Fluid Flow Equations Research Articles

Performance analysis of annular disks with non-Newtonian Rabinowitsch fluid model: Influence of squeeze film pressure, surface roughness, porosity and viscosity variation

In various systems of energy, such as turbo-machinery and viscous-lock system, squeeze film plays a crucial role for improving transport systems. The importance of the aforementioned applications led us to explore the physical behavior of Rabinowitsch fluid flow for several values of viscosity, [Formula: see text], load capacity [Formula: see text], nonlinear factor, [Formula: see text], dimensionless film thickness [Formula: see text], dimensionless radius of the capillary tube [Formula: see text], dimensionless thickness of porous pad [Formula: see text]), inner and outer radius ratio [Formula: see text], roughness parameter [Formula: see text] through a squeeze film with two rough porous annular discs. Five-point Gauss quadrature integral formula has been used to examine the characteristics of annular discs, and small perturbation method has been used to discretize the governing Rabinowitsch fluid flow (RFF) equations. The modified Reynolds equation for rough, porous and viscosity change with film thickness is developed using Christensen’s stochastic theory, modified Darcy’s theory and the association between viscosity and film thickness agreement. The impacts of [Formula: see text] on the behavior of porosity and viscosity variability of RFF have been visually depicted in terms of film pressure, load capacity and squeeze reaction time of annular discs. It has been determined that for RFF with variable viscosity and roughness parameter on a porous media, the performance of the annular disks enhances for [Formula: see text] (dilatant lubricant), and diminishes for [Formula: see text] (pseudo-plastic lubricant). It has been found that the film pressure in case of dilatant lubricant is increased by the influence of Rabinowitsch fluids on porous walls, but the behavior of pseudo-plastic lubricants has an opposite tendency relative to Newtonian lubricants. It has also been found that the impact of variability in porosity, roughness and viscosity diminishes the load-carrying capacity (LCC) with increased film thickness than the case of nonporous, smooth surface and constant viscosity of Rabinowitsch fluid.

Thermo-poroelastodynamic response of a borehole in a saturated porous medium subjected to a non-hydrostatic stress field

A thermo-poroelastodynamic (TPED) model incorporating the thermo-osmosis and thermal-filtration phenomena is used to analyze the transient response of a borehole subject to a non-hydrostatic stress field. The coupled TPED model is developed on the basis of equations of motion, fluid flow, heat transfer and constitutive equations so as to couple the stress wave, pressure wave and thermal wave in early times. Poroelastodynamics (PED) and thermo-elastodynamics (TED) models correspond two particular cases of the current model. The borehole problem is decomposed into the superposition of an axisymmetric mode and an asymmetric mode, corresponding to axisymmetric and deviatoric loadings, respectively. The semi-analytical solutions in the Laplace transform space are inverted numerically to obtain time-dependent displacement, pore pressure, temperature and stresses. Comparisons are performed among different sub-models to examine the influence of thermal effect on the response of pore pressure and stresses as well as the influence of pore water on the response of stress components. A direct comparison among the dynamic, quasi-static and static models shows that inertial effect is very important in the early times and it contributes to the early-time wave behavior.

Analysis of Williamson nanofluid MHD over a nonlinear stretchable sheet by velocity slip and newtonian heating

In this paper, the influence of velocity slip and the Newtonian heating parameters on Williamson Nanofluid across a variable stretching sheet is studied. The basic fluid flow equations are turned into a system of coupled ordinary differential equations with appropriate conditions, which are solved numerically with the Keller-box approach. The outcomes show that, increasing Williamson parameter, due to the viscosity velocity of the fluid slowdown. As a result, it is noticed that the velocity slip parameter reduces Williamson Nanofluid velocity, while increasing shear stress, according to the results. Furthermore, it has been discovered that unlike velocity profiles, the slip influence is more pronounced on temperature and concentration profiles. Furthermore, the Newtonian heating parameter may be shown to have rising functions for velocity and dimensionless temperature. Newtonian heating has uses in the petroleum industry, heat exchangers, conjugate heat transfer around fins and solar radiation. It is discovered that the Joule heating parameter improves the thermal boundary layer thickness and slows down the velocity of the fluid, which aids in maintaining system temperature. Additionally, it has been found that the temperature profile exhibits a stronger slip effect compared with the velocity profile. To prepare toffee, chocolate and other delicacies as efficiently as possible, it is crucial to investigate the flow of Williamson fluids (such as drilling muds, clay coatings, various suspensions and some lubricating oils, thermoplastic melts, and a variety of colloids) in the incidence of heat transfer. The novelty of this paper is the effect of joule heating, Newtonian heating parameter and velocity slip on Williamson fluid velocity and temperature gradient. Furthermore, the heat and mass transfer rates are calculated for various parameters numerically and shown in tabular form.

