Navier-Stokes Equations Of Fluid Dynamics Research Articles

The Effect of Porous Media Grain Size on non-Darcy Flow Behavior using Pore Scale Simulation

The impact of grain size of the porous media of the 3D images of a beadpack on the permeability values and non-Darcy flow behaviour is analyzed using pore-scale flow modelling utilizing the finite volume method. In this study, the sample used was a 3D image from random tight packing of spheres (beadpack) with a size of 250×250×250 voxels and a porosity of 0.36. Variation in grain size is carried out by upscaling the 3D image sample, so that this treatment affects the specific surface area value but does not change the porosity and tortuosity of the sample. Several variations of grain diameter include 100 μm, 125 μm, 150 μm, 175 μm, and 200 μm. In this study, we adopted PIMPLE algorithm, which combines SIMPLE and PISO algorithms to accomplish the Navier-Stokes equations of fluid dynamics in such porous media. The simulation results show that the permeability value exhibits an inverse relationship with the specific surface area. This correlation is in accordance with the Kozeny-Carman equation. In addition, the initiation of non-Darcy flow occurs at the reference length-based Reynolds number of 2.06 and the permeability-based Reynolds number of 0.049.

Numerical Simulation for Cooling of Integrated Toroidal Octagonal Inductor Using Nanofluid in a Microchannel Heat Sink

This paper presents a comprehensive numerical simulation study focused on the cooling of integrated toroidal octagonal inductor using nanofluids within a microchannel heat sink. The investigation utilizes COMSOL Multiphysics 6.0 integrated with the Fluid Flow and Conjugate Heat Transfer Module. The primary objective is to explore and understand fluid flow and heat transfer characteristics within the integrated inductor. The study involves testing three distinct fluids, water, CuO-water nanofluid, and Al2O3-water nanofluid, under laminar flow conditions within microchannels. The choice of fluid plays a significant role in heat transfer, interacting with the microchannel geometry to optimize performance. Three-dimensional computational fluid dynamics (CFD) models are meticulously developed; focusing on toroidal inductors equipped with micro pin fins heat sinks. The study commences by detailing the geometry of the micro coil and the integrated heat sink. The simulation encompasses a mathematical model that captures the intricate interplay between the governing Navier-Stokes equations for fluid dynamics and the heat transfer equations within the integrated inductor. As φ increases, temperature, viscosity, and pressure decrease. CuO-water and Al2O3-water nanofluids play a significant role in influencing laminar flow and key thermal parameters in the toroidal inductor. These nanofluids, which consist of base fluids (water) with dispersed nanoparticles (CuO or Al2O3), are employed as cooling agents to enhance heat transfer. The presence of nanoparticles in the fluid alters its thermal properties, leading to changes in the flow dynamics and overall heat dissipation within the toroidal inductor.The laminar flow characteristics are affected by the nanofluid's viscosity, density, and thermal conductivity. Additionally, the Nusselt number, Reynolds number, and thermal resistance are key thermal parameters that reflect the performance of the cooling system. The nanofluid's influence on these parameters is crucial for understanding and optimizing the thermal management of the integrated toroidal inductor. The enhancement of heat dissipation in the toroidal inductor is achieved through improved thermal properties of the nanofluid. Higher nanoparticle concentrations result in better heat transfer rates, leading to lower temperatures in the toroidal inductor. This, in turn, improves the overall efficiency and performance of the cooling system. The viscosity of the nanofluid is influenced by the presence of nanoparticles. The pressure within the microchannels is also affected by the nanoparticle concentration. An increase in φ can lead to changes in pressure drop along the microchannels. Understanding these variations is crucial for designing an effective cooling system.

Accounting for adsorption and desorption in lattice Boltzmann simulations

We report a Lattice-Boltzmann scheme that accounts for adsorption and desorption in the calculation of mesoscale dynamical properties of tracers in media of arbitrary complexity. Lattice Boltzmann simulations made it possible to solve numerically the coupled Navier-Stokes equations of fluid dynamics and Nernst-Planck equations of electrokinetics in complex, heterogeneous media. With the moment propagation scheme, it became possible to extract the effective diffusion and dispersion coefficients of tracers, or solutes, of any charge, e.g., in porous media. Nevertheless, the dynamical properties of tracers depend on the tracer-surface affinity, which is not purely electrostatic and also includes a species-specific contribution. In order to capture this important feature, we introduce specific adsorption and desorption processes in a lattice Boltzmann scheme through a modified moment propagation algorithm, in which tracers may adsorb and desorb from surfaces through kinetic reaction rates. The method is validated on exact results for pure diffusion and diffusion-advection in Poiseuille flows in a simple geometry. We finally illustrate the importance of taking such processes into account in the time-dependent diffusion coefficient in a more complex porous medium.

