Governing Navier-Stokes Equations Research Articles

Double diffusive thermogravitational convection in a convex U-shaped porous chamber filled with radiative ternary hybrid nanoliquid

This research deals with the intricate dynamics of double diffusive thermogravitational convection within a convex U-shaped porous chamber and sheds light on the use of a radiative ternary hybrid nanoliquid. In this configuration, the lower flat boundary is assumed to be thermally hot and densely concentrated while the curved lateral boundaries remain cold and dilute. The other boundaries of the enclosure are kept under adiabatic conditions. The governing Navier–Stokes equations along with thermal and species equations are effectively solved by employing a higher order compact technique. The developed in-house program has been rigorously verified against experimental and computational benchmark results. The research meticulously examines the impact of several pivotal parameters, including the Lewis number (1≤Le≤20), buoyancy ratio (0≤N≤10), Darcy number (10−4≤Da≤10−2), Rayleigh number (104≤Ra≤106), volumetric heat source/sink coefficient (−10≤q≤10), radiation parameter (1≤Rd≤5), aspect parameter of the U-shaped chamber (0.2≤AR≤0.6), and solid particles concentration (0.0≤ϕthnp≤0.04) of the ternary hybrid nanofluid. The findings are eloquently portrayed through graphical representations by showcasing streamlines, iso-solutals, isotherms, and the dimensionless Nusselt (Nuavg) and Sherwood (Shavg) parameters. Our investigation demonstrates that the ternary hybrid nanofluid outperforms both hybrid and mono nanofluids in facilitating double diffusion processes. Moreover, optimal heat transfer efficiency is achieved under conditions characterized by an aspect ratio of AR = 0.2, Rayleigh number Ra=106, Darcy number Da=10−2, buoyancy ratio N = 10, Lewis number Le = 1, and solid volume fraction ϕthnp=0.04.

Describing geophysical turbulence with a Schrödinger–Coriolis equation in velocity space

In this paper, we examine the predictions of the scale-relativity approach for a turbulent fluid in rotation. We first show that the time derivative of the governing Navier–Stokes equation in the usual x-space can be transformed into a Schrödinger-like equation in velocity space with an external vectorial field to account for the rotation, together with a local velocity harmonic oscillator (VHO) potential in the v-space. The coefficients of this VHO are given by the second order x-derivatives of the pressure. We can then give formulas for the velocity and acceleration probability distribution functions (PDF). Using a simple model of anisotropic harmonic oscillator, we compare our predictions with relevant data from both direct numerical simulations (DNS) and oceanic drifter velocity measurements. We find a good agreement of the predicted acceleration PDF with that observed from drifters and some possible support in DNS for the existence of gaps in the local velocity PDF, expected in the presence of a Coriolis force.

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.

BE-BDF2 Time Integration Scheme Equipped with Richardson Extrapolation for Unsteady Compressible Flows

In this work we investigate the effectiveness of the Backward Euler-Backward Differentiation Formula (BE-BDF2) in solving unsteady compressible inviscid and viscous flows. Furthermore, to improve its accuracy and its order of convergence, we have equipped this time integration method with the Richardson Extrapolation (RE) technique. The BE-BDF2 scheme is a second-order accurate, A-stable, L-stable and self-starting scheme. It has two stages: the first one is the simple Backward Euler (BE) and the second one is a second-order Backward Differentiation Formula (BDF2) that uses an intermediate and a past solution. The RE is a very simple and powerful technique that can be used to increase the order of accuracy of any approximation process by eliminating the lowest order error term(s) from its asymptotic error expansion. The spatial approximation of the governing Navier–Stokes equations is performed with a high-order accurate discontinuous Galerkin (dG) method. The presented numerical results for canonical test cases, i.e., the isentropic convecting vortex and the unsteady vortex shedding behind a circular cylinder, aim to assess the performance of the BE-BDF2 scheme, in its standard version and equipped with RE, by comparing it with the ones obtained by using more classical methods, like the BDF2, the second-order accurate Crank–Nicolson (CN2) and the explicit third-order accurate Strong Stability Preserving Runge–Kutta scheme (SSP-RK3).

