the composition of human sweat (97% water and 3% minerals) and employ a hygroscopic substance near the outlet of an artificial sweat duct rig. This leads to complete diffusion and dehydration of sweat, forming a natural mineral plug within the artificial sweat duct that halts the flow. Chapter 4 examines the behavior of room temperature water droplets spreading on a flat icy substrate. The use of flat ice, as opposed to cold substrates, eliminates the nucleation energy barrier, enabling freeze front initiation as soon as the bulk temperature of the spreading drop reaches 0 C. Through scaling analysis, we identify distinct thermo-hydrodynamic regimes with varying Weber numbers. Chapter 5 presents a novel construct for lubricant-impregnated surfaces (LIS). To date, most of the investigations characterizing the wettability of LIS have focused on droplet mobility. We pioneer a lubricant-impregnated fiber (LIF) which exhibits unique droplet dynamics due to simultaneous exploitation of both, high mobility and high adhesion. Chapter 6 proposes an innovative approach for defrosting by exploiting the polarizability and natural thermo-voltage of frost sheets. By placing an actively charged electrode near the frost sheet, we observe that frost dendrites migrate towards the electrode. This technique, termed Electrostatic Defrosting (EDF), effectively removes up to 75% of the frost mass for superhydrophobic surfaces and 50% of the frost mass for untreated surfaces in less than 100 s.
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    Nonhomogeneous Initial Boundary Value Problems for Two-Dimensional Nonlinear Schrodinger Equations
    Ran, Yu (Virginia Tech, 2014-05-08)
    The dissertation focuses on the initial boundary value problems (IBVPs) of a class of nonlinear Schrodinger equations posed on a half plane R x R+ and on a strip domain R x [0,L] with Dirichlet nonhomogeneous boundary data in a two-dimensional plane. Compared with pure initial value problems (IVPs), IBVPs over part of entire space with boundaries are more applicable to the reality and can provide more accurate data to physical experiments or practical problems. Although there is less research that has been made for IBVPs than that for IVPs, more attention has been paid for IBVPs recently. In particular, this thesis studies the local well-posedness of the equation for the appropriate initial and boundary data in Sobolev spaces H^s with non-negative s and investigates the global well-posedness in the H^1-space. The main strategy, especially for the local well-posedness, is to derive an equivalent integral equation (whose solution is called mild solution) from the original equation by semi-group theory and then perform the Banach fixed-point argument. However, along the process, it is essential to select proper auxiliary function spaces and prepare all the corresponding norm estimates to complete the argument. In fact, the IBVP posed on R x R+ and the one posed on R x [0,L] are two independent problems because the techniques adopted are different. The first problem is more related to the initial value problem (IVP) posed on the whole plane R^2 and the major ingredients are Strichartz's estimate and its generalized theory. On the other hand, the second problem can be studied as an IVP over a half-line and periodic domain, which is established on the analysis for series inspired by Bourgain's work. Moreover, the corresponding smoothing properties and regularity conditions of the solution are also considered.
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    Numerical Simulations of Interactions of Solid Particles and Deformable Gas Bubbles in Viscous Liquids
    Qin, Tong (Virginia Tech, 2013-01-11)
    Studying the interactions of solid particles and deformable gas
    bubbles in viscous liquids is very important in many applications,
    especially in mining and chemical industries. These interactions
    involve liquid-solid-air multiphase flows and an
    arbitrary-Lagrangian-Eulerican (ALE) approach is used for the direct
    numerical simulations. In the system of rigid particles and
    deformable gas bubbles suspended in viscous liquids, the
    Navier-Stokes equations coupled with the equations of motion of the
    particles and deformable bubbles are solved in a finite-element
    framework. A moving, unstructured, triangular mesh tracks the
    deformation of the bubble and free surface with adaptive refinement.
    In this dissertation, we study four problems. In the first three
    problems the flow is assumed to be axisymmetric and two dimensional
    (2D) in the fourth problem.

    Firstly, we study the interaction between a rising deformable bubble
    and a solid wall in highly viscous liquids. The mechanism of the
    bubble deformation as it interacts with the wall is described in
    terms of two nondimensional groups, namely the Morton number (Mo)
    and Bond number (Bo). The film drainage process is also
    considered. It is found that three modes of bubble-rigid wall
    interaction exist as Bo changes at a moderate Mo.
    The first mode prevails at small Bo where the bubble deformation
    is small. For this mode, the bubble is
    hard to break up and will bounce back and eventually attach
    to the rigid wall. In the second mode, the bubble may break up after
    it collides with the rigid wall, which is determined by the film
    drainage. In the third mode, which prevails at high Bo, the bubble
    breaks up due to the bottom surface catches up the top surface
    during the interaction.

    Secondly, we simulate the interaction between a rigid particle and a
    free surface. In order to isolate the effects of viscous drag and
    particle inertia, the gravitational force is neglected and the
    particle gains its impact velocity by an external accelerating
    force. The process of a rigid particle impacting a free surface and
    then rebounding is simulated. Simplified theoretical models are
    provided to illustrate the relationship between the particle
    velocity and the time variation of film thickness between the
    particle and free surface. Two film thicknesses are defined. The
    first is the thickness achieved when the particle reaches its
    highest position. The second is the thickness when the particle
    falls to its lowest position. The smaller of these two thicknesses
    is termed the minimum film thickness and its variation with the
    impact velocity has been determined. We find that the interactions
    between the free surface and rigid particle can be divided into
    three regimes according to the trend of the first film thickness.
    The three regimes are viscous regime, inertial regime and jetting
    regime. In viscous regime, the first film thickness decreases as the
    impact velocity increases. Then it rises slightly in the inertial
    regime because the effect of liquid inertia becomes larger as the
    impact velocity increases. Finally, the film thickness decreases
    again due to Plateau-Rayleigh instability in the jetting regime.
    We also find that the minimum film thickness corresponds to an
    impact velocity on the demarcation point between the viscous and
    inertial regimes. This fact is caused by the balance of viscous
    drag, surface deformation and liquid inertia.

    Thirdly, we consider the interaction between a rigid particle and a
    deformable bubble. Two typical cases are simulated: (1) Collision of
    a rigid particle with a gas bubble in water in the absence of
    gravity, and (2) Collision of a buoyancy-driven rising bubble with a
    falling particle in highly viscous liquids. We also compare our
    simulation results with available experimental data. Good agreement
    is obtained for the force on the particle and the shape of the
    bubble.

    Finally, we investigated the collisions of groups of bubbles and
    particles in two dimensions. A preliminary example of the oblique
    collision between a single particle and a single bubble is conducted
    by giving the particle a constant acceleration. Then, to investigate
    the possibility of particles attaching to bubbles, the interactions
    between a group of 22 particles and rising bubbles are studied. Due
    to the fluid motion, the particles involved in central collisions
    with bubbles have higher possibilities to attach to the bubble.