Equations Of Fluid Mechanics Research Articles

Complex flow field analysis in Multi-Shaft stirred Reactors: Dynamics of Wave-Vortex coupling revealed by POD and DMD methods

Insufficient understanding of complex flow structures in multi-shaft stirred reactors hinders industrial adoption. In this work, Proper Orthogonal Decomposition (POD) and Dynamic Mode Decomposition (DMD) methods were used to analyze the Large-Eddy Simulation (LES) results of the stationary and rotating zones for three types of stirred reactors, including multi-shaft configurations. The results indicate that the flexible spatial distribution of the impellers in multi-shaft stirred reactors enhances fluid interactions, leading to the superior performance in energy cascading, flow field self-regulation, and wave-vortex coupling. Frequency analysis of single modes underscores DMD’s efficacy in interpreting complex flow phenomena. Based on this, the DMD modal coefficients were fitted to equations, clarifying the development process of wave-vortex coupling within the stirred reactors. Additionally, by integrating fundamental fluid mechanics equations with the outcomes of the DMD method, a wave-vortex coupling model was developed, which is anticipated to yield a more accurate representation of the flow field.

Chien-physics-informed neural networks for solving singularly perturbed boundary-layer problems

A physics-informed neural network (PINN) is a powerful tool for solving differential equations in solid and fluid mechanics. However, it suffers from singularly perturbed boundary-layer problems in which there exist sharp changes caused by a small perturbation parameter multiplying the highest-order derivatives. In this paper, we introduce Chien’s composite expansion method into PINNs, and propose a novel architecture for the PINNs, namely, the Chien-PINN (C-PINN) method. This novel PINN method is validated by singularly perturbed differential equations, and successfully solves the well-known thin plate bending problems. In particular, no cumbersome matching conditions are needed for the C-PINN method, compared with the previous studies based on matched asymptotic expansions.

A formulation of quantum fluid mechanics and trajectories

A formalism of classical mechanics is given for time-dependent many-body states of quantum mechanics, describing both fluid flow and point mass trajectories. The familiar equations of energy, motion, and those of Lagrangian mechanics are obtained. An energy and continuity equation is demonstrated to be equivalent to the real and imaginary parts of the time dependent Schrödinger equation, respectively, where the Schrödinger equation is in density matrix form. For certain stationary states, using Lagrangian mechanics and a Hamiltonian function for quantum mechanics, equations for point-mass trajectories are obtained. For 1-body states and fluid flows, the energy equation and equations of motion are the Bernoulli and Euler equations of fluid mechanics, respectively. Generalizations of the energy and Euler equations are derived to obtain equations that are in the same form as they are in classical mechanics. The fluid flow type is compressible, inviscid, irrotational, with the nonclassical element of local variable mass. Over all space mass is conserved. The variable mass is a necessary condition for the fluid flow to agree with the zero orbital angular momentum for s states of hydrogen. Cross flows are examined, where velocity directions are changed without changing the kinetic energy. For one-electron atoms, the velocity modification gives closed orbits for trajectories, and mass conservation, vortexes, and density stratification for fluid flows. For many body states, under certain conditions, and by hypotheses, Euler equations of orbital-flows are obtained. One-body Schrödinger equations that are a generalization of the Hartree–Fock equations are also obtained. These equations contain a quantum Coulomb’s law, involving the 2-body pair function of reduced density matrix theory that replace the charge densities.

Rapid and Precise Computation of Fractional Flow Reserve from Routine Two-Dimensional Coronary Angiograms Based on Fluid Mechanics: The Pilot FFR2D Study.

