Modified Navier-Stokes Equations Research Articles

Effect of Al2O3-Cu/Water Hybrid Nanofluid on Natural Convection in an Inverted T-Shaped Enclosure

In the present study, the effect of Al2O3-Cu/water hybrid nanofluid (HNF) flow, governed by modified Navier-Stokes equations coupled with energy conservation on heat and mass transfer due to the natural convection in an inverted T shape domain is investigated numerically by solving coupled non-linear PDE model using finite element method. The effect of the governing parameters like source length, Rayleigh number (Ra), 1 ≤ Ra ≤ 104, breadth of the vertical tube, volume fraction of nanoparticle (Φ), 0 ≤ Φ ≤ 0.01, 1 ≤ Pr ≤ 50, and aspect ratio (AR), 0.5 ≤ AR ≤ 2, etc. on Nusselt number has been analyzed and presented through tabulated values and Matlab plotting. Also, the flow and heat transfer sensitivity to the the governing parameters are depicted through the streamline and isotherm contours.

Numerical simulation of molybdenum disulfide-carbon nanotubes/water hybrid nanofluid flow due to a stretching cylinder: Applications to industrial manufacturing processes

This study delves into the exploration of hybrid nanofluids within the context of a stretching cylinder, a domain that has captivated numerous researchers owing to its pivotal applications in industrial manufacturing processes, particularly in metal forming and stretch dies. The authors, recognizing the significance of these applications, have introduced a novel heat transfer fluid termed hybrid nanofluid, comprising molybdenum disulfide and carbon nanotubes suspended in the base liquid, water. The investigation focuses on the flow of the hybrid nanofluid within a stretching cylinder, considering various influential factors such as Joule heating, thermal radiation, porous medium, and magnetic field effects. To model this complex problem, we employed modified Navier–Stokes equations. Employing similarity transformations, assumptions, and non-dimensional parameters, the problem was effectively simplified. The MATLAB bvp4c technique, a well-established numerical approach, was then employed to solve the resulting mathematical formulation. Graphical representations are presented to illustrate various aspects of the flow, facilitating a comprehensive understanding of the system. Comparisons are drawn among the flow characteristics of mono nanofluid, and the developed hybrid nanofluid. It is noted from the current analysis that the temperature strength increases with higher values of the magnetic field parameter, curvature parameter, radiation parameter, and Eckert number in both fluid cases. The Nusselt number increases with higher values of Prandtl number and thermal relaxation parameter. The identified patterns in velocity distribution, temperature strength, and fluid behavior provide a valuable foundation for optimizing thermal efficiency in diverse industrial applications. By leveraging the insights gained from this research, manufacturers can make informed decisions to enhance heat transfer processes, particularly in areas such as metal forming and stretch dies.

Wong-Zakai approximations and attractors for stochastic three-dimensional globally modified Navier-Stokes equations driven by nonlinear noise

Wong-Zakai approximations and attractors for stochastic three-dimensional globally modified Navier-Stokes equations driven by nonlinear noise

Two-Phase Flow Through the PTL of PEM Water Electrolyzer: MRI Experiments and Numerical Modeling Using Phase-Field Theory

