Three-dimensional Incompressible Navier-Stokes Equations Research Articles

CT-FFR by expanding coronary tree with Newton-Krylov-Schwarz method to solve the governing equations of CFD.

A new model of computational fluid dynamics (CFD)-based algorithm for coronary CT angiography (CCTA)-derived fractional flow reserve (FFR) (CT-FFR) analysis by expanding the coronary tree to smaller-diameter lumen (0.8 mm) using Newton-Krylov-Schwarz (NKS) method to solve the three-dimensional time-dependent incompressible Navier-Stokes equations has been developed; however, the diagnostic performance of this new method has not been sufficiently investigated. The aim of this study was to determine the diagnostic performance of a novel CT-FFR technique by expanding the coronary tree in the CFD domain. Six centres enrolled 338 symptomatic patients with suspected or known coronary artery disease (CAD) who prospectively underwent CCTA and FFR. Stenosis assessment in CCTA and CT-FFR analysis were performed in independent core laboratories. Haemodynamically significant stenosis was defined by a CT-FFR and FFR ≤ 0.80, and anatomically obstructive CAD was defined as a CCTA with stenosis ≥ 50%. Diagnostic performance of CT-FFR was evaluated against invasive FFR using receiver operating characteristic (ROC) curve analysis. The correlation between CT-FFR and invasive FFR was analysed using the Spearman correlation coefficient and Bland-Altman analysis. Intra-observer and inter-observer agreements were evaluated utilizing the intraclass correlation coefficient (ICC). In this study, 338 patients with 422 targeted vessels were investigated, revealing haemodynamically significant stenosis in 31.1% (105/338) of patients and anatomically obstructive stenosis in 54.1% of patients. On a per-vessel basis, the area under the ROC curve for CT-FFR was 0.94 vs. 0.76 for CCTA (P < 0.001). Per-vessel accuracy, sensitivity, specificity, positive predictive value, and negative predictive value were 89.8%, 89.3%, 90.0%, 79.0%, and 99.2%, respectively, for CT-FFR and were 68.4%, 82.8%, 62.3%, 48.1%, and 89.6%, respectively, for CCTA stenosis. CT-FFR and FFR were well correlated (r = 0.775, P < 0.001) with a Bland-Altman bias of 0.0011, and limits of agreement from -0.1509 to 0.1531 (P = 0.770). The ICCs with CT-FFR for intro- and inter-observer agreements were 0.919 (95% CI: 0.866-0.952) and 0.909 (95% CI: 0.851-0.945), respectively. The average computation time for CT-FFR analysis was maintained at 11.7 min. This novel CT-FFR model with the inclusion of smaller lumen provides high diagnostic accuracy in detecting haemodynamically significant CAD. Furthermore, the integration of the NKS method ensures that the computation time remains within an acceptable range for potential clinical applications in the future.

Global regularity and decay behavior for Leray equations with critical-dissipation and its application to self-similar solutions

In this paper, we show the global regularity and the optimal decay of weak solutions to the generalized Leray problem with critical dissipation. Our approach hinges on the maximal smoothing effect, L p L^{p} -type elliptic regularity of linearization, and the action of the heat semigroup generated by the fractional powers of Laplace operator on distributions with Fourier transforms supported in an annulus. As a by-product, we construct a self-similar solution to the three-dimensional incompressible Navier-Stokes equations. Most notably, we prove the global regularity and the optimal decay without the need for additional requirements found in existing literatures.

Monge–Ampère geometry and vortices

We introduce a new approach to Monge–Ampère geometry based on techniques from higher symplectic geometry. Our work is motivated by the application of Monge–Ampère geometry to the Poisson equation for the pressure that arises for incompressible Navier–Stokes flows. Whilst this equation constitutes an elliptic problem for the pressure, it can also be viewed as a non-linear partial differential equation connecting the pressure, the vorticity, and the rate-of-strain. As such, it is a key diagnostic relation in the quest to understand the formation of vortices in turbulent flows. We study this equation via an associated (higher) Lagrangian submanifold in the cotangent bundle to the configuration space of the fluid. Using our definition of a (higher) Monge–Ampère structure, we study an associated metric on the cotangent bundle together with its pull-back to the (higher) Lagrangian submanifold. The signatures of these metrics are dictated by the relationship between vorticity and rate-of-strain, and their scalar curvatures can be interpreted in a physical context in terms of the accumulation of vorticity, strain, and their gradients. We show explicity, in the case of two-dimensional flows, how topological information can be derived from the Monge–Ampère geometry of the Lagrangian submanifold. We also demonstrate how certain solutions to the three-dimensional incompressible Navier–Stokes equations, such as Hill’s spherical vortex and an integrable case of Arnol’d–Beltrami–Childress flow, have symmetries that facilitate a formulation of these solutions from the perspective of (higher) symplectic reduction.