Objective: To present a novel pipeline for rapid and precise computation of fractional flow reserve from an analysis of routine two-dimensional coronary angiograms based on fluid mechanics equations (FFR2D). Material and methods: This was a pilot analytical study that was designed to assess the diagnostic performance of FFR2D versus the gold standard of FFR (threshold ≤ 0.80) measured with a pressure wire for the physiological assessment of intermediate coronary artery stenoses. In a single academic center, consecutive patients referred for diagnostic coronary angiography and potential revascularization between 1 September 2020 and 1 September 2022 were screened for eligibility. Routine two-dimensional angiograms at optimal viewing angles with minimal overlap and/or foreshortening were segmented semi-automatically to derive the vascular geometry of intermediate coronary lesions, and nonlinear pressure-flow mathematical relationships were applied to compute FFR2D. Results: Some 88 consecutive patients with a single intermediate coronary artery lesion were analyzed (LAD n = 74, RCA n = 9 and LCX n = 5; percent diameter stenosis of 45.7 ± 11.0%). The computed FFR2D was on average 0.821 ± 0.048 and correlated well with invasive FFR (r = 0.68, p < 0.001). There was very good agreement between FFR2D and invasive-wire FFR with minimal measurement bias (mean difference: 0.000 ± 0.048). The overall accuracy of FFR2D for diagnosing a critical epicardial artery stenosis was 90.9% (80 cases classified correctly out of 88 in total). FFR2D identified 24 true positives, 56 true negatives, 4 false positives, and 4 false negatives and predicted FFR ≤ 0.80 with a sensitivity of 85.7%, specificity of 93.3%, positive likelihood ratio of 13.0, and negative likelihood ratio of 0.15. FFR2D had a significantly better discriminatory capacity (area under the ROC curve: 0.95 [95% CI: 0.91-0.99]) compared to 50%DS on 2D-QCA (area under the ROC curve: 0.70 [95% CI: 0.59-0.82]; p = 0.0001) in predicting wire FFR ≤ 0.80. The median time of image analysis was 2 min and the median time of computation of the FFR2D results was 0.1 s. Conclusion: FFR2D may rapidly derive a precise image-based metric of fractional flow reserve with high diagnostic accuracy based on a single two-dimensional coronary angiogram.

Using Hydrodynamic Similarity as a Verification Method for Impact Cratering Simulations in the FLAG Hydrocode

Hydrodynamic codes (hydrocodes) are common tools for modeling hypervelocity impacts to provide insight into the physical phenomenon. Hydrocodes can simulate impacts from micrometer to kilometer spatial scales and reach impact velocities difficult to achieve in experimental settings. However, numerical models are approximations, and demonstrating that a numerical method is capable of providing physical results for these models is essential. In this work, we employ a hydrocode verification technique that leverages hydrodynamic similarity, a mathematical property of the conservation equations of fluid mechanics that form the basis for hydrocode models. Using the FLAG hydrocode, we simulate aluminum (Al) and basalt projectiles and targets at spatial scales spanning 7 orders of magnitude (hundreds of micrometers to kilometers). These materials were chosen because Al-6061 is a common material in spacecraft and satellites and basalt is a useful approximation of rocky astronomical bodies. Our results show that hydrodynamic similarity holds for each material model used and across spatial scales. We show that under certain conditions hydrodynamic similarity can apply in the presence of gravity and that similarity does not hold in the presence of strength models. We conclude that the FLAG hydrocode preserves important mathematical properties of fluid dynamics in hypervelocity impacts of Al-6061 and basalt.

Symmetry-based closed-form solutions and conserved quantities of the new (2+1)-dimensional Bogoyavlensky-Konopelchenko equation in fluid mechanics

In this investigation, we explore the mathematical intricacies of the novel (2+1)-dimensional Bogoyavlensky-Konopelchenko equation, a model with practical applications in elucidating the dynamics of internal waves within deep water. The equation’s significance spans various scientific domains, including plasma physics, nonlinear optics, and fluid dynamics. Employing a comprehensive analytical approach, specifically Lie symmetry analysis, we aim to unravel the underlying complexities of this equation and obtain new analytical solutions. To further scrutinize the equation, we apply various methods, namely Kudryashov’s method, the (G′/G)-expansion method, the simplest equation technique, and the power series method, all of which have not been applied to the equation before. Through these techniques, we successfully derive solutions in diverse functional forms, encompassing rational, trigonometric, exponential, hyperbolic, and Jacobi elliptic functions. To enhance comprehension, we present our findings visually using three-dimensional and two-dimensional plots density plots via the Mathematica tool. These graphical representations effectively communicate the intricate characteristics and nuances inherent in the solutions. Our visual representations reveal a spectrum of patterns, including periodic, singular periodic, kink-shaped structures. Additionally, our investigation extends to the determination of conserved quantities associated with the new (2+1)-dimensional Bogoyavlensky-Konopelchenko equation. This involves the application of the multiplier method and Ibragimov’s theorem, two potent techniques for identifying and understanding the conservation laws governing the model.