In 2021, estimation shows that the worldwide annual hydrogen production is around 94 Mt. Exploitation of native hydrogen being not mature, it is obtained by its separation from other elements by different methods such as steam methane reforming, and electrolysis of water. The latter supplied by a renewable power source will take part in the development of a green hydrogen economy [1]. In this context, Proton Exchange Membrane (PEM) Electrolyzer is a promising technology due to its flexibility and very quick adaptation to load variations. However, its current development still confronts some limitations at a large industrial scale. For instance, efficiency and durability are directly impacted by the mass transport and electrical transfer within the porous materials at the anode side. Limitation of the water supply to the catalyst layer happens once there is a poor oxygen evacuation which decreases the performance of the device by inducing high overpotential. Furthermore, the PTL has an important role as an electrical conductor for charge transfer from the catalyst layer [1]. The efficiency of all the transport phenomena through the PTL and porous electrode assembly depends on their transport properties which are related to their microstructure and operating conditions. For instance, PTL with a large pore size allows good water and gas transport, while produced electrons choose a long-distance path generating an electrical resistance higher than that in the case of PTL with a small pore size [1]. So, controlled and optimum porosity and pore size could contribute to efficient water, gas, and electron transport. To define the best porous layers morphologies, a deep and accurate understanding of the phenomena is developed in this work by combining modeling and experiments.The magnetic resonance imaging (MRI) technique was used to quantify the water content within the porous layer during the two-phase flow. Instead of the real PTL made of titanium which is paramagnetic and cannot be used in the MRI, borosilicate filters with thickness, porosity, and pore size similar to the PTL were used. After positioning the sample in the 600 MHz vertical imager (Figure 1-A), a constant water flow rate is introduced while the gas flow rate is varied. The saturation profiles measured through the porous material depend on the gas flow rate and a semi-dryness of the sample occurs (Figure 1-B) with a residual quantity trapped between pores (minimum stable water content). The water flow rate variation in the channel does not affect saturation, but a higher gas flow rate is needed to reach a minimum stable water content for higher water flow rate.The gas pressure drop through the porous medium was measured and bubble formation in the channel was also analyzed. The results show that the pressure drops and types of flow (slug, annular, and bubble flow) depend on orientation of the water channel (horizontally and vertically) and flow direction (up or downward), and on the water and gas flow rates.To reach a better understanding of the dynamic characteristics of water and oxygen transport over the PTL, the phase-field model based on the Cahn-Hilliard theory was used to simulate the two-phase flow through a porous medium [2]. In this model, the modified Navier-Stokes equations for two phases are coupled with a phase-field equation for describing the diffuse interface. Numerical simulations performed in the COMSOL® multiphysics software were carried on 2D geometries composed of spherical solid grains of different sizes, having properties similar to the PTL used in the MRI experiments. Gas is injected on one side of the sample and flows through the porous medium initially saturated and evacuated on another side in contact with the water channel (Figure 1-C). Gas flow in the porous medium and bubble formation in the water channel are studied while varying the gas and water flow rates. The simulation results give information about the gas pathways within the porous medium and the saturation profiles over time, depending on the gas/water flow rates, which will be compared with experimental results.[1] J. Parra-Restrepo, “Caractérisation des hétérogénéités de fonctionnement et de dégradation au sein d’un électrolyseur à membrane échangeuse de protons (PEM),” Université de Lorraine, 2020.[2] J. W. Cahn and J. E. Hilliard, “Free Energy of a Nonuniform System. I. Interfacial Free Energy,” The Journal of Chemical Physics, vol. 28, no. 2, pp. 258–267, Feb. 1958. Figure 1

An Extended Model for Short Wavelength Peristaltic Flow

Peristaltic flow theory is often based on the long wavelength assumption, which can be approximated by small wave number cases. This paper presents a unique approach to analyze the peristaltic flow for both short and long wavelength domains. The governing equation considering wave number is solved by combining eigen function expansion and generalized inversion methods, which is first proposed to solve peristaltic flow problems. In particular, the modified Navier-Stokes equations, considers the effects of the Trouton’s elongational viscosity. The result indicates that this computational method efficiently reduces the dimensions and condition number of the matrix. The sketch of stream function shows that a trapped bolus also occurs in short wave peristaltic flow. The validity of this computational approach is demonstrated by comparing these results with existing theories.

Stochastic 3D Globally Modified Navier–Stokes Equations: Weak Attractors, Invariant Measures and Large Deviations

Stochastic 3D Globally Modified Navier–Stokes Equations: Weak Attractors, Invariant Measures and Large Deviations

Numerical analysis of finite bearing operating under non-Newtonian lubricant and flow turbulence effect

