Geometric Flow Research Articles

Partial Differential Equations

The workshop covered topics in nonlinear elliptic and parabolic Partial Differential Equations as well as topics in Geometric Measure Theory, touching topics such as geometric variational problems and minimal surfaces, geometric flows, free boundaries and the structure of nodal sets of eigenfunctions as well as real and complex Monge–Ampère equations.

Space Target Dynamic Identification by Exploiting Geometrical Feature Flow From ISAR Image Sequences

Dynamic identification is crucial for space target investigation and awareness, which is especially challenging for uncooperative and uncontrolled targets. This article presents a dynamic identification algorithm using the time-variant geometric feature flow extracted from an inverse synthetic aperture radar (ISAR) image sequence. The range-Doppler (RD) projection, along with the sequential radar light of slight of the target component, intrinsically relies on the targets’ dynamic states. With the accommodation of a long short-time memory (LSTM) network, reliable dynamic identification of space targets can be implemented by exploiting the geometrical feature flow from the ISAR sequence. The superiority of the proposed method is demonstrated through comparison experiments.

Alternative flow model of anisotropic porous media

Porous flow is typically analyzed by the Kozeny–Carman equation using geometric variables, such as hydraulic diameter and tortuosity. The hydraulic tortuosity was first described by Kozeny, and later redefined by Carman as the tortuosity square term in the Kozeny–Carman equation. However, the revised term correlating the flow velocity with path length square would be physically ambiguous. Moreover, the hydraulic diameter, which is directly correlated to the permeability and interstitial velocity in the Kozeny–Carman equation, is an isotropic constant property; thus, it should be verified whether the isotropic hydraulic diameter can reasonably correlate each anisotropic directional flow feature passing through heterogeneous complex media. Accordingly, the Kozeny–Carman equation was theoretically examined and experimentally verified in this study to obtain the proper correlation based on the definitions of truly equivalent diameter and tortuosity. Therefore, the effective variables of porous media were presented, and it confirmed that the effective diameter corresponded to the physically equivalent diameter of anisotropic porous media. Moreover, using the mass conservation relation, it was verified that Kozeny's tortuosity is exactly associated with the truly equivalent flow model, and then the Kozeny constant must be differently defined from the original Kozeny equation. Accordingly, the Kozeny–Carman equation was improved by appertaining either effective diameter or tortuosity, and the momentum conservation relation was used to verify it. The pore-scale simulations using 5-sorts of 25-series porous media models were performed to test the validity of derived effective variables and revised equations. Finally, the alternative flow model of anisotropic porous media was presented using equivalent geometric and frictional flow variables. Subsequently, their practical and more accurate estimations were achieved by introducing the concentric annulus flow model under the special hydraulic condition. The new variables and relations are expected to be usefully applied to various porous flow analyses, such as interstitial velocity estimations, geometric condition variations, flow regime changes, and anisotropic heat and multiphase flows.

Structure and dynamics of the turbulent flow through a central baffle

Central baffle flume (CBF) can be utilized as a control structure to measure flow discharge in irrigation channels under free and submerged flow conditions. Stage-discharge relationship has been extensively studied for various geometrical parameters and flow conditions, whereas internal structure of the flow around a baffle has not been investigated in the literature. To address this need, the present work investigates the turbulent flow around a central baffle through high-resolution numerical simulations using an open source computational model. Velocity measurements were conducted in a laboratory flume to setup and validate the numerical model. Comparison of the numerical results with the experimental measurements proves that the present numerical model can predict water depth and velocity field. Longitudinal distance from the apex to the intersection point of water and critical depths can be estimated as Lxc = 2Le, where Le is the longitudinal length of the guide walls. A horseshoe vortex system identified in front of the baffle produces a significant bump on the free-surface and rib vortices generated from the baffle extend up to the sidewalls of the channel. The vertical separation layer observed downstream of the baffle results in a reverse flow and a vortex pair is formed by the impingement of the resulting reverse flow on the back of the baffle. Reverse flow, plunging flow structure, splash and rebounding wave events observed at the downstream produce substantial hydrodynamic effects on the baffle. Geometry of the central baffle was modified to suppress recirculation effects based on the insights into the complete flow structure around the baffle. Eventually, vortex structures were suppressed and the length of the recirculation zone was reduced by 76%.