On a Hirota equation in oceanic fluid mechanics: Double-pole breather-to-soliton transitions

In oceanic fluid mechanics, a Hirota equation describing the propagation of the deep ocean broader-banded waves is studied in this paper. With respect to the envelope of the wave field, we simultaneously take the multi-pole phenomena and breather-to-soliton transitions into account to investigate the double-pole breather-to-soliton transitions via the second-order generalized Darboux transformation. Under certain parameter conditions, double-pole anti-dark solitons, periodic waves, W-shaped solitons and multi-peak solitons are derived, analyzed and graphically illustrated. We perform the asymptotic analysis on the double-pole anti-dark solitons to investigate their interaction properties, e.g., amplitudes, characteristic lines, slopes, phase shifts and position differences. Different from the known double-pole solitons of some other models, the double-pole anti-dark solitons, hereby, indicate that the two soliton components own unequal amplitudes.

Hydrodynamic density functional theory of simple dissipative fluids

In this paper, a statistical physical derivation of thermodynamically consistent fluid mechanical equations is presented for non-isothermal viscous molecular fluids. The coarse-graining process is based on (i) the adiabatic expansion of the one-particle probability density function around local thermodynamic equilibrium, (ii) the assumption of decoupled particle positions and momenta, and (iii) the variational principle. It is shown that there exists a class of free energy functionals for which the conventional thermodynamic formalism can be naturally adopted for non-equilibrium scenarios, and describes entropy monotonic fluid flow in isolated systems. Furthermore, the analysis of the general continuum equations revealed the possibility of a non-local transport mode of energy in highly compressible dense fluids.

On Λ-Fractional fluid mechanics

Λ-fractional analysis has already been presented as the only fractional analysis conforming with the Differential Topology prerequisites. That is, the Leibniz rule and chain rule do not apply to other fractional derivatives; This, according to Differential Topology, makes the definition of a differential impossible for these derivatives. Therefore, this leaves Λ-fractional analysis the only analysis generating differential geometry necessary to establish the governing laws in physics and mechanics. Hence, it is most necessary to use Λ-fractional derivative and Λ-fractional transformation to describe fractional mathematical models. Other fractional “derivatives” are not proper derivatives, according to Differential Topology; they are just operators. This fact makes their application to mathematical problems questionable while Λ-derivative faces no such problems. Basic Fluid Mechanics equations are studied and revised under the prism of Λ-Fractional Continuum Mechanics (Λ-FCM). Extending the already presented principles of Continuum Mechanics in the area of solids into the area of fluids, the basic Λ-fractional fluid equations concerning the Navier-Stokes, Euler, and Bernoulli flows are derived, and the Λ-fractional Darcy’s flow in porous media is studied. Since global minimization of the various fields is accepted only in the Λ-fractional analysis, shocks in the Λ-fractional motion of fluids are exhibited.

Integration of Energy Conservation Principles Across Mechanical Engineering Curriculum: A Work-in-Progress Project

The conservation of energy, mass, and momentum stands as fundamental laws in physics, resonating deeply within engineering education. This ongoing project, now in its third year of implementation, aims to seamlessly integrate energy conservation principles across the mechanical engineering undergraduate curriculum. Across various courses, including dynamics, fluid dynamics, and thermodynamics, students delve into the diverse forms of mechanical energy. From kinetic and potential energy in dynamics to fluid flow energies and thermodynamic principles in fluid dynamics and thermodynamics courses respectively, the overarching principle remains: energy is neither created nor destroyed but instead transfers between different forms, maintaining a constant total within a fixed domain. The project emphasizes the energy balance equation introduced in the first Thermodynamics course, laying the groundwork for subsequent exploration. Students are guided through the transition from the first law of thermodynamics to Bernoulli's equation in Fluid Mechanics, bridging theoretical concepts with practical applications. This integration continues into Heat Transfer and elective courses such as Industrial Hydraulics and Aerodynamics. To assess student comprehension, direct and indirect assessments are conducted, measuring understanding through principle and practical examples. Feedback and questionnaire responses indicate enhanced understanding of energy conservation principles through the synchronization of energy balance concepts across multiple courses. This presentation showcases the evolution of our work-in-progress project, initially presented at the 2022 and 2023 WVAS meetings. By fostering a holistic understanding of energy conservation principles, our aim is to empower students with a comprehensive foundation to tackle real-world engineering challenges.