The purpose of this work is to evaluate the combined influence of turbulence and shear-thinning/shear-thickening lubricant on the steady-state characteristics of hydrodynamic journal bearings. The modified Navier–Stokes equation for linear turbulence and non-Newtonian (shear-thinning as well as shear-thickening aspect) lubricants are solved using a finite element method to acquire load capacity, attitude angle, friction coefficient variable, total oil flow, and temperature rise variable for distinct combined values of the flow behavior index and Reynolds number. This work utilizes the Navier–Stokes equation with Ng–Pan's linear turbulence and shear-thinning/shear-thickening lubricant models, which is very helpful in predicting the performance characteristics of hydrodynamic bearings under harsh industrial operating conditions. The results show that the combined presence of non-Newtonian lubricant and turbulence in the flow significantly affects the steady-state performance parameters of such type of bearing compared to existence of individual presence of turbulence and non-Newtonian lubricant in the clearance space. This work helped to provide better insight into the operation of journal bearings in a severe environment owing to the high speed and changes in the nature of lubricant.

Simulation study of effective pumping speed of divertor pumping system for CFETR with COMSOL Multiphysics

In this study, COMSOL Multiphysics is used in the numerical analysis of a neutral gas in the divertor pumping system of the China Fusion Engineering Test Reactor (CFETR). The slip boundary condition modified Navier–Stokes equation and angular coefficient methods are used for simulation. The corresponding divertor inlet pressures are 1 Pa under plasma discharge and 1×10−5Pa at the pumping limit. Additionally, the throughput and flow conductance of the divertor pumping system are calculated. The effective pumping speed (EPS) of a single cryopump in a vacuum chamber is also computed. Results indicate that the EPS values of the single cryopump from the divertor inlet are 33.35 and 16.91 m3/s, corresponding to the divertor inlet pressures of 1 and 1×10−5Pa, respectively. By comparing the simulation results with those of the direct simulation Monte Carlo and test particle Monte Carlo methods, the feasibility of COMSOL Multiphysics was verified with a maximum error of less than 5%. This study provides technical support to the subsequent optimization of CFETR divertor pumping channels and EPS evaluation.

Shear imposed falling liquid films on a slippery substrate with Marangoni effects: Effect of odd viscosity

We investigate the behavior of a thin fluid with disrupted time-reversal symmetry on a uniformly heated inclined surface under external shear stress using modified Navier–Stokes equations, an energy conservation equation, and incorporating a Navier slip condition. Critical conditions for instability onset are determined by a linear stability analysis within the Orr–Sommerfeld framework. We derive a first-order Benney-type evolution equation to study long-wave instabilities. We find slippery substrate, imposed shear stress along the flow direction, and Marangoni number consistently destabilize the flow, while odd viscosity and imposed counter-flow shear stabilize it. A weakly nonlinear analysis using multiple scales reveal distinct zones of instability. Marangoni number, slip length, odd viscosity, and imposed shear direction significantly impact stability and instability regions. Numerical simulations of the free surface evolution equation of a flow system under consideration provide clear evidence of the contributions of thermocapillary, slip length, odd viscosity, and imposed shear direction. Furthermore, our analysis of linear and weakly nonlinear stability, as well as our numerical simulations, exhibit remarkable consistency.

Random dynamics and limiting behaviors for 3D globally modified Navier–Stokes equations driven by colored noise

AbstractThis paper is mainly concerned with the long‐term random dynamics for the nonautonomous 3D globally modified Navier–Stokes equations with nonlinear colored noise. We first prove the existence of random attractors of the nonautonomous random dynamical system generated by the solution operators of such equations. Then we establish the existence of invariant measures supported on the random attractors of the underlying system. Random Liouville‐type theorem is also derived for such invariant measures. Moreover, we further investigate the limiting relationship of invariant measures between the above equations and the corresponding limiting equations when the noise intensity approaches to zero. In addition, we show the invariant measures of such equations with additive white noise can be approximated by those of the corresponding equations with additive colored noise as the correlation time of the colored noise goes to zero.

The Periodic and Limiting Behaviors of Invariant Measures for 3D Globally Modified Navier–Stokes Equations

The Periodic and Limiting Behaviors of Invariant Measures for 3D Globally Modified Navier–Stokes Equations

Large deviation for a 3D globally modified Cahn–Hilliard–Navier–Stokes model under random influences

In this article, we derive a large deviation principle of the strong solution of the 3D globally modified Cahn–Hilliard–Navier–Stokes model under random influences. The model consists of the stochastic globally modified Navier–Stokes equations for the velocity, coupled with a Cahn–Hilliard equation for the order (phase) parameter. The proof relies on the weak convergence method that was introduced in Budhiraja et al. (2011) and based on a variational representation on infinite-dimensional Brownian motion.