An accurate analytical model for squirt flow in anisotropic porous rocks — Part 2: Complex geometry

Seismic waves exhibit strong attenuation and velocity dispersion when they propagate in porous rocks saturated with a fluid. The main cause of such energy dissipation is fluid flow in the pore space, so called squirt flow. Squirt flow takes place between interconnected pores and cracks or grain-to-grain contacts. The corresponding theory can be used to characterize porous rocks in the subsurface with noninvasive seismic methods. We extend the analytical model for a classical pore geometry presented in part 1 of our paper to more complex geometries of the pore space, where the crack edge is partially connected to multiple pores. This pore geometry is much more closely representative of rocks than the classical geometry where the crack edge is fully connected to a toroidal pore. We develop an approach to calculate the model compliance taking into account the interconnectivity of the crack and pores. We redefine a squirt flow length parameter which takes into account the geometrical configuration of the multiple connections between a crack and the surrounding pores. This configuration will control the geometrical flow pattern and thus the diffusion length scale or, in other words, the characteristic frequency. We validate our analytical model against inherently accurate 3D numerical solutions. The analytical and numerical results are in excellent agreement for a range of different pore geometries. Published analytical solutions expect the user to know the dry stiffness (e.g., from laboratory measurements), but in our work, we also provide a way to calculate analytically the dry stiffness for the precise geometry that we consider. The new analytical model redefines the quantitative and qualitative description of seismic attenuation and velocity dispersion due to squirt flow. We provide the MATLAB and symbolic Maple routines to reproduce our main results.

Investigation of thermal-hydraulic performance of metal-foam heat sink using machine learning approach

Metal foam heat sinks (MFHSs) are often used for electronic cooling because of their high specific heat transfer surface area, fluid mixing capability, and lightweight. Thermal-hydraulic performance of MFHS is known to be related to the morphological parameters of the metal foam (MF), physical dimensions of the heat sink, and coolant flow condition. Moreover, the highly random and complex geometrical nature of MFHS makes full-scale numerical simulation and experimental analysis expensive. In this paper, we have utilized machine learning (ML) methods to predict the thermal-hydraulic performance of MFHS, considering MF morphology, heat sink dimension, and coolant flow condition as input features. Considering local thermal non-equilibrium conditions for MFHS, a database of 1000 data points is generated utilizing a validated high-fidelity computational fluid dynamics/heat transfer (CFD/HT) model. The data sets consist of two non-dimensional descriptors, i.e., Nusselt number (Nu) and friction factor (f), along with the four input features such as MF morphological parameters, i.e., porosity and pore density, heat sink geometrical dimensions, i.e., hydraulic diameter (Dh) and flow condition, i.e., Reynolds number (Re). Five different ML-based regression models, namely k-nearest neighbor (KNN), random forest (RF), extreme gradient boosting (XGBoost), support vector regressor (SVR), and artificial neural networks (ANN), have been developed. Hyper-parameters of each ML model have been systematically optimized, and optimized model's performance have been compared using statistical score metrics and considering the initialization effect. Additionally, in order to demonstrate the robustness of the developed ML models, individual model performance has been evaluated by independent datasets. Results indicate that for testing datasets, all ML algorithms, except KNN, can predict the thermal-hydraulic performance of MFHS reasonably well, with a mean absolute percent error (MAPE) of about 4.59% and a 1.24 standard deviation (SD). However, SVR and ANN outperform other models for independent datasets with a MAPE of less than 3.1%.

Positive Hermitian curvature flow on special linear groups and perfect solitons

We study invariant solutions to the positive Hermitian curvature flow, introduced by Ustinovskiy, on complex Lie groups. We show in particular that the canonical scale-static metrics on the special linear groups, arising from the Killing form, are dynamically unstable. This disproves a conjecture of Ustinovskiy. We also construct certain perfect Lie groups that admit at least two distinct invariant solitons for the flow, only one of which is algebraic. This is the second known example of a geometric flow with non-algebraic, homogeneous solitons. The first being the G2-Laplacian flow.

Numerical Study of an Improved Non-Circular MC Fuelled with H/Air for Thermo-Photovoltaic (TPV) Applications

In recent years, researchers have paid special attention to wall uniformity temperature and improvement in a micro-combustor (MC) for applications as thermos-photovoltaic (TPV) since they have a direct influence on feasibility and efficiency of the required conversion of energy. Numerous experimentation and numerical studies in the field of micro-combustion have been carried out and published in this respect. This study focuses on the flame and flow behaviour in an improved MC. Utilizing the famous approach of fluid dynamics being computational (CFD), the impact of geometrical flow parameters behaviour and temperature of wall is explored and measured. The generated model is being used to evaluate the effects of Hydrogen (H) mass flow rate (MFR) and H/air equivalency ratios on the flame pattern and outer temperature of wall of conventional and enhanced MCs. Moreover, the influence of various MC cross sections (CS), such as square, circular, and flat, on temperature of wall is examined. The findings reveal that when the MFR rises, the wall outside temperature rises as well. The MC with flat cross-sections (CSs) is found to be more effective in terms of power of emitter and efficiency of emitter.