Mixed Convection in a Casson Fluid Flow towards a Heated Shrinking Surface

In this paper, an extensive analysis of mixed convection effects on steady two-dimensional stagnation point flow of a Casson fluid over a heated horizontal sheet has been numerically investigated. The governing Navier-Stokes equations of the present problem are transformed into nonlinear ordinary differential equations by applying a similarity transformation. A numerical solution of the problem has been obtained by employing the linearization technique along with the finite difference discretization. The impact of the Casson fluid parameter, the thermo-radiative parameter, the mixed convection parameter, the slip parameter, and the Prandtl number on the fluid motion and temperature is studied through graphical data. The convection parameter, the slip parameter and the Casson fluid parameter tends to accelerate the flow. The temperature distribution is however reduced by the convection parameter, slip parameter, thermal radiation and the Prandtl number. This work is based on [25] in which micropolar fluid flow over a heated surface was investigated by using the homotopy analysis method. We have extended the problem by considering the combined impact of mixed convection and thermal radiation on the Casson fluid flow towards a heated shrinking sheet, by using a numerical method.

Simulation of magneto mixed convective flow and thermal behavior of hybrid nanofluid in a partially heated wavy cavity

This study is an attempt to address numerically the thermal performance of magneto mixed convection in a wavy cavity filled with Cu-Al2O3(50%–50%) hybrid nanoparticles suspended in water under the influence of magnetic field dependent viscosity. A focus is made on the nonuniform partial heating from the left. The role of geometric parameters on the thermal performance is scrutinized thoroughly by changing surface waviness of the right wall, heating location and the direction of motion of the top and bottom walls. Based on these lid motion, the analysis is conducted by considering four geometrical set up. The transformed governing Navier–Stokes equations in dimensionless form are written in stream function-vorticity and energy transport equations. These equations are solved by using an inhouse computing code written in the compact finite difference approach. This study encompasses the effects of different physical pertinent parameters such as Richardson number, Hartmann number, magnetic orientation, solid concentrations, magnetic number. Simulations show the significant enhancement of thermal performance with the increase of magnetic orientation, undulation of the waviness and solid concentration of nanoparticles while it decreases with the increase in magnetic intensity. It is also seen that Case-I (single wall motion) and Case-III (parallel wall motion) can be an optimal configuration to get the strongest heat transfer inside the partially heated wavy enclosure.

Impacts of low oscillating magnetic field on ferrofluid flow over upward/downward moving rotating disk with effects of nanoparticle diameter and nanolayer

The current research investigates the impacts of nanoparticle diameter and liquid–solid nanolayer on ferrofluid (Fe3O4-water) flow over an upward/downward moving rotating disk. Additionally, a low-oscillating magnetic field is employed to study the hydrothermal aspects of fluid flow. The governing Navier–Stokes equations are well supported by Shliomis theory to include the effects of the oscillating magnetic field in the flow domain. The effects of the pertinent non-dimensional parameters on flow and heat transport characteristics are done through tables and graphical illustrations using the fifth-order Runge–Kutta–Fehlberg scheme. The study’s primary outcome is that the presence of a low oscillating magnetic field with the downward motion of the rotating disk enhances the heat transport rate more efficiently than the stationary rotating disk, as the upward and downward movement of the disk imparts contrasting features. The nanoparticle concentration and effective magnetization parameter escalate the skin friction. Moreover, the nanolayer and base fluid conductivity ratio enhances heat transport by 19% when the disk moves upward with rotation, while it is a 12% increment when the disk is stationary rotating. The current study is calibrated in its reduced form to an already published literature to validate the model.

Thermosolutal hydromagnetic mixed convective hybrid nanofluid flow in a wavy walled enclosure

This paper deals with the effects of an inclined magnetic field and hybrid nanoliquid on thermosolutal mixed convection in a partially heated wavy walled enclosure. Two distinct flow features are considered by performing a variant of heating and concentrating locations in both the vertical walls. The upper border is in motion with the same speed in the right direction. The governing Navier–Stokes equations are modeled to describe thermosolutal phenomena. These equations are solved by reconstructing a recently developed compact scheme. Results are displayed in terms of streamlines, isotherms, iso-concentrations, average Nusselt and Sherwood numbers to evoke the thermosolutal phenomena for various physical parameters. Furthermore, the significance of well-defined parameters influencing the fluid rotation and thermosolutal transfer, namely Richardson number (0.1≤Ri≤10), Lewis number (1≤Le≤5), thermal Grashof number (GrT=104), Buoyancy ratio (−1≤N≤1), Hartmann number (0≤Ha≤40), orientation angle of magnetic field (00≤γ≤900) and solid volume fraction (0.0≤ϕhnp≤0.04) of the hybrid nanofluid are performed generously. Results reveal that the heating areas and the angle γ play a vital role in flow topology, thermal and solutal transport inside the wavy enclosure. In addition, thermal and solutal transfer enhancement are noted with increasing values of γ while thermal and solutal transfer reduction are noted with the increasing Hartmann number. We have noticed that for the change in the concentration of Cu-Al2O3/water hybrid nanofluid (ϕhnp) from 1% to 4%, average Nusselt number is enhanced by 7.46% in Case-I and 5.09% in Case-II whereas average Sherwood number is decreased by 2.42% and 1.58% for Case-I and Case-II respectively. It is also concluded that the single nanofluid is more effective than the hybrid nanofluid in the case of thermosolutal transport phenomena.