Self-similar temporal turbulent boundary layer flow

Direct numerical simulations of temporally evolving boundary layer flows are considered with solutions restricted to self-similar profiles. A new set of modified Navier–Stokes equations is solved with periodic boundary conditions in the streamwise direction, and solutions reach statistically steady states, independent of the initial conditions. The results are presented for different cases, with and without a pressure gradient, and found to be in agreement with existing results of spatial turbulent boundary layer flows. Thus, self-similar temporal solutions are able to reproduce the general features of turbulent boundary layer flows. Finally, the model is applied to simulate a turbulent spot in equilibrium, which is difficult to obtain otherwise.

A novel approach for wall-boundary immersed flow simulation (proposal of modified Navier-Stokes equation)

A novel approach for wall-boundary immersed flow simulation (proposal of modified Navier-Stokes equation)

Random attractors and invariant measures for 3D stochastic globally modified Navier-Stokes equations with time-dependent delay and coefficient

In this article, we investigate the dynamics of three-dimensional, non-autonomous, and globally modified Navier-Stokes equations with time-dependent delay and additive noise. Based on the well-posedness of solutions, we demonstrate that the solution operators of such equations generate a non-autonomous random dynamical system. We further prove the existence and uniqueness of random attractors as well as the periodicity and the upper semicontinuity for this random dynamical system. We also prove the existence of invariant measures supported on the random attractors. Moreover, we rigorously find that every limit of a sequence of invariant measures must be an invariant measure of the corresponding limiting equations as the noise intensity $ \varepsilon\rightarrow0 $ or the modification parameter $ N\rightarrow N_0\in (0, +\infty) $.

Asymptotic behaviour of solutions to stochastic three-dimensional globally modified Navier–Stokes equations

We consider 3D stochastic globally modified Navier–Stokes equations in bounded domains with homogeneous Dirichlet boundary conditions and infinite dimensional Wiener process. We study stability properties of stationary solutions. We also show that one can stabilize an unstable stationary solution by using a multiplicative Itô noise of sufficient intensity or a linear internal feedback controller with support large enough.

Dynamics of chiral liquid crystals between cylinders using Landau-de Gennes theory

ABSTRACT This study employs the Landau-de Gennes theory to investigate the behaviour of chiral liquid crystals (CLCs) between concentric and eccentric cylinders under different flow conditions. This theory was implemented in dynamic finite element simulations to solve the evolution of the microstructure of CLCs and couple it with linear momentum balance equation (modified Navier–Stokes equation) to capture the structure of CLCs. The study focused on the microstructure formation of CLCs and their performance as lubricants under various chirality strengths (θ), Deborah numbers (De), and eccentricity of eccentric cylinders. The hexagonal structure of the CLCs was found between cylinders at low De (0.001), where the chiral term predominates. Moreover, the number of hexagonal structures between cylinders was shown to be significantly correlated to the chiral strength. We showed that increasin De resulted in vanishing hexagonal pattern of CLCs and the De value at which the hexagonal pattern vanishes depends on the chiral strength. We also observed that a higher eccentricity ratio significantly increased the pressure force per length on the inner cylinders and caused more non-uniformity in the structure. Based on the findings, we concluded that the performance of CLCs as lubricants is greater at higher θ and eccentricity and lower De.