Three Dimensional Homogeneous Hyperbolic Ricci Solitons

In this paper, we consider the self-similar solutions to the hyperbolic geometric flow, called hyperbolic Ricci solitons. Also, we investigate hyperbolic Ricci solitons on three-dimensional homogeneous manifolds with Riemannian and Lorentzian metrics and obtain some hyperbolic Ricci solitons on such manifolds.

A review of physics of moving contact line dynamics models and its applications in interfacial science

Despite several decades long research on moving contact line dynamics due to its remarkable role in interfacial science and technological fields, there is still a significant lack of comprehensive knowledge about the true nature of the contact line. The moving contact line is much more complex than just a geometric flow condition. Given the role of moving contact line physics in numerous emerging advanced technologies including bioprinting, tissue engineering, pharmaceutical industry, smart functional coatings, flexible electronics, and healthcare, it is needed for scientific community to direct their focus in advancement on better understanding of interfacial science and physics of moving contact line. Therefore, this review primarily concentrates on the current physical theories which describe the moving contact line dynamics. Moreover, numerous potential and ongoing applications of moving contact line dynamics linked to the interfacial science are thoroughly discussed. Finally, this review presents the future perspectives related to the current challenges in moving contact line dynamics.

Analytical study of fluid flow and thermal characteristics of flow in micro channel heat sinks with working fluid as water and alumina & water nanofluid

The electronic industry is looking at microchannel based MEMS technology to resolve the issue of increasing heat dissipation from electronic circuits. Micro-channels play an important role in development of small scale fluid flow devices. The internal flow configuration represents a convenient geometry for heat transfer between fluids used in chemical processing, environmental control, and energy conversion technologies. In order to fabricate such micro devices effectively, it is extremely important to understand the fundamental mechanisms involved in fluid flow and heat transfer characteristics in microchannels. This paper presents an analytical study of flow mechanism in microchannels, with a focus on effect of geometric dimensions and flow rate on the various flow parameters of a Rectangular microchannel. An attempt is also made to determine optimum microchannel dimensions based on the analysis. Basic flow equations are used for modelling the flow and determining various parameters. Further, the study analyses the effect of using Alumina & Water nanofluid on flow and thermal characteristics as compared to using water alone. The results observed show a deeper and wider microchannel would keep the pressure drop across the microchannel lower. Also using nanofluid as working fluid would result in lower pressure drop.

Classification of Transformed and Geometrically Distorted Images using Convolutional Neural Network

ConvolutionalNeural Network is a deep learning method that is used in many image-relatedapplications, such as image recognition and classification, it has achievedgreat performance in these fields, but it still suffers from some shortcomings.One of these shortcomings is not being able to be invariant to the input datadue to some image transformations like translation, rotation, scaling, andgeometric distortions such as skewness, perspective distortion and pincushiondistortion. This study presents anoptimized CNN which uses the Geometric Heat Flow (GHF) to improve theperformance of the CNN regarding the invariant limitation and classificationaccuracy. GHF is a partial differential equation that expresses how the heatwould diffuse on a surface concerning time in a specific location. GHF isinvariant to image transformations and geometric distortions if it was takenconcerning the object's arc length which will lead to an invariant CNN. Theexperiments show that GHF improves the performance of the CNN, and the proposedwork achieves an accuracy of 98.09% on the MNIST handwritten dataset, 92.58% onthe MNIST-Fashion dataset, and 86.09% on the CIFAR-10 dataset.

Optimal Design and Operation of Dual-Ejector PEMFC Hydrogen Supply and Circulation System

A proton exchange membrane fuel cell (PEMFC) system requires an adequate hydrogen supply and circulation to achieve its expected performance and operating life. An ejector-based hydrogen circulation system can reduce the operating and maintenance costs, noise, and parasitic power consumption by eliminating the recirculation pump. However, the ejector’s hydrogen entrainment capability, restricted by its geometric parameters and flow control variability, can only operate properly within a relatively narrow range of fuel cell output power. This research introduced the optimal design and operation control methods of a dual-ejector hydrogen supply/circulation system to support the full range of PEMFC system operations. The technique was demonstrated on a 70 kW PEMFC stack with an effective hydrogen entrainment ratio covering 8% to 100% of its output power. The optimal geometry design ensured each ejector covered a specific output power range with maximized entrainment capability. Furthermore, the optimal control of hydrogen flow and the two ejectors’ opening and closing times minimized the anode gas pressure fluctuation and reduced the potential harm to the PEMFC’s operation life. The optimizations were based on dedicated computational fluid dynamics (CFD) and system dynamics models and simulations. Bench tests of the resulting ejector-based hydrogen supply/circulation system verified the simulation and optimization results.