Combined Marangoni and Buoyancy Convection in a Porous Annular Cavity Filled with Ag-MgO/Water Hybrid Nanofluid

Background: This article numerically examines the effect of buoyancy and Marangoni convection in a porous enclosure formed by two concentric cylinders filled with Ag-MgO water hybrid nanofluid. The inner wall of the cavity is maintained at a hot temperature, and the outer vertical wall is considered to be cold. The adiabatic condition is assumed for the other two boundaries. The effect of the magnetic field is considered in radial and axial directions. The Brinkman-extended Darcy model has been adopted in the governing equations. Methods: The finite difference scheme is employed to work out the governing Navier-Stokes equations. The numerically simulated outputs are deliberated in terms of isotherms, streamlines, velocityand average Nusselt number profiles for numerous governing parameters. Results: Except for a greater magnitude of axial magnetic field, our results suggest that the rate of thermal transport accelerates as the nanoparticle volume fraction grows. Also, it is observed that there is an escalation in the profile of average Nusselt numberwith an enhancement in Marangoni number. Conclusion: Furthermore, the suppression of heat and fluid flow in the tall annulus is mainly due to the radial magnetic field whereas in the shallow annulus, the axial magnetic field profoundly affects the flow field and thermal transfer.

Mixing in density- and viscosity-stratified flows

The lock-exchange problem is used extensively to study the flow dynamics of density-driven flows, such as gravity currents, and as a canonical problem to mixing in stratified flows. Opposite halves of a domain are filled with two fluids of different densities and held in place by a lock-gate. Upon release, the density difference drives the flow causing the fluids to slosh back and forth. In many scenarios, density stratification will also impose a viscosity stratification (e.g., if there are suspended sediments or the two fluids are distinct). However, numerical models often neglect variable viscosity. This paper characterizes the effect of both density and viscosity stratification in the lock-exchange configuration. The governing Navier–Stokes equations are solved using direct numerical simulation. Three regimes are identified in terms of the viscosity ratio μ2/μ1=(1+γ) between the dense and less dense fluids: when γ≪1, the flow dynamics are similar to the equal-viscosity case; for intermediate values (γ∼1), viscosity inhibits interface-scale mixing leading to a global reduction in mixing and enhanced transfer between potential and kinetic energy. Increasing the excess viscosity ratio further (γ≫1) results in significant viscous dissipation. Although many gravity or turbidity current models assume constant viscosity, our results demonstrate that viscosity stratification can only be neglected when γ≪1. The initial turbidity current composition could enhance its ability to become self-sustaining or accelerating at intermediate excess viscosity ratios. Currents with initially high excess viscosity ratio may be unable to dilute and propagate long distances because of the decreased mixing rates and increased dissipation.

Insight into the motion of Water-Copper nanoparticles over a rotating disk moving upward/ downward with viscous dissipation

Some of the fundamental properties of the nanofluids affect not only the transport phenomenon but also enhance the the heat transfer characteristics. The advancement in Rotatory machine technology can also be attributed mainly to a lot of research work that had been done on rotating disk flows with addition of nanoparticles in the base fluid. Taking cue of these developments, we examined the nanofluid (Cu+H2O) flow over a rotating disk moving upward/ downward with viscous dissipation by reducing the governing Navier–Stokes equations into ordinary differential equations using appropriate transformations and then numerically evaluated them by the BVP Mid-rich scheme in Maple software. The study of velocity and thermal profiles is carried out and examined graphically. The results show that the upward movement of the disk escalates the radial and azimuthal velocity profiles along with the thermal gradient. In contrast, the addition of nanoparticles decreases the heat transfer rate at a constant disk movement.