Prediction of the fuel spray characteristics in the combustion chamber with methane and TiO2 nanoparticles via numerical modelling

• Effect of the fuel injection velocity on the combustion has been examined. • Numerical model was proposed for investigating the fuel injection velocities varying from 100 m/s to 200 m/s under the influence of nanoparticles. • Spray velocity increases the turbulence and formation of vortices in the combustion chamber. • No significant changes in mass fractions of both fuel and oxidiser were observed at higher injection velocities. In this study the methane combustion was analysed with the TiO 2 nanoparticles. A series of the simulation runs were performed by varying the fuel inlet velocity. However, the oxidizer and the nanoparticles spray were maintained constant for the entire run. The spray velocity varied from 100 m/s to 200 m/s with titanium dioxide (TiO 2 ) nanoparticles. Using the series of the governing equation and modified Navier Stokes equation the model has been developed with the aid of numerical workbench. Three different domains are generated for fuel, oxidizer and nanoparticles. The velocity of the air and nanoparticles were maintained at constant levels and varying only the spray velocity of the fuel. Based on the findings, the mass fraction of both fuel and formation of the CO 2 were dependent on the spray velocity. As the spray velocity increases the turbulence in the combustion chamber increases which ensures the higher mixing of both air–fuel and nanoparticles. From the procured findings 175 m/s and 200 m/s were the ideal range for better combustion efficiency compared to 100 m/s and 150 m/s. The simulation results have ascertained the role of the spray velocity on the emissions and the combustion efficiency of the engine. It is hoped that obtained results can provide directions to work on the combustion of the methane with the nanoparticles at the optimized spray velocity.

Reservoir end wall effects on bivariate ion and fluid transport in micro/nano-nozzles for effective electroosmotic mixing

The aim of this study was to develop an efficient convection diffusion-based mathematical model for the species transport and mixing in different shaped (i.e., nozzle, diffuser, diffuser–nozzle, nozzle–diffuser) micro/nano-channels connected to large reservoirs. Both analytical and numerical studies are performed to illustrate the impact of inertial and contact angles for the generation of complex flow patterns due to different aspect ratio specified transformations. The hydrodynamics of the ion and fluid transport are analyzed through the Poisson–Nernst–Plank-based Navier–Stokes model subjected to specified system of forces endured by the reservoir fluids. The numerical results for pressure velocity correlations are obtained when the transport mechanism of the domain is changed from nozzle to diffuser. Mixing efficiency is evaluated for different geometric configurations and compared with a rectangular slit channel when the parallel reservoirs are connected. The role of Debye–Hückel parameter, conical angles or slope, and reservoir height/width on the transport of ions and enhancement of mixing are discussed. The mixing efficiency is found to attain a higher value after considering the reservoir connected to a nozzle without involving any hurdles or heterogeneous zeta potential along the channel wall. Closed-form analytical solutions of the electric potential are obtained through the linearized Poisson–Boltzmann model and further incorporated for the pressure evaluation. The axial and transverse velocities are evaluated from the modified Navier–Stokes equation including electric body force term and are validated with the experimental results. Effective nonlinear coupling responses of ion transport are found to be more pronounced in nozzle compared with diffuser resulting a higher mixing. Also, the solutions of velocity resulting in a low torque satisfy the equilibrium conditions and are optimized in terms of adversion of frictional factor and viscous dissipation resulting in an effective mixing. The findings manifest the species patterns with high accuracy and versatility, which could possibly help to handle the technical challenges associated with the design of pumpless actuated microfluidic devices.

Numerical investigation of viscoelastic boundary layer in forced convection flow on surface under prescribed heat flux

AbstractThe paper intends to analyze numerically the effect of forced convection with viscoelastic boundary on porous media subjected to constant heat flux on the surface. The modified Navier–Stokes equations in nondimensional forms are formulated and modeled with Darcy–Forchheimer–Brinkman to solve the equations using bvp4c with MATLAB package. The numerical results are found for local Nusselt number, shear stress, temperature distribution, and finally the velocity of the boundary layer over a horizontal plate. The analysis includes the influence of several parameters, such as Forchheimer, porous media, Nusselt number, and viscoelastic on distributions of velocity and temperature. Also, the effects comprise the friction and heat transfer coefficients. The numerical results under constant heat flux indicate an increase in shear stress and boundary layer velocity as the Darcy parameter increases and at the same time, the values of temperature and Nusselt number are decreased. A significant increase was observed for the values of Nusselt number, temperature, and shear stress under large values of the Forchheimer parameter. The numerical results show a good agreement with available previous works.