Solitons, nonlinear wave transitions and characteristics of quasi-periodic waves for a (3+1)-dimensional generalized Calogero–Bogoyavlenskii–Schiff equation in fluid mechanics and plasma physics

A (3+1)-dimensional generalized Calogero–Bogoyavlenskii–Schiff equation describing many nonlinear phenomena in fluid dynamics and plasma physics is considered. The N-solitons and breathers are obtained by basing on its Hirota’s bilinear form and taking the complex conjugate condition on parameters of N-solitons. What is more, breathers can be transformed into a series of nonlinear localized waves by the mechanism of breather transformation. Then through the multi-dimensional Riemann-theta function and the bilinear method, the high-dimensional complex three-periodic wave solutions are constructed systematically, which are the generalization of one-periodic wave and two-periodic wave solutions. By a limiting procedure, the asymptotic relations between the quasi-periodic waves and solitons are strictly established. Additionally, a novel analytical method of characteristic line is introduced to analyze statistically the dynamical characteristics of the quasi-periodic waves. The analytical method employed in this paper can be further extended to investigate the other complex high-dimensional nonlinear integrable equations.

Lump–kink and hybrid solutions of the extended (3+1)-dimensional potential KP equation in fluid mechanics

In this paper, the extended (3+1)-dimensional potential KP equation in fluid mechanics is studied through Hirota bilinear method. Many types of hybrid solutions, such as the lump–kink solution, lump-two kink solution and periodic lump solution are obtained by assuming different functions in the bilinear equation. The interaction solution between lump and triangular periodic wave is also derived by combining sine and cosine functions with quadratic functions. Dynamical structures of these exact solutions are depicted by presenting the corresponding three-dimensional, two-dimensional structures and density graphs. These diverse interaction solutions could be helpful for understanding physical phenomena in fluid mechanics.

MHD boundary layer micropolar fluid flow over a stretching wedge surface: Thermophoresis and brownian motion effect

To investigate the consequence of thermophoresis and Brownian diffusion on convective boundary layer micropolar fluid flow over a stretching wedge-shaped surface. The effects of non-dimensional parameters namely coupling constant parameter (0.01 ≤ B1 ≤ 0.05), magnetic parameter (1.0 ≤ M ≤ 15.0), Grashof number (0.3 ≤ Gr ≤ 0.9), modified Grashof number (0.3 ≤ Gm ≤ 0.8), micropolar parameter (2.0 ≤ G2 ≤ 7.5), vortex viscosity constraint (0.02 ≤ G1 ≤ 0.2), Prandtl number (7.0 ≤ Pr ≤ 15.0), thermal radiation parameter (0.25 ≤ R ≤ 0.50), Brownian motion parameter (0.2 ≤ Nb ≤ 0.62), thermophoresis parameter (0.04 ≤ Nt ≤ 0.10), heat generation parameter (0.1 ≤ Q ≤ 0.5), Biot number (0.65 ≤ Bi ≤ 1.0), stretching parameter (0.2 ≤ A ≤ 0.5), Lewis number (3.0 ≤ Le ≤ 7.0), and chemical reaction parameter (0.2 ≤ K ≤ 0.7) on the steady MHD heat and mass transfer is investigated in the present study. The coupled non-linear partial differential equations are reduced into a set of non-linear ordinary differential equations employing similarity transformation. Furthermore, by using the Runge-Kutta method followed by the shooting technique, the transformed equations are solved. The main goal of this study is to investigate the numerical analysis of nanofluid flow within the boundary layer region with the effects of the microrotation parameter and velocity ratio parameter. The novelty of this paper is to propose a numerical method for solving third-order ordinary differential equations that include both linear and nonlinear terms. To understand the physical significance of this work, numerical analyses and tabular displays of the skin friction coefficient, Nusselt number, and Sherwood number are shown. The new approach of the present study contributes significantly to the understanding of numerical solutions to non-linear differential equations in fluid mechanics and micropolar fluid flow. Micropolar fluids are becoming even more of a focus due to the desire for engineering applications in various fields of medical, mechanical engineering, and chemical processing.