A comprehensive approach to prediction of fractional flow reserve from deep-learning-augmented model

The underuse of invasive fractional flow reserve (FFR) in clinical practice has motivated research towards non-invasive prediction of FFR. Although the non-invasive derivation of FFR (FFRCT) using computational fluid dynamics (CFD) principles has become a common practice, its clinical application has been limited due to the considerable time required for computation of resulting changes in haemodynamic conditions. An alternative to CFD technology is incorporating a neural network into the computational process to reduce the time necessary for running an effective model.In this study we propose a cascade of data-driven and physic-based neural networks (DP-NN) for predicting FFR (DL-FFRCT). The first network of cascade network DP-NN includes geometric features, and the second network includes physical features. We compare the differences between data-driven neural network (D-NN) and DP-NN for predicting FFR. The training and testing datasets were obtained by solving the three-dimensional incompressible Navier–Stokes equations. Coronary flow and geometric features were used as inputs to train D-NN. In DP-NN the training process involves first training a D-NN to output resting ΔP as one input feature to the DP-NN. Secondly, the physics-based microcirculatory resistance as another input feature to the DP-NN.Using clinically measured FFR as the “gold standard", we validated the computational accuracy of DL-FFRCT in 77 patients. Compared to D-NN, DP-NN improved the prediction of ΔP (R2 = 0.87 vs. R2 = 0.92). Statistical analysis demonstrated that the diagnostic accuracy of DL-FFRCT was not inferior to FFRCT (85.71 % vs. 88.3 %) and the computational time was reduced by a factor of approximately 3000 (4.26 s vs. 3.5 h).DP-NN represents a near real-time, interpretable, and highly accurate deep-learning network, which contributes to the development of high-performance computational methods for haemodynamics. We anticipate that DP-NN will enable near real-time prediction of DL-FFRCT in personalized narrow blood vessels and provide guidance for cardiovascular disease treatments.

Existence of global solution to 3D density-dependent incompressible Navier-Stokes equations

&lt;abstract&gt;&lt;p&gt;In this article, we are committed to studying the three-dimensional incompressible Navier-Stokes equations, where the viscosity depends on density according to a power law. We investigate the Cauchy problem by constructing an approximated solution and bootstrap argument. Finally, we establish the existence of a global strong solution under the conditions of small initial data and the compatibility condition. Meanwhile, the algebraic decay-in-time rates for the solution are also obtained. It is worth pointing out that the degradation of viscosity is allowed.&lt;/p&gt;&lt;/abstract&gt;

Inertial wave attractors in librating cuboids

Perturbed rapidly rotating flows are dominated by inertial oscillations, with restricted group velocity directions, due to the restorative nature of the Coriolis force. In containers with some boundaries oblique to the rotation axis, the inertial oscillations may focus upon reflections, whereby their energy increases whilst their wavelength decreases and their trajectories focus onto attractor regions. In a linear inviscid setting, these attractors are Delta-like distributions. The linear inviscid setting is obtained formally by setting both Ekman number ${E}$ (ratio of inertial to viscous time scales) and Rossby number ${Ro}$ (non-dimensional amplitude of the forcing that drives the inertial oscillations) to zero. These settings raise fundamental questions, in particular concerning the nature of energy dissipation in the vanishing Ekman number regime. Here, we consider a simple container geometry, a rectangular cuboid, in which the direction of the rotation axis is oblique to four of its walls, subject to librational forcing (small-amplitude harmonic oscillations of the rotation rate). This geometry allows for accurate and efficient direct numerical simulations of the three-dimensional incompressible Navier–Stokes equations with no-slip boundary conditions using a spectral-Galerkin spatial discretisation along with a third-order temporal discretisation. Solutions with Ekman and Rossby numbers as small as ${E}={Ro}=10^{-8}$ reveal many details of how the inertial oscillations focus, at the libration frequency considered, onto attractors, and how the focusing leads to increased localised nonlinear and dissipative processes as ${E}$ and ${Ro}$ are reduced. Even for extremely small forcing amplitudes, nonlinear effects have important dynamic consequences for the attractors.

