Teaching Theology and Religion

Abstract We derive the incompressible Navier--Stokes equations from the lattice Boltzmann equation using the Chapman--Enskog expansion and the Sone expansion and clarify the differences between the two approaches. In the Chapman--Enskog expansion, we first derive the compressible Navier—Stokes equations on the multiple time scales (the acoustic and diffusive time scales). Then the incompressible Navier--Stokes equations are derived under the conditions of low Mach number flows and small density variations on the diffusive time scale. If the acoustic time scale remains in the analysis of the derived macroscopic equations, the incompressible Navier--Stokes equations are recovered with only first-order spatial accuracy. On the other hand, in the Sone expansion we can derive the incompressible Navier--Stokes equations under the condition of low Mach number flows on the diffusive time scale. Despite some differences between the two approaches, we obtain the same result that the flow velocity and the pressure satisfy the incompressible Navier--Stokes equations with the second-order spatial accuracy in low Mach number flows on the diffusive time scale. The accuracy is verified through simulating a generalized Taylor--Green problem.

Abstract In this study, the squirmer model with a prescribed tangential velocity is used as a model for swimming microorganisms where its geometric center is offset from the center of mass (bottom-heavy). The settling behavior and interactions of two bottom-heavy squirmers in a vertical channel are simulated numerically under low Reynolds number. Five settling modes, i.e. stable vertical settling, stable inclined settling, wall-attracting oscillatory, oscillatory, and chaotic motion are identified. In addition to the swimming Reynolds number Re s [0.1,1.0], density ratio γ [1.1,2.1], and swimming strength β [−7,7], another bottom-heavy parameter ER (the ratio of the distance from the center of mass to the geometric center relative to the radius, in the range of [0a 0,0.75a 0] is introduced. The effects of these parameters on the settling modes of bottom-heavy squirmers, terminal Reynolds number Re t, and interactions of the two bottom-heavy squirmers are discussed. The results showed that a pair of neutral bottom-heavy squirmers more easily achieved a stable structure at the channel center. In contrast, a pair of bottom-heavy pushers were more likely to be captured by the channel walls, leading to a stable structure near the walls. The stable symmetric structure of a pair of bottom-heavy pullers was disturbed, resulting in turbulence. Increasing the swimming strength β accelerates the settling of a pair of pushers. For different ER, the settling speed of two bottom-heavy pushers is greater than that of two bottom-heavy pullers. Additionally, the difference in settling speed between two bottom-heavy squirmers becomes more pronounced with an increase in Re s. As γ increases, the settling behavior of bottom-heavy squirmers with high β differs from that of those with low β. Moreover, Re t of a pair of pushers gradually approaches that of neutral bottom-heavy squirmers.

Abstract In this article, three-dimensional lid-driven flows in cavities with a semicircular round bottom are studied. We have first focused on lid-driven flow in a semicircular cavity with a unit square moving lid and height 1/2. The critical Reynolds number R e cr for the transition from steady flow to unsteady one has been obtained. Based on the averaged velocity field in one cycle of fluid flow motion, the flow difference between the averaged one and velocity field, called oscillation mode, at several time instances in such cycle shows an almost identical pattern for several Reynolds numbers close to R e cr . This similarity indicates the oscillation mode associated with the Hopf bifurcation originated at Re less than R e cr . For lid-driven flow in a cavity with a semicircular round bottom and height one, its oscillation mode shows a periodic change of local secondary flows associated with Hopf bifurcation and pairs of Taylor–Görtler-like vortices are obtained.

Abstract The Darcy-Forchheimer model extends Darcy's law to account for nonlinear inertial effects in fluid dynamics. This paper integrates fuzzy logic and variable exponent sequence spaces into the model to address the inherent uncertainties and heterogeneities in porous media. Through Fredholm operator theory and fixed-point theorems, we establish the existence and uniqueness of solutions to the nonlinear system. Numerical investigations highlight the impact of varying porous media conditions on fluid behavior. As a main outcome, we observe that solutions exhibit a smooth behaviour for moderate levels of permeability and porosity in the media, while for increased fuzzy norms in both media properties, the fluid velocity profile exhibits fluctuations due to the influence of the fuzzy terms.

Abstract The flow of a film induced by gravity is not only widespread in nature but also has significant applications in various industrial
technologies such as coating techniques, nanotechnology, microfluidic chips, and heat exchangers. The film exhibits various nonlinear dynamic phenomena due to the interactions between surface tension, gravity, and other forces, making the study of this type of flow of great importance. However, the theoretical derivation of thin film fluid dynamics is complex, with diverse working conditions, making numerical solutions difficult, time-consuming, and labor-intensive. Therefore, it is of significant importance to seek an approach that differs from theoretical derivation or numerical solutions for the study of thin film fluid dynamics. With the rapid development of deep learning, this paper employs physics-informed neural networks (PINNs) algorithm, in conjunction with the partial differential equation (PDE) governing the fluid film thickness, to conduct research on forward prediction of the film thickness variation over time and space, and inverse problem solving to determine unknown parameters in the governing equation from data. The study investigates the governing equation for the thickness of a falling film along an inclined plane and applies the PINN method to solve it. The research predicted the film thickness for three different characteristic waveforms and conducted a comparative analysis with solutions obtained using the commercial software COMSOL. Additionally, the unknown parameters in the governing equation were inversely solved using limited data, and the differences in prediction results with and without noisy data were compared.