On the space-time analyticity of the inhomogeneous heat equation on the half space with Neumann boundary conditions

We consider the inhomogeneous heat equation on the half-space R + d with Neumann boundary conditions. We prove a space-time Gevrey regularity of the solution, with a radius of analyticity uniform up to the boundary of the half-space. We also address the case of homogeneous Robin boundary conditions. Our results generalize the case of homogeneous Dirichlet boundary conditions established by Kukavica and Vicol [A direct approach to Gevrey regularity on the half-space. In: Partial differential equations in fluid mechanics. Cambridge: Cambridge Univ. Press; 2018. p. 268–288. (London math. soc. lecture note ser.; vol. 452).].

Numerical model for simulating the hydraulic parameters of the aeration system ensuring equal oxygenation of the compost heap

The aim of the work was to develop a mathematical model using equations of fluid mechanics that describe the dynamics of air flow in a part of the compost aerating system integrated with a stationary reactor. The results of the simulation show that adjusting the flow resistance along the entire length of the compost aerating duct, depending on the distance from the connection of the duct with the fan's pressure conduit pipe through gradually increasing the air outflow area by increasing the number of repeatable gaps, yields a uniform pressure distribution above the grate. The process parameters used for computation were relevant to composting a subscreen fraction separated from mixed municipal waste using 80 mm mesh screen (Fr&amp;lt;80 mm) under real conditions. Microsoft EXCEL 2010 software and STATISTICA version 13.3 by StatSoft were used for numerical and statistical analysis of the test results. The research results are presented in four tables and five figures and discussed in the text of the article. During tests performed in real conditions, various variants were tested for reactor filling level and air outflow active surfaces in subsequent grate parts (Fc (i)). It was found that the target waste layer thickness i.e. 3.0 m and Fc (i) changes, in accordance with the values of the developed model, result in a stable pressure distribution pd, amounting to 1506 Pa and 1495 Pa at the grate front and end part.

Kinetic energy correction coefficient for rectangular drainage channels

In urban water supply and drainage systems, rainwater channels or pipes are rectangular in design to help control the flow rate and adapt well to limited space. When the Bernoulli equation in fluid mechanics is used to solve the head loss of rectangular pipelines, the velocity parameter used in the kinetic energy term is usually the instantaneous or average velocity of the section at a certain point. Given that this velocity parameter is in exponential form, the smaller the error is, the greater the impact on the result will be. Thus, the kinetic energy term must be corrected. This study focuses on establishing a cross section velocity distribution model in a rectangular pipe and deriving the kinetic energy correction coefficient through the velocity distribution. Based on the Navier–Stokes equation, the partial differential equation describing the velocity distribution is further refined and simplified. Combined with the boundary conditions of the pipeline, the method of separating variables and Fourier transform are used to solve the equation. An example shows how to establish the velocity distribution model and find the analytical solution. Finally, the analytical formula of the kinetic energy correction factor of different cross section parameters and fluid properties is derived. To verify the accuracy of the analytical formula, the Fluent numerical simulation software is used for empirical verification, and then the Deming regression method is used to analyze the error between the theoretical and experimental values. The regression results of the kinetic energy correction coefficient prediction model established in this study are consistent with the actual values, and the confidence interval reaches 95%. This work provides strong guidance for the prediction of the kinetic energy correction coefficient in fluid mechanics and has an important theoretical and practical value.

Interactions of certain localized waves for an extended (3+1)-dimensional Kadomtsev-Petviashvili equation in fluid mechanics

Currently, there is much interest in the study on the localized waves and their interactions in fluid mechanics. An extended (3+1)-dimensional Kadomtsev-Petviashvili equation is considered here. By means of the Hirota method, we obtain the N-soliton solutions, with N as a positive integer. The higher-order breather solutions are obtained from the N-soliton solutions through the complex conjugated transformations. Making use of the long-wave limit method, we determine the higher-order lump solutions via the N-soliton solutions. Besides, some hybrid solutions are presented. Three kinds of the localized waves, namely, the solitons, breathers and lumps, along with their interactions, are investigated via the above solutions. Amplitudes, shapes and velocities of those localized waves remain invariant after the interactions, which indicate that the interactions are elastic. Fluid-mechanically meaningful, all our results rely on the coefficients in that equation.