Interactions between internal forces, body stiffness, and fluid environment in a neuromechanical model of lamprey swimming.

Animal movements result from a complex balance of many different forces. Muscles produce force to move the body; the body has inertial, elastic, and damping properties that may aid or oppose the muscle force; and the environment produces reaction forces back on the body. The actual motion is an emergent property of these interactions. To examine the roles of body stiffness, muscle activation, and fluid environment for swimming animals, a computational model of a lamprey was developed. The model uses an immersed boundary framework that fully couples the Navier-Stokes equations of fluid dynamics with an actuated, elastic body model. This is the first model at a Reynolds number appropriate for a swimming fish that captures the complete fluid-structure interaction, in which the body deforms according to both internal muscular forces and external fluid forces. Results indicate that identical muscle activation patterns can produce different kinematics depending on body stiffness, and the optimal value of stiffness for maximum acceleration is different from that for maximum steady swimming speed. Additionally, negative muscle work, observed in many fishes, emerges at higher tail beat frequencies without sensory input and may contribute to energy efficiency. Swimming fishes that can tune their body stiffness by appropriately timed muscle contractions may therefore be able to optimize the passive dynamics of their bodies to maximize peak acceleration or swimming speed.

Differential geometry based multiscale models.

Large chemical and biological systems such as fuel cells, ion channels, molecular motors, and viruses are of great importance to the scientific community and public health. Typically, these complex systems in conjunction with their aquatic environment pose a fabulous challenge to theoretical description, simulation, and prediction. In this work, we propose a differential geometry based multiscale paradigm to model complex macromolecular systems, and to put macroscopic and microscopic descriptions on an equal footing. In our approach, the differential geometry theory of surfaces and geometric measure theory are employed as a natural means to couple the macroscopic continuum mechanical description of the aquatic environment with the microscopic discrete atomistic description of the macromolecule. Multiscale free energy functionals, or multiscale action functionals are constructed as a unified framework to derive the governing equations for the dynamics of different scales and different descriptions. Two types of aqueous macromolecular complexes, ones that are near equilibrium and others that are far from equilibrium, are considered in our formulations. We show that generalized Navier-Stokes equations for the fluid dynamics, generalized Poisson equations or generalized Poisson-Boltzmann equations for electrostatic interactions, and Newton's equation for the molecular dynamics can be derived by the least action principle. These equations are coupled through the continuum-discrete interface whose dynamics is governed by potential driven geometric flows. Comparison is given to classical descriptions of the fluid and electrostatic interactions without geometric flow based micro-macro interfaces. The detailed balance of forces is emphasized in the present work. We further extend the proposed multiscale paradigm to micro-macro analysis of electrohydrodynamics, electrophoresis, fuel cells, and ion channels. We derive generalized Poisson-Nernst-Planck equations that are coupled to generalized Navier-Stokes equations for fluid dynamics, Newton's equation for molecular dynamics, and potential and surface driving geometric flows for the micro-macro interface. For excessively large aqueous macromolecular complexes in chemistry and biology, we further develop differential geometry based multiscale fluid-electro-elastic models to replace the expensive molecular dynamics description with an alternative elasticity formulation.

Mean Field Theory for Non-Abelian Gauge Theories and Fluid Dynamics: A Brief Progress Report

We review the long standing problem of ‘mean field theory’ for non-abelian gauge theories. As a consequence of the AdS/CFT correspondence, in the large N limit, at strong coupling, and high temperatures and density, the ‘mean field theory’ is described by the Navier-Stokes equations of fluid dynamics. We also discuss and present results on the nonconformal fluid dynamics of the D1 brane in 1 + 1 dim.