Scalable methods for computing sharp extreme event probabilities in infinite-dimensional stochastic systems

We introduce and compare computational techniques for sharp extreme event probability estimates in stochastic differential equations with small additive Gaussian noise. In particular, we focus on strategies that are scalable, i.e. their efficiency does not degrade upon temporal and possibly spatial refinement. For that purpose, we extend algorithms based on the Laplace method for estimating the probability of an extreme event to infinite dimensional path space. The method estimates the limiting exponential scaling using a single realization of the random variable, the large deviation minimizer. Finding this minimizer amounts to solving an optimization problem governed by a differential equation. The probability estimate becomes sharp when it additionally includes prefactor information, which necessitates computing the determinant of a second derivative operator to evaluate a Gaussian integral around the minimizer. We present an approach in infinite dimensions based on Fredholm determinants, and develop numerical algorithms to compute these determinants efficiently for the high-dimensional systems that arise upon discretization. We also give an interpretation of this approach using Gaussian process covariances and transition tubes. An example model problem, for which we provide an open-source python implementation, is used throughout the paper to illustrate all methods discussed. To study the performance of the methods, we consider examples of stochastic differential and stochastic partial differential equations, including the randomly forced incompressible three-dimensional Navier–Stokes equations.

Global strong solution of 3D inhomogeneous incompressible Navier-Stokes equations with degenerate viscosity

In this paper, we investigate the Cauchy problem of the three-dimensional inhomogeneous incompressible Navier-Stokes equations with degenerate viscosity. For the case μ(ρ)=ρ, we prove the global existence of strong solution for small initial data satisfying the compatibility condition. Furthermore, we also obtain the large time algebraic decay rates of the velocity. The innovation of this article is that it does not require a positive lower bound for viscosity.

Global-in-time probabilistically strong and Markov solutions to stochastic 3D Navier–Stokes equations: Existence and nonuniqueness

We are concerned with the three-dimensional incompressible Navier–Stokes equations driven by an additive stochastic forcing of trace class. First, for every divergence free initial condition in L2 we establish existence of infinitely many global-in-time probabilistically strong and analytically weak solutions, solving one of the open problems in the field. This result, in particular, implies nonuniqueness in law. Second, we prove nonuniqueness of the associated Markov processes in a suitably chosen class of analytically weak solutions satisfying a relaxed form of an energy inequality. Translated to the deterministic setting, we obtain nonuniqueness of the associated semiflows.

Numerical investigation of distinct flow modes in a wide-gap spherical annulus using OpenMP

Spherical Couette flow (SCF) between two rotating concentric spheres has many applications in different fields of Mathematics, Physics, and Engineering. The transitions occurring in this flow phenomena have the relevancy with geophysical motions. In this paper, we use the time-dependent three-dimensional incompressible Navier–Stokes equations (3D-INSEs) to investigate distinct flow modes for a wide-gap clearance ratio (CR) . An Artificial Compressibility method has been used along with three implicit solvers, i.e. alternating direction implicit (ADI), line Gauss–Seidel (LGS), and point Gauss–Seidel (PGS). We apply the basic techniques of OpenMP to our Fortran code. First, we make a performance comparison of the three implicit solvers running on a single central processing unit (CPU) with that of the multi-core CPUs using Open Multi-Processing (OpenMP) to choose the best solver for the simulation of the SCF problem. The comparison shows that the ADI solver is the most efficient solver compared to LGS and PGS solvers. Using the most efficient parallel ADI solver, we compute distinct flow modes for this wide gap in a range of Reynolds number between . For this wide gap, first we obtain the basic 0-vortex flow (0-VF) at . By increasing the the flow becomes symmetric 1-vortex flow (1-VF) and asymmetric spiral 1-VF at and respectively. The flow returns back to super critical basic flow at . Further increasing the the flow becomes spiral 0-VF with m = 4 and m = 5 spiral waves at and respectively. Increasing the further the flow finally becomes turbulent. For this CR, we have first time numerically simulate these flow transitions from basic flow to super critical spiral basic flow with increasing . Abbreviations: ACM: artificial compressibility method; ADI: alternating directionimplicit; CPU: central processing unit; CR: clearance ratio; CVD: circumferentialvelocity distribution; INSEs: incompressible Navier–Stokes equations; LGS: lineGauss–Seidel; OpenMP: open multi-processing; PGS: point Gauss–Seidel; Re: Reynoldsnumber; SCF: spherical Couette flow; SG: spherical-gaps; TG: Taylor–Gortler; TVF:Taylor vortex flow; 0-VF: 0-vortex flow; 1-VF: 1-vortex flow; WENO: weightedessentially non-oscillatory.

