Navier-Stokes Momentum Equations Research Articles

Study of multilayer flow of two immiscible nanofluids in a duct with viscous dissipation

Numerical simulations for the mixed convective multilayer flow of two different immiscible nanofluids in a duct with viscous heating effects were performed in this study. The left and right faces of the duct are maintained to be isothermal, while other side faces are insulated. The mathematical governing system for each layer consists of an incompressibility condition equation, the Navier–Stokes momentum equation, and the conservation of energy equation. At the interface of the immiscible layer, the continuity of velocity, shear stress, temperature, and heat flux are considered. The dimensionless equations governing each layer were numerically integrated using the finite difference method and the Southwell-over-relaxation method. A mesh independence test is conducted. Furthermore, a parametric study is performed to analyze how the different nanoparticle volume fractions and viscous heating affect the transport characteristics of engine oil–copper and mineral oil–silver nanofluids. The study also examined the effects of various types of nanoparticles and base fluids. The results demonstrated that heat transport could be efficiently controlled by considering the viscous heating aspect. Moreover, the effects of different nanoparticles on heat transport were found to be more significant than those of base fluids. Finally, a point-wise comparison of our numerical results demonstrates a good agreement with existing studies in the literature.

The Rayleigh–Bénard problem for water with maximum density effects

Linear stability and weakly nonlinear stability analyses are developed for Rayleigh–Bénard convection in water near 3.98 °C subject to isothermal boundary conditions. The density–temperature relationship (equation of state) is approximated by a cubic polynomial, including linear, quadratic, and cubic terms. The continuity equation, the Navier–Stokes momentum equation, the equation of state, and the energy equation constitute the governing system. Linear stability analysis is used to investigate how the maximum density property of water affects the onset of convective instability and the choice of unstable wave number for four different types of boundary conditions. Then, a weakly nonlinear stability study is done using the spectral Fourier method for isothermal tangential stress-free boundary conditions to quantify the heat transport of the system and demonstrate the transition from regular/periodic convection to chaotic convection. A Stuart-Ginzburg–Landau equation is obtained using the multiscale expansion method. Streamlines and isotherms are presented and analyzed. The influence of maximum density has been shown to delay the onset of instability and is, therefore, a stabilizing mechanism for thermal instability. Due to the maximum density, the onset of chaotic convection is also delayed. Among four different boundaries, the impermeable rigid boundaries require the highest Rayleigh number for instability to begin. Increasing boundary temperatures advance the onset of chaotic convection and improve the heat transport situation.

Numerical validation of Gassmann’s equations

Derived several decades ago, Gassmann’s equations continue to be widely used in applied geophysics. Gassmann’s equations allow us to calculate the elastic moduli of a fully saturated rock from dry rock moduli knowing the porosity, fluid bulk modulus, and bulk modulus of the solid grains. These equations are treated as exact in the scientific community, but there is a lack of comprehensive numerical validation. Furthermore, recently several publications have appeared in the literature postulating a logical error in the derivation of Gassmann’s equations. Therefore, I develop a numerical validation of Gassmann’s equations. For that, I use a 3D finite-element approach to resolve the conservation of linear momentum that is coupled with the stress-strain relations for the solid phase and the quasistatic linearized compressible Navier-Stokes momentum equation for the fluid phase. Finally, a convergence study validating the correctness of Gassmann’s equations for a particular yet arbitrarily chosen “generic” pore geometry is presented. The arbitrary model geometry is simple as compared with real rocks; however, it is sufficiently complex with elements resembling wider pore bodies and narrower pore throats to, in general, validate Gassmann’s equations. MATLAB routines to reproduce the presented results are provided.

An investigation of flow characteristics between different mediums

In this research topic, two-dimensional tunnels made of different mediums (including the contrast of gas-solid condition and solid-solid condition and the contrast of solid-solid condition with or without a bump) are designed to compare how they influence the velocity and the pressure of the fluid when the fluid is flowing through them. The Navier-Stokes momentum equation is used to guide the project and set up the equations of motion for these models. At the same time, Newton’s law of viscosity is applied to set the boundary conditions. Numerical methods involving finite differences are used to help discretize those equations and simulate the models in computers. The result shows that the velocity and the pressure of the fluid are influenced significantly by different tunnels settings. A nozzle with or without a bump can also significantly affect the velocity and pressure of the fluid. As the materials of tunnels change, the dependence of these flow characteristics in terms of x and y-direction will change, and the shape of the tunnel will also influence these traits.