Fractal diffusion from a geometric Ricci flow

Fractal diffusion from a geometric Ricci flow

A Process-Based Pore Network Model Construction for Granular Packings Under Large Plastic Deformations

We propose a process-based method for constructing a pore network model of granular packings under large deformations. The method uses the radical Voronoi tessellation and constructive solid geometry operations on meshes of deformed particles, to construct a three-dimensional solid of the pore network and estimate its geometric characteristics, CTS-model parameters and flow transport properties, and it uses the particle mechanics approach to model consolidation of powders under large deformations. This process-based method thus explicitly simulates not only a packing of grains but also its corresponding consolidation process, which in this work is restricted to powder die compaction up to a relative density close to one (i.e., to die filling, compaction to low porosity, unloading and ejection). The efficacy of the proposed method is borne out by studying granular packings with the same composition, namely a 50–50 binary mixture of two monodisperse systems comprised by elasto-plastic spheres with bonding strength, but with microstructures which are topologically different, namely a random packing, a bilayer and core-shell structures. These simulations reveal that topological differences affect the formation and evolution of the pore space statistical signature during consolidation and, therefore, showcase that process-based approaches for constructing PNM are of paramount relevance to understanding architectured granular material systems.

Convexity estimates for hypersurfaces moving by concave curvature functions

We study fully nonlinear geometric flows that deform strictly k-convex hypersurfaces in Euclidean space with pointwise normal speed given by a concave function of the principal curvatures. Specifically, the speeds we consider are obtained by performing a nonlinear interpolation between the mean and the k-harmonic mean of the principal curvatures. Our main result is a convexity estimate showing that, on compact solutions, regions of high curvature are approximately convex. In contrast to the mean curvature flow, the fully nonlinear flows considered here preserve k-convexity in a Riemannian background, and we show that the convexity estimate carries over to this setting as long as the ambient curvature satisfies a natural pinching condition.

Smooth solutions to the Gauss image problem

In this paper we study the the Gauss image problem, which is a generalization of the Aleksandrov problem in convex geometry. By considering a geometric flow involving Gauss curvature and functions of normal vectors and radial vectors, we obtain the existence of smooth solutions to this problem.

Regularity for parabolic systems with critical growth in the gradient and applications

Motivated by problems arising in geometric flows, we prove several regularity results for systems of local and nonlocal equations, adapting to the parabolic case a neat argument due to Caffarelli. The geometric motivation of this work comes from recent works arising in the theory harmonic maps with free boundary in particular. We prove Hölder regularity of weak solutions.

DNA-templated programmable excitonic wires for micron-scale exciton transport

<h2>Summary</h2> Rationally designed photonic complexes promoting the efficient collection and harnessing of excitation energy are key to artificial photosynthesis and optoelectronics. The precise control over the geometric arrangement and energy flow of photonic materials is in great demand. Mimicking natural light-harvesting systems in terms of multi-pigment complexes well organized by protein scaffolds, herein, we report programmable photonic materials directed by structural DNA templates. Four-helix-bundle DNA origami was used to guide the assembly of the cyanine dye K21 to form closely packed dye aggregates exhibiting strong excitonic coupling between chromophores. This enables sub-micron-scale exciton migration, demonstrated by spectroscopic measurements and theoretical modeling. The DNA-templated dye aggregates acting as "excitonic wires" could mediate directional energy transfer over a half-micrometer distance and were further programmed to achieve geometric complexity and modular bottom-up fabrication of higher order photonic architectures. This work offers a rich toolbox to design and create complex photonic devices and excitonic networks.

Homogeneous Einstein metrics on Euclidean spaces are Einstein solvmanifolds

We show that homogeneous Einstein metrics on Euclidean spaces are Einstein solvmanifolds, using that they admit periodic, integrally minimal foliations by homogeneous hypersurfaces. For the geometric flow induced by the orbit-Einstein condition, we construct a Lyapunov function based on curvature estimates which come from real GIT.