Abstract The onset of phototaxis-driven bioconvection in an anisotropic (forward) scattering algal suspension illuminated from above by diffuse/scattered sunlight is investigated in the proposed work. Linear stability analysis is performed to investigate the onset of bioconvection in the proposed study and the resulting eigenvalue problem is solved using a fourth-order accurate, finite-difference scheme based on the Newton Raphson Kantorovich iteration. The study demonstrates that forward scattering enhances suspension stability, energy transfer to deeper regions resulting significant biomass contribution and variation in it allows the bioconvective solution to shift from mode 1 to mode 2 at fluid dynamic instability. Moreover, the bioconvective flow patterns of the proposed model via perturbed algal concentration are implicated in key ecological phenomena, including blooms. The findings of this study show some resemblance to gyrotactic bioconvection via dismissal of the collimated beam. Furthermore, the outcomes of the proposed work include evidence of some interesting phenomena, such as the existence of limit cycles (and/or orbits) via bifurcation analysis.

Abstract This article explores the steady and uniform motion of two spherical particles submerged in an infinite fluid with Brinkman-couple stress properties, where the particles are entirely immersed and surface slippage is disregarded. While the particles may vary in size, they are composed of the same material and move along the line that connects their centers, influenced by different velocities or external forces. The analysis is performed under low Reynolds number conditions, which signify laminar flow dominated by viscous forces. Using boundary collocation methods, semi-analytical solutions to the governing differential equations are derived, facilitating a thorough investigation of the system. The study employs normalized drag force to evaluate the effects of particle interactions, presenting calculated values graphically and discussing them with several key factors, such as particle size and separation distance, thereby enhancing the understanding of particle dynamics in complex fluid environments. The analysis of how varying the permeability parameter affects flow patterns around two spheres in a couple of stress fluids is shown.

Abstract The effects of pivot location ( x p / c ) and spacing ratio ( λ = d s / c ) on enhancing the lift generated by rigid biplane NACA 0012 airfoils at R e = 1.35 × 10 5 are studied through the numerical method. SST k − ω turbulence model is adopted to solve URANS equations with overset grids. Results reveal that C L hysteresis exist phase lag under five fixed pivot locations x p / c = 0 , 0.25, 0.5, 0.75 and 1 when biplane airfoils tilt just like pitching stroke. C M curves drop regularly as the pivot point moves towards the spacing direction. However, λ decreasing plays a important role in aerodynamics: C L , lower curve family significantly collapse and C L , upper hysteresis rise overall owing to intensified low-pressure distribution in the gap. Additionally, these performances can be explained from the view of vorticity evolution, particularly, high angles of attack around the stall induce violent turbulence structures. With rearward movement of the pivot point, resemble vortex details appear in the further upstream, indicating aerodynamic characteristics have high similarity under different x p / c regardless of λ. And thus a concept of effective angle of attack for biplane airfoils is proposed where a new factor λ must be introduced. Data assimilation results and vorticity contours are exhibited to prove the feasibility of α eff .

Abstract We investigated the inertial migration of deformable particles in a two-dimensional channel flow. This study analyzes the effects of the channel Reynolds number (Re), channel blockage ratio (k), particle number (Np ) and reduced bending modulus (Eb ) on the formation of staggered particle trains. The results show that the stable normalized distance d p e q / H between two staggered particles is influenced by Re, k and Eb , where H is the channel width. As k increases or Eb decreases, d p e q / H decreases. The value of d p e q / H initially increases and then decreases with the increase of Re; when Eb is large and k is small, d p e q / H continuously increases with increasing Re. With the increase of Np , the closely arranged staggered particle trains evolve into five distinct migration Regimes. We explain the conditions for the formation of each Regime and explore the mechanisms of their interconversion. The findings of this study contribute to a better understanding of the self-organization process of deformable particles in channel flow.

Abstract An analytic solution of Navier–Stokes flow past a sphere in the region of intermediate Reynolds number

Abstract The free fall of an ellipse in an infinite linearly-stratified fluid is investigated using a linear two-dimensional, Boussinesq, diffusionless, inviscid model. The oscillations of the ellipse decay because of radiation damping, but unlike the case of a circular cylinder, the ellipse can also rotate and move horizontally. The resulting equations are solved analytically for some simple cases, for which there is little or no rotation. Motions with rotation are studied numerically using a spectral method to solve for the wave field in the fluid.