Numerical analysis of the SIS infectious disease model with spatial heterogeneity

PurposeThe susceptible-infectious-susceptible (SIS) infectious disease models without spatial heterogeneity have limited applications, and the numerical simulation without considering models’ global existence and uniqueness of classical solutions might converge to an impractical solution. This paper aims to develop a robust and reliable numerical approach to the SIS epidemic model with spatial heterogeneity, which characterizes the horizontal and vertical transmission of the disease.Design/methodology/approachThis study used stability analysis methods from nonlinear dynamics to evaluate the stability of SIS epidemic models. Additionally, the authors applied numerical solution methods from diffusion equations and heat conduction equations in fluid mechanics to infectious disease transmission models with spatial heterogeneity, which can guarantee a robustly stable and highly reliable numerical process. The findings revealed that this interdisciplinary approach not only provides a more comprehensive understanding of the propagation patterns of infectious diseases across various spatial environments but also offers new application directions in the fields of fluid mechanics and heat flow. The results of this study are highly significant for developing effective control strategies against infectious diseases while offering new ideas and methods for related fields of research.FindingsThrough theoretical analysis and numerical simulation, the distribution of infected persons in heterogeneous environments is closely related to the location parameters. The finding is suitable for clinical use.Originality/valueThe theoretical analysis of the stability theorem and the threshold dynamics guarantee robust stability and fast convergence of the numerical solution. It opens up a new window for a robust and reliable numerical study.

New in the hydraulics of channels with permeable walls: indicator of the relative value of the dissipation

The factors that have led to the creation of the scientific direction “hydraulics of variable mass” to study the fluid movement laws in channels with permeable walls are indicated. The results of applying of the dynamics of a variable mass point for describing the flow in such pipelines are presented. It is noted unjustifiability of the second Newton’s law generalization to the case of motion of a variable mass point for hydrodynamics problems. The functionality of one-dimensional and multidimensional models of fluid motion in permeable channels based on the classical equations of fluid and gas mechanics is characterized. It is substantiated the dominance of one-dimensional models in engineering computational practice, and a number of contradictions in the description of fluid dynamics are shown (with the flow visualization). On the base of the new kinematic image (instead of the generally accepted “solid jet”, when fluid particles are separated or joined), it has been obtained a one-dimensional equation of fluid motion in a permeable channel in which the friction coefficient is an indicator of the relative magnitude of the energy dissipation of the flow. The dependence of the Coriolis coefficient on the flow regime is constructed. The structure of the friction drag coefficient of a permeable channel has been studied using the vector dimension of length. It is shown that the dissipation of the flow energy in a permeable channel is higher both during outflow and inflow of liquid than in channels with solid walls at the same flow rates. The results are in demand in the development of chemical technology devices, nuclear reactors with microfuel elements, filters and heat exchangers containing channels with permeable walls.

Numerical Study of Possion Equation in Subsonic and Supersonic Airfoil Flow

In the context of subsonic and supersonic airfoil flows in fluid mechanics, this study aims to numerically simulate the Poisson equation. The focus is on analyzing the variation of fluid flow under stable conditions of incompressible fluid. The research classifies the flow into four distinct cases based on the Mach number and its constancy and employs Python for intricate numerical computation. The culmination of the study involves generating Mach number contour plots and streamlining diagrams for each case using Python, which visually depicts the nuanced evolution of the flow field. Moreover, the study conducts a rigorous comparative analysis of the simulation results for the different cases, exploring the intricate differences in flow behavior across a spectrum of Mach numbers and flow conditions. The discussion section further broadens the scope of inquiry by addressing potential limitations and charting out prospective research directions. By providing a comprehensive exploration of the widespread application of the Poisson equation in fluid mechanics, this paper stands as a beacon for further investigations in the domain and presents invaluable insights for the progressive advancement of fluid mechanics studies in the future.