Corotational Hookean Models of Dilute Polymeric Fluids: Existence of Global Weak Solutions, Weak-Strong Uniqueness, Equilibration, and Macroscopic Closure

We prove the existence of global weak solutions to the corotational Hookean dumbbell model, a system of PDEs arising in the kinetic theory of dilute polymers, involving the unsteady incompressible Navier–Stokes equations in a bounded domain coupled to a Fokker–Planck type parabolic equation including a center-of-mass diffusion term, satisfied by the probability density function, modeling the evolution of the configuration of noninteracting polymer molecules in a viscous incompressible solvent. The micro-macro interaction is manifested by the presence of a corotational drag term in the Fokker–Planck equation and the divergence of a polymeric extra-stress tensor on the right-hand side of the Navier–Stokes momentum equation. We also analyze certain properties of weak solutions to this system of PDEs: we use the relative energy method to deduce a weak-strong uniqueness type result, and derive the macroscopic closure of the kinetic model; a corotational Oldroyd-B model with stress-diffusion. Finally, we discuss the existence and uniqueness of global weak solutions to this class of corotational Oldroyd-B models with stress-diffusion.

Study of flow of Buongiorno nanofluid in a conical gap between a cone and a disk

The cone–disk apparatus consists of a cone that touches the disk at its apex and is used in medical evices, viscosimeters, conical diffusers, etc. Theoretically, a three-dimensional flow of a nanofluid in a conical gap of a cone–disk apparatus is studied for four different physical configurations. Buongiorno nanofluid model, consisting of thermophoresis and Brownian diffusion mechanisms, is used to describe the convective heat transport of the nanofluid. The continuity equation, the Navier–Stokes momentum equation, the heat equation, and the conservation of nanoparticle volume fraction equation constitute the governing system for the flow of nanofluids. The Lie group approach is used to obtain self-similar equations. Solutions are computed for an appropriate rotational Reynolds number and four different gap angles to examine flow, mass, and heat transport features. The skin friction coefficients and torque are computed and analyzed. Multivariate nonlinear regression analysis is also performed. A co-rotating disk and cone configuration has been shown to produce less torque due to the increased centrifugal force. Of the four cone–disk apparatus configurations, the maximum heat/mass transport occurs for a rotating disk with a static cone for all selected gap angles, and the least drag in the radial direction is attained for a rotating cone with a static disk. In addition, there is a minimal drag along the tangential direction for the counter-rotating disk and cone configuration. Brownian diffusion and thermophoresis of the nanoparticles lead to a higher fluid temperature and, thus, lower Nusselt numbers are obtained.

Numerical simulation of multiphase oil behaviors in ice-covered nearshore water

There has been an increase in marine transportation in cold regions, which in turn has led to an increasing risk of oil spills in these areas. To better support risk assessment and pollution control of oil spills, it is important to have a good understanding of oil transport in the environment. This information is essential to manage response priorities and help prepare contingency and mitigating measures. This study aims to simulate 3D wave propagation in shallow water with different broken-ice aerial coverage percentages to assess the fate and transport of oil spill in a nearshore area under different conditions. Based on the Reynolds-averaged Navier-Stokes momentum equations for an incompressible viscous fluid and the Volume of Fluid (VOF) method that is coupled with Six Degree of Freedom (6-DOF) model, a 3D numerical model of three-phase transient flow was developed. It was found that the presence of ice makes the spreading of spilled oil slower in the horizontal direction since the ice can build natural barriers to oil movement. The higher the ice concentration, the slower spilled oil migrates in all directions. The maximum oil volume fraction varies with increasing ice coverage on the water surface area. The wave frequency, the averaged flow velocity, and oil properties affect the oil spread extent and the oil volume fraction. The dumping effect of the wave due to the presence of ice makes the impact of this factor less critical than those in open water.