Numerical Simulation of Microscale Oblique Droplet Impact on Liquid Film

The oblique impact of microscale water droplets on liquid film is numerically investigated. Two-phase flow problems are simulated using three-dimensional incompressible Navier-Stokes equations, and the level-set method is employed for capturing the gas-liquid interface. The numerical model is verified using experimental results from a normal and oblique impact via the qualitative comparison of crown profile features and quantitative contrast of the crown height and radius varying with time. The article discusses the influence of tangential impact velocity, water film thickness, Reynolds number, and Weber number on the shape characteristics, tangential momentum, and kinetic energy of the annular crown. The results show that the decreasing momentum in the tangential direction can be divided into three clear stages: rapid decrease, slight increase, and continuous decrease. In addition, film thickness and Weber number have significant effects on the momentum decay rate.

A highly parallel fully implicit domain decomposition method for the simulation of the hemodynamics of a patient-specific artery at the full-body scale

The numerical simulation of blood flows in the human body with a certain level of clinical accuracy is important for the understanding of the human physiology. The success of the modeling relies on a robust numerical method with the corresponding software that can handle the complex geometry, the complex fluid flows and run efficiently on a supercomputer. In this work, we introduce a highly parallel domain decomposition method to solve the three-dimensional incompressible Navier-Stokes equations on a patient-specific artery at the full-body scale from neck to feet with 222 outlets and a minimum diameter around 1.0 mm. A locally refined, unstructured mesh is used to resolve the complex fluid flow. Moreover, a two-level method is introduced to determine the model parameters in the Windkessel outlet boundary condition to guarantee clinically correct flow distributions to 14 major regions. A fully implicit Newton-Krylov-Schwarz method is used to solve the nonlinear algebraic system at each time step and numerical experiments show that the proposed method is robust with respect to the complex geometry, the graph-based partition of the complex mesh, the ill-conditioned sparse systems with locally dense blocks, and different model parameters and is scalable with up to 15,360 processor cores. With the proposed method, one simulation of the blood flow in a full-body arterial network can be obtained in about 8 hours per cardiac cycle, which enables its potential use in a wide range of clinical scenarios.

A forced convection of water aluminum oxide nanofluid flow and heat transfer study for a three dimensional annular with inner rotated cylinder

The article examines a water alumina nanofluid and heat transfer through the three-dimensional annular. The annular is constructed by the two concentric cylinders in which the inner cylinder can rotate along the tangential direction at a constant speed. A slip boundary condition will be imposed to vanish the viscous effect in the vicinity of the outer cylinder wall. Moreover, the rotating cylinder is kept at a hot temperature, and the outer one is at a cold temperature. A three-dimensional incompressible Navier Stokes and energy equations were carried in cylindrical coordinates. The simulation was observed using the emerging computational tool of COMSOL Multiphysics 5.6, which implements Least Square Galerkin's scheme of finite element method. The parametric study will be done by altering the speed of rotation of the inner cylinder from 1 to 4, volume fraction from 0.001 to 0.9, and the aspect ratio from 0.4 to 0.6 for a fixed Reynolds number of 35,000. The results will be displayed with graphs and tables for average values of the Nusselt number, the percentage change in the temperature, and the skin friction at the middle plan. It was found that the average Nusselt number at the middle of the annular increases before the volume fraction of 0.2 and then decreases for all values of the volume fraction for a fixed rotation of the inner cylinder. The average percentage change relative to the inner cylinder's hot temperature decreases with the volume fraction increase for the fixed rotation. Also, it was found that the quantity of nanoparticles in the domain is improving the average skin friction in the middle of the channel, and it can be reduced by improving the rotation of the inner cylinder by about 10–23% strictly depending upon the aspect ratio for a particular case.

Stabilized reduced-order models for unsteady incompressible flows in three-dimensional parametrized domains