Heat Transfer Past a Rotationally Oscillating Circular Cylinder in Linear Shear Flow

Abstract This study investigates the unsteady, two-dimensional flow and heat transfer past a rotationally oscillating circular cylinder in linear shear flow. A higher order compact (HOC) finite difference scheme is used to solve the governing Navier–Stokes equations coupled with the energy equation on a nonuniform grid in polar coordinates. The hydrodynamic and thermal features of the flow are mainly influenced by the shear rate (K), Reynolds number (Re), Prandtl number (Pr), and the cylinder oscillation parameters, i.e., oscillation amplitude (αm), the frequency ratio (fr). The simulations are performed for Re=100,Pr=0.5−1.0, 0.0≤K≤0.15, and 0.5≤αm≤2.0. The numerical scheme is validated with the existing literature studies. Partial and full vortex suppression is observed for certain values of shear parameter K. The connection between heat transfer and vortex shedding phenomenon is examined where a pronounced increase in the heat transfer is observed for certain values of oscillation parameter, relative to the nonshear flow case.

On the flow characteristics of a four square-cylinder array at a 45° incidence angle and low Re

The four-cylinder model is widely applied in many engineering projects immersed in unsteady flow. However, its flow hydrodynamic characteristics are not fully understood. In the present work, the finite volume method is adopted in spatial discretization, while the SIMPLE algorithm in association with the non-iterative fractional step method is utilized to address the governing Navier–Stokes equation. Four equilaterally arranged cylinders at a flow incidence angle of 45° (α = 45°) are numerically simulated, and the corresponding Reynolds number ranges from 40 to 160. The accuracy and applicability of the numerical method employed in this investigation are verified by flow over a single cylinder. In the current work, the hydrodynamic characteristics, flow field characteristics, and structural vibration characteristics of cylinders are systematically investigated by changing the spacing ratio L/D (L/D = 1.2–4.0) and the Reynolds number. Numerical results show that the four cylinders have respective features and interaction effects. There are approximately eight wake flow patterns influenced by the spacing ratio. In addition, the numerical results also indicate that fluid dynamics are determined by flow patterns. These results can be a reference in engineering design, such as for ocean platforms with four cylindrical columns.

Turbulent Structures in Gap Flow

A computational study was carried out to investigate the effect of the gap between the body and the bed on the wake characteristics in shallow flows. A sharp-edged bluff body with two different clearances from the bed is employed in this investigation. The transient three-dimensional governing Navier-Stokes equations are numerically solved using a finite volume formulation with improved delayed detached-eddy simulation as a turbulence model. The volume of fluid approach is used in the current study as a free-surface modelling technique for locating and tracking the free surface. It is found that the existence and size of the gap influence the wake size and the location where the free surface is restored to its original and flat shape. This study also reveals that the fluid structures lose their symmetry about the center of the wake as the gap size increases.

Peridynamic differential operator-based Eulerian particle method for 2D internal flows

This paper proposes a non-local Eulerian particle method for two-dimensional internal flows. To balance computational accuracy and efficiency, the governing Navier–Stokes equations for this specific problem are recast in their integral form using a second-order peridynamic differential operator (PDDO). The particle method is then derived by applying a symmetric and uniform particle distribution to the solution of such an integral equation, yielding a readily accessible discretized algorithm. To stabilize and accelerate the proposed method, a pressure correction and a strategy of particle refinement are introduced with only a marginal loss of accuracy. The main procedure of the proposed method is detailed, as well as some other computational issues, such as time step and boundary conditions. Finally, several benchmark numerical examples, including Couette flow, shear-driven cavity, and internal flow around a cylinder, are carried out using the proposed method, demonstrating the level of its efficiency, accuracy, and capabilities.

Combined Marangoni and buoyancy convection in a porous annular cavity filled with Ag-MgO/water hybrid nanofluid

Background: This article numerically examines the effect of buoyancy and Marangoni convection in a porous enclosure formed by two concentric cylinders filled with Ag-MgO water hybrid nanofluid. The inner wall of the cavity is maintained at a hot temperature and the outer vertical wall is considered to be cold. The adiabatic condition is assumed for other two boundaries. The effect of magnetic field is considered in radial and axial directions. The Brinkman-extended Darcy model has been adopted in the governing equations. Methods: The finite difference scheme is employed to work out the governing Navier-Stokes equations. The numerically simulated outputs are deliberated in terms of isotherms, streamlines, velocityand average Nusselt number profiles for numerous governing parameters. Results: Except for a greater magnitude of axial magnetic field, our results suggest that the rate of thermal transport accelerates as the nanoparticle volume fraction grows.Also, it is observed that there is an escalation in the profile of average Nusselt numberwith an enhancement in Marangoni number. Conclusion: Furthermore, the suppression of heat and fluid flow in the tall annulus is mainly due to the radial magnetic field whereas in shallow annulus, the axial magnetic field profoundly affects the flow field and thermal transfer.