A Low Prandtl Number Haemodynamic Oscillatory Flow through a Cylindrical Channel using the Power Series Method

This article details the use of the power series method to solve a haemodynamics problem in a cylindrical channel with a low Prandtl number. The process involves modifying the Navier-Stokes momentum equation and energy equation with radiation absorption to represent flow through a cylindrical channel; the governing models are made dimensionless with the help of some dimensionless quantities; and the flow is subjected to no-slip boundary conditions. It is true that the flow through biological vessels is thought to be oscillatory due to the pulsatile nature of the heart. The solutions were thought to be associated with an oscillatory frequency term. The dimensionless models were perturbed using the oscillatory term, and the partial differential equations were reduced to ordinary differential equations. Wolfram Mathematica, version 12, was used to code the analytical solutions, which included biophysical parameters such as the Prandtl number, oscillatory frequency parameter, Hartmann number, radiation absorption parameters, and dimensionless wall temperature. It was discovered that changes in biophysical parameters caused changes in both the velocity and temperature profiles, which is extremely important for scientists and clinicians. It is recommended that we pay attention to some of the parameters mentioned above in order to achieve the best results when studying blood flow through a vessel.

Research on the Flow and Transmission Performance of Magnetorheological Fluid between Two Discs

The viscoplastic flow of magnetorheological fluid in a disc was analyzed based on the Navier–Stokes Momentum Equation, and the yielded and unyielded decomposition surfaces were obtained. For the shear-thinning phenomenon of magnetorheological fluid, the magnetorheological properties of the magnetorheological fluid were described based on the Herschel–Bulkley model. Then, the relationship between torque, magnetic field, material, size and motion was established, and the magnetic field and working gap were optimized and analyzed. The results show that in the unyielding region, the magnetorheological fluid flows rigidly. In the yielding region, it flows as a viscous fluid. The degree of error of the proposed torque equation decreased gradually with the increase to current, as observed by experimental comparison.

Numerical Study Using Statistical and Quantum Approaches for Solving Energy and Navier Stokes Momentum Equations (PDEs)

Statistical and Quantum numerical method was implemented in this study to solve various cases in partial differential equations (PDEs) in engineering applications. One-dimensional with two lattices arrangements as well as two-dimensional with nine lattices arrangements is employed. The stability and the accuracy have been investigated either using statistical technique or using Euler’s method. The numerical limitations of using LBM method have been obtained and compared with those obtained by Euler’s method finite difference method. The main goal of this study is to investigate the ability of a statistical method in solving various ODEs or PDEs in energy and momentum equations and comparing them with those obtained by a classical numerical technique. The results show the ability of the statistical method for solving ODEs and PDE’s with more stable and accurate results. Therefore, the motivation of utilizing the statistical technique is the stability and it is easy for a complex fluid flow application.

Pressure from data-driven-estimated velocity fields using snapshot PIV and fast probes

This work explores the use of data-driven techniques to retrieve time-resolved information from snapshot PIV by exploiting the information from synchronized high-repetition rate sensors measuring flow quantities in few points, and to compute from it the instantaneous pressure field leveraging the Navier-Stokes momentum equation of the flow. This work focus on a technique rooted in the Extended Proper Orthogonal Decomposition, which already proven good performances in estimating time-resolved velocity fields from a finite number of probes synchronized with field measurements. The performances of the technique and its robustness to noise are tested on 2 synthetic dataset, a laminar one and a turbulent one, and compared to the most commonly applied technique to retrieve time-resolved information from snapshot PIV which exploits Taylor’s hypothesis.

Multiphase CFD simulation of the nearshore spilled oil behaviors

Oil spills are a serious environmental problem. To better support risk assessment and pollution control for oil spills, a good understanding of oil transport in the environment is required. This study focused on the numerical simulation of the nearshore oil behaviors based on computational fluid dynamics. Based on the Navier-Stokes momentum equations for an incompressible viscous fluid and volume of fluid (VOF) method, a 3D numerical model of three-phase transient flow was developed. The wave number, averaged flow velocity and oil properties would affect the oil spread extent and the oil volume fraction. The higher the averaged flow velocity and wave number, the lower the oil concentration, and the faster the horizontal movement of the oil. The spilled oil may move to contact the seafloor by increasing the averaged flow velocity at the inlet boundary. Through increasing the wave number, the oil would stay near the water surface. In the nearshore, where the wave is the main seawater motion, the oil containment boom should be set preferentially to the direction of wave transmission for oil cleaning. This study shows that by doubling the wave number and increasing the averaged flow velocity (ten times) at the same time, the maximum oil volume fraction would be reduced by around 32%. Finally, the water temperature had no significant impact on the oil migration, and the impact of evaporation should be considered in the simulation.