In this work we derive a parametric reduced-order model (ROM) for the unsteady three-dimensional incompressible Navier–Stokes equations without additional pre-processing on the reduced-order subspaces. Concerning the high-fidelity, full-order model, we start from a streamline-upwind Petrov–Galerkin stabilized finite element discretization of the equations using P1−P1 elements for velocity and pressure, respectively. We rely on Galerkin projection of the discretized equations onto reduced basis subspaces for the velocity and the pressure, respectively, obtained through Proper Orthogonal Decomposition on a dataset of snapshots of the full-order model. Both nonlinear and nonaffinely parametrized algebraic operators of the reduced-order system of nonlinear equations, including the projection of the stabilization terms, are efficiently assembled exploiting the Discrete Empirical Interpolation Method (DEIM), and its matrix version (MDEIM), thus obtaining an efficient offline–online computational splitting. We apply the proposed method to (i) a two-dimensional lid-driven cavity flow problem, considering the Reynolds number as parameter, and (ii) a three-dimensional pulsatile flow in stenotic vessels characterized by geometric and physiological parameter variations. We numerically show that the projection of the stabilization terms on the reduced basis subspace and their reconstruction using (M)DEIM allows to obtain a stable ROM with coupled velocity and pressure solutions, without any need for enriching the reduced velocity space, or further stabilizing the ROM. Additionally, we demonstrate that our implementation allows to compute the ROM solution about 20 times faster than the full order model.

Simulation of drop impact on substrate with micro-wells

In this paper, we numerically investigate drop impact on a micro-well substrate to understand the phenomena of non-wettability. The simulation is carried out by solving three-dimensional incompressible Navier–Stokes equations using a density projection method and an adaptive grid refinement algorithm. A very sharp interface reconstruction algorithm, known as the moment-of-fluid method, is utilized to identify the multi-materials and multi-phases present in the computation domain. Our simulations predicted that a micro-well with a deep cavity can significantly reduce a solid–liquid contact in the event of drop impact. The results from the drop impact on the micro-well substrate are compared with results from drop impact on a flat substrate. Significant differences are observed between these two cases in terms of wetted area, spreading ratio, and kinetic energy. Our simulation shows that under the same conditions, a drop is more apt to jump from a micro-well substrate than from a flat surface, resulting in smaller wetted area and shorter contact time. Based on the simulation results, we draw a drop jumping region map. The micro-well substrate has a larger region than the flat surface substrate. Finally, we present a comparative analysis between a flat substrate and a substrate constructed with a dense array of micro-wells and, therefore, show that the array of micro-wells outperforms the smooth substrate with regard to non-wettability and drop wicking capability.

Numerical Simulation of the Force Acting on the Riser by Two Internal Solitary Waves

An internal wave is a typical dynamic process. As an internal wave, an internal solitary wave usually occurs between two layers of fluids with different densities. Compared with general internal waves, internal solitary waves have large amplitudes, fast propagation speeds, short-wave periods, and often have tremendous energy. The propagation causes strong convergence and divergence of seawater and generates a sudden strong current. Due to its various characteristics, the propagation of internal solitary waves can cause serious harm to offshore engineering structures. Therefore, studying the effects of internal solitary waves on risers is vital in preventing environmental pollution caused by riser damage. Although the research on internal solitary waves has achieved very fruitful results, the research on structures is mostly focused on a single condition, and the occurrence of internal solitary wave, as a complex ocean phenomenon, is often accompanied by many situations. Therefore, this paper constructs a numerical simulation of the interaction between two columns of internal solitary waves and risers. This study explores the force and flow field changes of the riser under the condition of multiple internal solitary waves using the Star-CCM+ software in the simulation. The improved K-epsilon turbulence model was adopted to close the three-dimensional incompressible Navier–Stokes equation, and the solitary wave solution of the eKdV equation was used as the initial and boundary conditions. The interaction between single and double internal solitary waves and a riser was calculated, compared, and analyzed using numerical analysis. The experiment results indicate that the conditions of two internal solitary waves differ from those of a single internal solitary wave. After colliding at the riser, the waves gradually merge into a single wave, and the flow field reaches its minimum velocity. Under the two-wave condition, the horizontal force on the riser as a whole is less than the single-wave condition. As the amplitude difference between the two internal solitary waves gradually decreases, the horizontal opposing force received by the riser first increases and then decreases, while the horizontal positive force gradually decreases.

Spontaneous symmetry breaking for extreme vorticity and strain in the three-dimensional Navier-Stokes equations.

We investigate the spatio-temporal structure of the most likely configurations realizing extremely high vorticity or strain in the stochastically forced three-dimensional incompressible Navier–Stokes equations. Most likely configurations are computed by numerically finding the highest probability velocity field realizing an extreme constraint as solution of a large optimization problem. High-vorticity configurations are identified as pinched vortex filaments with swirl, while high-strain configurations correspond to counter-rotating vortex rings. We additionally observe that the most likely configurations for vorticity and strain spontaneously break their rotational symmetry for extremely high observable values. Instanton calculus and large deviation theory allow us to show that these maximum likelihood realizations determine the tail probabilities of the observed quantities. In particular, we are able to demonstrate that artificially enforcing rotational symmetry for large strain configurations leads to a severe underestimate of their probability, as it is dominated in likelihood by an exponentially more likely symmetry-broken vortex-sheet configuration.This article is part of the theme issue ‘Mathematical problems in physical fluid dynamics (part 2)’.