Abstract The lift generation mechanisms of dragonflies have been extensively and deeply studied. As research advances, controlling the lift coefficient faces significant challenges. How the lift coefficient varies and whether a unified model can predict lift tendency remains unresolved. In this study, we propose a flapping amplitude partial advanced model (FAPAM) to control the linear variation of lift in dragonfly vertical flight. The FAPAM model can predict the average lift coefficient by the spatial plane and control the dragonfly lift coefficient over a large range. In this model, the maximum lift coefficient is 2.01 times higher than the weight of a dragonfly. The control parameters of the FAPAM are flapping amplitude (FA) and partial lead percent (PLP). Any linear combination of FA and PLP ensures a linear variation of the average lift coefficient. When FA and advanced rotation angle (ARA) increase by one degree, respectively, the increased lift coefficient of FA is 5.38~9.52 times higher than that of ARA, which is closely related to the leading-edge vortex, trailing-edge vortex, and positive pressure zone. The FAPAM model seamlessly integrates vertical ascending mode and vertical climbing mode by introducing transition mode. Additionally, FAPAM can effectively simulate the lift coefficient required for the vertical undulating motion of dragonflies during their oviposition process on water. Most importantly, the FAPAM model can maximize the energy efficiency of different motion modes.

Abstract Based on the original Miller-Robert-Sommeria theory, we explicitly compute a statistical equilibrium of two-dimensional turbulent flow on a sphere for a generic initial vorticity field introduced in a previous study. The macroscopic vorticity field corresponding to the obtained statistical equilibrium has a quadrupole structure. The resulting quadrupole structure is topologically consistent with the final state of the long-term time integration of the vorticity equation. However, the statistical equilibrium does not predict the formation of concentrated vortices as seen in the time integration. We also calculate statistical equilibria for the initial vorticity field with a planetary vorticity term, and find a change of statistical equilibria from quadrupole states to zonally symmetric states as the angular velocity of the sphere increases. The quadrupole statistical equilibria show nearly linear relations between the macroscopic vorticity and the macroscopic stream function, implying that higher-order Casimir invariants are virtually ineffective even when all Casimir invariants are considered. The discrepancy between the equilibria and the time integration results emphasizes the importance of mixing barriers, which prevent the relaxation of the evolving vorticity field to the statistical equilibria and allow the point-vortex-like dynamics of coherent vortices to persist.

Abstract The investigation focuses on the Maxwell–Cattaneo (MC) effect on the thermal convection instability in a vertical porous layer saturated with an Oldroyd-B fluid. The MC effect modifies the conventional Fourier’s law for temperature by incorporating the upper convective Oldroyd derivative. The flow through the porous layer is modeled by the Darcy‒Oldroyd model. Using the Chebyshev collocation method addresses an Orr-Sommerfeld eigenvalue problem. Analysis of temporal growth rates reveals that the MC effect causes the originally stable flow to become unstable. Furthermore, the study finds double impacts of the MC effect on convection instability depending on whether it is primarily influenced by the fluid or the solid phase. Neutral stability curves highlight a critical threshold for the averaged Cattaneo number (Ca ) of both solid and fluid. When Ca falls below this critical value, instability is suppressed, but when it exceeds this value, instability is magnified. The analysis also reveals that viscoelasticity parameters can impact the system by either stabilizing or destabilizing it. A rise in the retardation time parameter (λ 2) exerts a stabilizing influence, whereas an increase in the relaxation time parameter (λ 1) exhibits a destabilizing effect.

Abstract Microfluidic devices are increasingly valuable for their compact size and ability to handle tiny fluid volumes, making efficient mixing at this scale (micromixing) a critical focus. This research aims to optimize micromixer geometries to improve mixing efficiency while controlling pressure drop, providing a method that balances performance and computational cost. Building on previous work, we introduce a novel optimization framework in microfluidics combining computational fluid dynamics (CFD) and machine learning (ML) techniques, particularly Gaussian process (GP) modeling with Genetic Algorithm (GA) optimization. Inspired by a Y-type micromixer design with cylindrical grooves on the main channel’s surface and internal obstructions, our study examines the impact of circular obstructions on mixing percentage and pressure drop under varying obstruction diameter and offset. Simulations conducted using OpenFOAM software generate data for a reduced-order GP model, which provides model uncertainty. The geometry is then optimized using the GA algorithm on the reduced model. Results indicate that medium-sized obstructions (137 mm diameter, 10 mm offset) near the channel wall achieve optimal mixing and pressure performance, closely aligning with previous studies. The uncertainties, recorded as 3.9% and 21.5% for mixing percentage and pressure drop, respectively, further validate the robustness of our model. This study highlights an effective, uncertainty-quantified optimization process that leverages CFD and ML integration, setting a foundation for efficient microfluidic design strategies.