Rigorous investigation of the Navier–Stokes momentum equations and correlation tensors

An analytical solution to the Navier–Stokes momentum equations for a compressible flow with volume and small shear viscosities as well as external friction is presented while the dynamic viscosity is set to zero. The demonstrated methodology holds in d dimensions. However, in this study, the three-dimensional case is considered in detail. The analytical solution blows up at finite times T, which is determined by a cubic relation if the initial flow velocity is not divergence-free. The existence of T is a necessary and sufficient condition for implementing a singularity. Nonetheless, for external friction μe>T−1, all analytical expressions are smooth while the averaged expressions are smooth for all times.

Erratum in “On the Holographic Principle Description for the Viscous Incompressible Fluids” and “The Mathematical Description of Homogeneous Turbulence for Incompressible Fluids

This note is to acknowledge errors made in previous article: “On the Holographic Principle Description for the Viscous Incompressible Fluids” due to the Laplacian of the Lagrangian fails to be zero as expected when a using G.I. Taylor's 2-D solution of the Navier-Stokes momentum equations. Unfortunately, the article “The Mathematical Description of Homogeneous Turbulence for Incompressible Fluids” was also affected due to the use of the Laplacian of the Lagrangian in order to obtain the power spectrum relationship of the pressure which obviously now is incorrect.

Using a porous-media approach for CFD modelling of wave interaction with thin perforated structures

This work presents the use of a porous-media approach for computational fluid dynamics (CFD) modelling of wave interaction with thin perforated sheets and cylinders. The perforated structures are not resolved explicitly but represented by a volumetric porous zone where a volume-averaged pressure gradient in the form of a drag term is applied to the Navier–Stokes momentum equation. The horizontal force on the structures and the free-surface elevation at wave gauges around the cylinder model have been analysed for a range of porosities and regular wave conditions. The CFD results are verified against results from a linear potential-flow model and validated against experimental results. The applied pressure gradient formulation produces good agreement for all porosity values, wave frequencies and wave steepnesses investigated. It is demonstrated that an isotropic macroscopic porosity representation used for large volumetric granular material can also be used for thin perforated structures. This approach offers greater flexibility in the range of wave conditions that can be modelled compared to approaches based on linear potential-flow theory and requires a smaller computational effort compared to CFD approaches which resolve the flow through the openings. The approach can therefore be an efficient alternative for engineering problems where large-scale effects such as global forces and the overall flow-behaviour are of the main interest.

Frequency-dependent attenuation and dispersion caused by squirt flow: Three-dimensional numerical study

Seismic waves may exhibit significant dispersion and attenuation in reservoir rocks due to pore-scale fluid flow. Fluid flow at the microscopic scale is referred to as squirt flow and occurs in very compliant pores, such as grain contacts or microcracks, that are connected to other stiffer pores. We have performed 3D numerical simulations of squirt flow using a finite-element approach. Our 3D numerical models consist of a pore space embedded into a solid grain material. The pore space is represented by a flat cylinder (a compliant crack) whose edge is connected with a torus (a stiff pore). Grains are described as a linear isotropic elastic material, whereas the fluid phase is described by the quasistatic linearized compressible Navier-Stokes momentum equation. We obtain the frequency-dependent effective stiffness of a porous medium and calculate dispersion and attenuation due to fluid flow from a compliant crack to a stiff pore. We compare our numerical results against a published analytical solution for squirt flow and analyze the effects of its assumptions. Previous interpretation of the squirt flow phenomenon based mainly on analytical solutions is verified, and some new physical effects are identified. The numerical and analytical solutions agree only for the simplest model in which the edge of the crack is subjected to zero fluid pressure boundary condition while the stiff pore is absent. For the more realistic model that includes the stiff pore, significant discrepancies are observed. We identify two important aspects that need improvement in the analytical solution: the calculation of the frame stiffness moduli and the frequency dependence of attenuation and dispersion at intermediate frequencies.

On The Steady Incompressible Laminar Saltwater Flow In Minkowski 2-D Subspace Continuum Through A Rectangular MHD Micro-pump

BackgroundThe problem of a steady, incompressible and fully developed laminar fluid flow is discussed. This study focuses on the predictions of the pumping performance of 1 M NaCl solution. MethodThe corresponding Navier-Stokes momentum (MHD) equations are solved analytically by introducing a new ansatz combining the wave-position characters of the flow modelled in Minkowski 2-D subspace continuum subjected to suitable boundary conditions. ResultsSolving for the velocity profile of the working fluid across the micro-channel; the obtained solutions which include the dimensional flow velocity, volumetric flow rate, average velocity and pressure gradient are plotted under various operating currents, magnetic flux densities and micro-channel aspect ratio values. ConclusionsOur results provide that increasing high ratios of the electrode to channel lengths serves to attain more coverable channel area. It is also noted that an excess of the applied fields tends to compensate the velocity degradation for deep channel configuration. Graphics show that the decreased friction force between the channel top and bottom is a preferable factor to improve the process in practice. Our results may be useful to shed light on studying the microfluidic systems flow.