Numerical Simulation of Slip effect on Lid-Driven Cavity Flow Problem for High Reynolds Number: Vorticity – Stream Function Approach

Incompressible 2-D Navier-stokes equations for various values of Reynolds number with and without partial slip conditions are studied numerically. The Lid-Driven cavity (LDC) with uniform driven lid problem is employed with vorticity - Stream function (VSF) approach. The uniform mesh grid is used in finite difference approximation for solving the governing Navier-stokes equations and developed MATLAB code. The numerical method is validated with benchmark results. The present work is focused on the analysis of lid driven cavity flow of incompressible fluid with partial slip conditions (imposed on side walls of the cavity). The fluid flow patterns are studied with wide range of Reynolds number and slip parameters.

Combined effect of roughness and suction on heat transfer in a laminar channel flow

A two-dimensional numerical simulation is carried with the aim to investigate the combined effects of roughness and wall suction on heat transfer performance of a steady-state laminar channel flow. The roughness considered as a square element mounted on the bottom wall of the channel and suction is applied on its walls. Different Reynolds numbers, different suction rates, and various locations of the suction area on the roughness elements are considered. The governing Navier-Stokes equations are solved using open source OpenFOAM software. The results are tested against theoretical results.It is shown that the simple addition of roughness element increases the local Nusselt number in comparison to that without roughness. That increase is further enhanced when localised suction is applied. In addition, it is observed that by applying suction strategically on the roughness, one can reduce the thickness of thermal boundary layer, and remove or at least weaken the vortical motion behind the back-face of element, which in turn leads to an increase in local Nusselt number, demonstrating that suction is an effective means for improving the heat transfer rate in a laminar channel flow. As continuation of study, the effect of space between two elements was investigated and the best space was obtained. It is found that this method is useful to enhance heat transfer performance, especially at high Reynolds numbers.

A Cartesian grid generation technique for 2-D non-Newtonian blood flow through a bileaflet mechanical heart valve

Non-Newtonian blood flow through an aortic bileaflet mechanical heart valve is investigated numerically using the Casson model for the shear thinning behavior of the blood. The objective of this study is to deal with the influence of the non-Newtonian property of blood as well as of geometry of the prosthesis on flow phenomena and wall shear stress during the peak systolic phase. A comparison of non-Newtonian and Newtonian results is also presented. A nonuniform Cartesian grid generation technique is presented to generate a two-dimensional grid for the irregular geometry of the Butterfly valve. The governing Navier–Stokes equations of the flow, written in a stream function-vorticity formulation, are solved by the finite difference method with hybrid differencing of the convective terms. The results obtained with the model show that the blood non-Newtonian behavior induces a flow deceleration with the existence recirculation and consequently stagnation causing thrombus formation, as well as an increase of the shear stress along the wall which contributes the hemolytic blood damage; this is in agreement with the clinical observations concerning thromboembolic problems.

Unsteady 3D micropolar nanofluid flow through a squeezing channel: application to cardiovascular disorders

Micropolar fluids had many industrial applications such as polymer solutions, lubricant fluids, and biological structures. However, the present study deals with the drug delivery in the cardiovascular system where the blood particles are considered as microparticles included their self-rotation (mainly red blood cells) and interaction. The nanoparticles may be used as drug carrier particles. In the present problem, nanoparticles are considered as metallic oxides, e.g., Alumina ( $${\text {Al}} _2{\text {O}} _3$$ ), Titania ( $${\text {TiO}} _2$$ ), and Magnetite ( $${\text {Fe}} _3{\text {O}} _4$$ ) with water as base fluid. The magnetohydrodynamic micropolar fluid flow between two parallel squeezing plates is considered. Further, the analysis is carried out in the presence of viscous dissipation and Joule heating effects. With the aid of a similarity transformation, the flow governing Navier–Stokes equations is transformed into a system of coupled nonlinear ordinary differential equations. Fourth-order Runge–Kutta method with shooting approach is used to solve the nonlinear coupled boundary value problem. The profiles of flow field variables are acquired for key parameters arising in the present problem. It is noticed that, when the plates are fixed, the viscous drag of the base fluid is the same as that of nanofluid. Further, it is observed that increasing volume fraction results a decrement in microrotation and thereby causing an increase in temperature of Titania–water nanofluid which is in contrast to the behavior of other nanofluids.