Non-Invasive Quantification of Fraction Flow Reserve Based on Steady-State Geometric Multiscale Models.

Background: The underuse of invasive fraction flow reserve (FFR) in clinical practice has motivated research towards its non-invasive prediction. The early attempts relied on solving the incompressible three-dimensional Navier–Stokes equations in segmented coronary arteries. However, transient boundary condition has a high resource intensity in terms of computational time. Herein, a method for calculating FFR based on steady-state geometric multiscale (FFRSS) is proposed. Methods: A total of 154 moderately stenotic vessels (40–80% diameter stenosis) from 136 patients with stable angina were included in this study to validate the clinical diagnostic performance of FFRSS. The method was based on the coronary artery model segmented from the patient’s coronary CTA image. The average pressure was used as the boundary condition for the inlet, and the microcirculation resistance calculated by the coronary flow was used as the boundary condition for the outlet to calculate the patient-specific coronary hyperemia. Then, the flow velocity and pressure distribution and the FFRss of each coronary artery branch were calculated to evaluate the degree of myocardial ischemia caused by coronary stenosis. Also, the FFRSS and FFRCT of all patients were calculated, and the clinically measured FFR was used as the “gold standard” to verify the diagnostic performance of FFRSS and to compare the correlation between FFRSS and FFRCT. Results: According to the FFRSS calculation results of all patients, FFRSS and FFR have a good correlation (r = 0.68, p < 0.001). Similarly, the correlation of FFRSS and FFRCT demonstrated an r of 0.75 (95%CI: 0.67–0.72) (p < 0.001). On receiver-operating characteristic analysis, the optimal FFRSS cut point for FFR≤0.80 was 0.80 (AUC:0.85 [95% confidence interval: 0.79 to 0.90]; overall accuracy:88.3%). The overall sensitivity, specificity, PPV, and NPV for FFRSS ≤0.80 versus FFR ≤0.80 was 68.18% (95% CI: 52.4–81.4), 93.64% (95% CI: 87.3–97.4), 82.9%, and 91.1%, respectively. Conclusion: FFRSS is a reliable diagnostic index for myocardial ischemia. This method was similar to the closed-loop geometric multiscale calculation of FFR accuracy but improved the calculation efficiency. It also improved the clinical applicability of the non-invasive computational FFR model, helped the clinicians diagnose myocardial ischemia, and guided percutaneous coronary intervention.

Deformation and acceleration of water droplet in continuous airflow

The present work investigates the deformation and acceleration of water droplets in continuous airflow. The numerical approach is based on the level set method for capturing the liquid–gas interface and the projection method for solving the three-dimensional incompressible Navier–Stokes equations. The effects of the incoming airflow velocity (10–100 m/s), initial droplet diameter (20–100 μm), and supercooling on water droplet deformation are investigated. The results indicate that the droplet enters the breakup regime at a critical Weber number of 10, which agrees with the published literature. A dimensionless deformation factor L is defined to describe the droplet deformation. The results confirm that the extreme values of L increase with increasing Weber number during droplet movement. Two models are proposed to predict the minimum deformation factor and the corresponding dimensionless time. The effect of supercooling on water droplet deformation is analyzed, and it is found that the deformation factor of supercooled droplets is lower than that of room-temperature droplets, while supercooled water droplets exhibit greater acceleration. Furthermore, based on the numerical results, a model governed by the Weber number and the Ohnesorge number is proposed for predicting droplet acceleration.

Global solutions to 3D incompressible Navier–Stokes equations with some large initial data

In this paper, we derive a new smallness hypothesis of initial data for the three-dimensional incompressible Navier–Stokes equations. More precisely, we prove that if ‖u01+u02‖Ḃp,13p−1+‖u03‖Ḃp,13p−1‖u01‖Ḃp,13p−1+‖u02‖Ḃp,13p−1×expC(‖u0‖Ḃ∞,2−12+‖u0‖Ḃ∞,1−1) is small enough, the Navier–Stokes equations have a unique global solution. As an application, we construct two examples of initial data satisfying the smallness condition, but whose Ḃ∞,∞−1(R3) norm can be arbitrarily large.