Simulation of dynamic stall using direct-forcing immersed boundary method at low Reynolds number

PurposeThe purpose of this study is simulation of dynamic stall behavior around the Eppler 387 airfoil in the low Reynolds number flow with a direct-forcing immersed boundary (DFIB) numerical model.Design/methodology/approachA ray-casting method is used to define the airfoil geometry. The governing continuity and Navier–Stokes momentum equations and boundary conditions are solved using the DFIB method.FindingsThe purposed method is validated against numerical results from alternative schemes and experimental data on static and oscillating airfoil. A base flow regime and different vortices patterns are observed, in accordance with other previously published investigations. Also, the effects of the reduced frequency, the pitch oscillation amplitude and the Reynolds number are studied. The results show that the reduced frequency has a major effect on the flow field and the force coefficients of the airfoil. On the other hand, the Reynolds number of the flow has a little effect on the dynamic stall characteristics of the airfoil at least in the laminar range.Practical implicationsIt is demonstrated that the DFIB model provides an accurate representation of dynamic stall phenomenon.Originality/valueThe results show that the dynamic stall behavior around the Eppler 387 is different than the general dynamic stall behavior understanding in the shedding phase.

Effects of Nanoparticle Enhanced Lubricant Films in Dynamic Properties of Plain Journal Bearings at High Reynolds Numbers

The aim of this paper is numerical analysis of plain journal bearing with nanoparticle added lubricating oil. The fluid film bearing is studied with the modified Reynolds equation which considered time dependent inertia effects at rotating speeds with the linearized turbulence. The research is performed for numerous nanofluids include nanoparticles of CuO, TiO2, Ag and Cu and SAE 20W50 as a base fluid. The time transient governing equations include the incompressible mass conservation equation, Navier-Stokes momentum equations for thin film lubrication, and full kinematic including shaft accelerations are solved numerically. Two cases of long and short bearings are studied with Sommerfeld and Gumbel boundary conditions. Linearized force coefficients such as mass, stiffness and damping factors are obtained for an ordinary journal bearing. The ordinary plain journal bearing velocity response and the rotor displacements when sudden forces applied by rigid rotor symmetrically is achieved. The results prove that using the nanoparticle enhanced lubricant causes higher mass coefficients, damping ratios and fluid effective stiffness.

Modeling the Aliso Canyon underground gas storage well blowout and kill operations using the coupled well-reservoir simulator T2Well

A blowout of the Sesnon Standard-25 well (SS-25; API 03700776) at the Aliso Canyon Underground Gas Storage Facility, first observed on October 23, 2015, eventually resulted in emission of nearly 100,000 tonnes of natural gas (mostly methane) to the atmosphere. Several thousand people were displaced from their homes as the blowout spanned 111 days. Seven attempts to gain pressure control and stop the gas flow by injection of heavy kill fluids through the wellhead failed, a process referred to as a “top kill.” Introduction of drilling mud when a relief well milled through the casing of SS-25 at a depth of ∼8 400 ft (“bottom kill”) succeeded in halting the gas flow on February 11, 2016. We carried out coupled well-reservoir numerical modeling using T2Well to assess why the top kills failed to control the blowout. T2Well couples a reservoir simulation in which porous media flow is described using Darcy's law with a discretized wellbore in which the Navier-Stokes momentum equation implemented via a drift-flux model (Shi et al., 2005) is used to describe multi-phase fluid transport to allow detailed process modeling of well blowouts and kill attempts. Modeling reveals the critical importance of well geometry in controlling flow dynamics and the corresponding success or failure of the kill attempts. Geometry plays a role in controlling where fluids can flow, e.g., when gas flow prevents liquid flow from entering the tubing from the annulus, but geometry also provides the opportunity for dead end regions to accumulate stagnant gas and liquid that can also affect kill attempts. Simulations show that follow-up fluid injections after the main kill attempts likely would have been effective to ensure that gas leakage remains stopped. T2Well is capable of simulating well kills and understanding the mechanisms behind well control failures and successes.