Large Scale Stochastic Dynamics

“Large Scale Stochastic Dynamics” is at the crossroad of probability theory and statistical physics. One central theme of statistical physics is the emergent behavior resulting from the interaction of many identical components, the paradigm being a fluid or a gas. On the atomistic scale they consist of a huge number of identical molecules. Their motion is governed by Newton's equations of classical mechanics (ignoring quantum effects). The emergent description, valid only for particular initial states and on a sufficiently coarse space-time scale, are the compressible Navier-Stokes equations of fluid dynamics. Roland Dobrushin (1929–1995) and Frank Spitzer (1926–1992) had the vision that in the context of stochastic dynamics with many identical components the issue of emergent behavior is both mathematically challenging and important in modelling applications. The latter judgement turned out to be more than true. Stochastic algorithms, such as kinetic Monte Carlo, importance sampling, Monte Carlo Markov chains, Glauber dynamics, and others, are daily practice. Their mathematical vision has evolved over the past twenty years into a rich, multifaceted research program. Our workshop is like a snap-shot of the current activities. A partial list of topics reads We had 49 participants from 11 countries, mostly probabilists, but also experts from partial differential equations and statistical physics. They all enjoyed tremendously the unique and stimulating atmosphere at the Mathematische Forschungsinstitut Oberwolfach and hope to return some day.

Numerical simulation of the landslide-induced tsunami of 1988 on Vulcano Island, Italy

On 20 April 1988 a landslide of approximately 200,000 m3 occurred on the northeastern flank of the volcano La Fossa on the island of Vulcano. The landslide fell into the sea, producing a small tsunami in the bay between Punte Nere and Punta Luccia that was observed locally in the neighbouring harbour called Porto Levante. The slide occurred during a period of unrest at the volcano that was monitored very accurately. The study of this event is composed of two parts, the simulation of the landslide and the simulation of the ensuing tsunami; the former is studied by means of a Lagrangian-type numerical model in which the landslide is seen as a multibody system, an ensemble of material-deforming blocks interacting together during their motion; the latter is simulated according to the Eulerian view by solving the shallow-water approximation to Navier-Stokes equations of fluid dynamics, with the incorporation of a forcing term depending on the slide motion. Technically, the slide evolution is computed first, and this result is then used to evaluate the excitation term of the hydraulic equations and to calculate the tsunami propagation. Computed wave fronts radiate both toward the open sea, with rapid amplitude decay, and along the shore, in the form of edge waves that lose energy slowly. Comparison between model outputs and observations can be carried out only in a qualitative way owing to the absence of tide-gauge records, and results are satisfactory.

The effect of the formulation of nonlinear terms on aliasing errors in spectral methods

The effect on aliasing errors of the formulation of nonlinear terms, such as the convective terms in the Navier-Stokes equations of fluid dynamics, is examined. A Fourier analysis shows that the skew-symmetric form of the convective term results in a reduced amplitude of the aliasing errors relative to the conservative and nonconservative forms. The three formulations of the convective term are tested for Burgers' equation and in large-eddy simulations of decaying compressible isotropic turbulence. The results for Burgers' equation show that, while in certain cases the nonconservative form has the lowest error, the skew-symmetric form is the most robust. For the turbulence simulations the skew-symmetric form gives the most accurate results, consistent with the error analysis.

Theory of friction and boundary lubrication.

When an external force acts on an adsorbate structure, the structure may slide or flow relative to the substrate. The mechanism behind this sliding motion is of fundamental importance for the understanding of friction and lubrication between two flat macroscopic surfaces, and is also related to the question of what boundary condition should be used for the velocity field at a solid-liquid interface when solving the Navier-Stokes equations of fluid dynamics. Here I study the friction which occurs when adsorbate structures slide on surfaces. I present results of simulations based on Langevin or Brownian-motion dynamics, where the dependence of the linear sliding friction on the temperature and on the coverage is studied. I also present results for the nonlinear (in the external driving force) sliding friction, which is found to exhibit hysteresis giving rise to the well-known phenomenon of ``stick-and-slip'' motion. The theory predicts that for a class of sliding systems the ratio ${\mathit{f}}_{\mathit{k}}$/${\mathit{f}}_{\mathit{s}}$ between the kinetic and the static friction coefficient should equal 1/2, in good agreement with experimental data.

Implementation of a three-dimensional Navier-Stokes code on the Symult Series 2010

The results of implementing a Navier-Stokes flow solver on the Symult Systems Series 2010 parallel processor are presented. The code solves the three-dimensional unsteady compressible Navier-Stokes equations of fluid dynamics for flow past arbitrary bodies, using an explicit finite-volume multi-stage Runge-Kutta time-stepping scheme. The implementation strategy on the Series 2010 parallel processor is discussed, and code-conversion issues such as domain decomposition and boundary condition implementation are high-lighted. The performance of the code is evaluated by calculating the transonic laminar viscous flow over a wing.