New Analytic Solutions for Fluid Flow Equations in Higher Dimensions Around an Offshore Structure Describing Bidirectional Wave Surfaces

New Analytic Solutions for Fluid Flow Equations in Higher Dimensions Around an Offshore Structure Describing Bidirectional Wave Surfaces

Electric field and EHD flow in longitudinal wire-to-plate DC and DBD electrostatic precipitators: A numerical study

In this study, the electrical and electrohydrodynamic (EHD) flow properties of a wire-to-plate electrostatic precipitator with a longitudinal wire electrode are investigated through a numerical simulation. The two modes of operation are considered: DC and AC supply voltages. The AC model features dielectric layers over collecting electrodes, hence a dielectric barrier discharge mechanism. To predict the ion concentration on the wire electrode surface, Kaptzov's hypothesis is employed. Both electrical and fluid flow equations are solved using the Finite Element Method. The EHD flow properties are estimated by simulating electric field, space charge density and fluid flow in the 3D precipitator channel. The results suggest that 1000 Hz frequency in AC supply generates EHD force with a 9 × 104 N/m3 maximum alue, which is an order of magnitude higher than the same value under DC supply. However, for 100 Hz supply voltage frequency the EHD force is lower than that for the DC supply and equal to 8 × 103 N/m3, although with a greater level of vorticity. The proposed models for DC and AC supply voltages display close characteristics to their experimental references with an average relative error of 13.5%.

Three-Dimensional High-Precision Numerical Simulations of Free-Product DNAPL Extraction in Potential Emergency Scenarios: A Test Study in a PCE-Contaminated Alluvial Aquifer (Parma, Northern Italy)

Chlorinated organic compounds are widespread aquifer contaminants. They are known to be dense non-aqueous phase liquids (DNAPLs). Therefore, they are denser than water and immiscible with other fluids. Their migration into the environment in variably saturated zones can cause severe damage. For this reason, optimizing those actions that minimize the negative impact of these compounds in the subsurface is essential. This paper presented a numerical model simulating the free-product DNAPL migration and extraction through a purpose-designed pumping well in a potential emergency scenario. The numerical simulations were performed using CactusHydro, a numerical code that uses a high-resolution shock-capturing flux conservative method to resolve the non-linear coupled partial differential equations of a three-phase immiscible fluid flow recently proposed in the literature, including the contaminant extraction at the base of the aquifer. We investigated the temporal (and spatial) evolution of its migration in the Parma (Northern Italy) porous alluvial aquifer following the saturation contour profiles of the three-phase fluid flow in variably saturated zones. The results indicated that this numerical approach can simulate the contaminant migration in the subsurface and the pumping of the free-product from a well screened at the base of the aquifer system. Moreover, the simulation showed the possibility of recovering about two-thirds of the free-product, in agreement with the scientific literature.

Transport-Equation-Based Shock Sensor for Shock–Turbulent Boundary-Layer Interaction

High-speed flows are often associated with shock–boundary-layer interaction (SBLI), which causes excessive vehicle surface heating and total pressure loss. Therefore, precise computational prediction of shock interactions becomes essential. Commonly used simulation techniques, including numerical schemes and turbulence models, give erroneous results at shock discontinuities. Local modifications are applied at shocks based on their location and strength. Most shock sensors in the literature predict shock location but do not give shock properties. We propose a shock function to locate shocks and estimate their strength in terms of density ratio by solving a transport equation, along with the governing equations of fluid flow. The sensitivity of the shock function to the transport equation’s parameters and grid resolution are explored. The shock function is used to define a shock sensor that can identify thin and sharp shocks. The shock function and shock sensor are then applied to an oblique shock impinging on a turbulent boundary layer. We use the shock unsteadiness modified model, which uses the estimated shock strength to predict the turbulence levels at shocks accurately. We study the effect of nonadiabatic wall temperature on the SBLI region, and the results are compared with available Direct Numerical Simulation (DNS) data.

Melting performance enhancement in a thermal energy storage unit using active vortex generation by electric field

Latent heat thermal energy storage (LHTES) devices aid in efficient utilization of alternate energy systems and improve their ability to handle supply–demand fluctuations. A numerical analysis of melting performance in a shell-and-tube LHTES unit in the presence of a direct current (DC) electric field has been performed. The governing equations of fluid flow, heat transfer, electric potential and charge conservation are solved using a customized finite-volume solver built in the open-source framework of OpenFOAM. Enthalpy-porosity method based fixed grid approach is used to track the melt interface. Primary objective of the study is to highlight the interface and flow morphology evolution in the presence of electric field induced flow and to evaluate the melting performance of the LHTES unit. The transient evolution of the melting process in the presence of electric field has been mapped in terms of total liquid fraction, kinetic energy density and mean Nusselt number. The charge injection from the tube surface generates multiple electrohydrodynamic (EHD) flow vortices in the liquid region. Thus, the inherent uni-cellular flow structure of the natural convection driven melting is disrupted. The multi-cellular flow structure with stronger velocity distribution enhances mixing and heat transfer. Melting performance at various levels of applied voltages (0≤V≤10kV) in both vertical and horizontal orientations of the LHTES unit has been quantified in terms of charging time and total power storage. The charging time gets shorter and total power storage gets higher with increasing applied voltages. In the vertical orientation, a maximum 82.52% reduction in charging time and 80.85% increase in net power storage is achieved. In the horizontal orientation, weaker buoyancy force leads to stronger influence of the electric field. A maximum of 89.61% reduction in charging time and 88.35% increase in power storage is achieved in the horizontal orientation. The results of this study aid in understanding the mechanism of EHD flow assisted melting and provide a reference for design of a shell-and-tube LHTES unit with improved performance.

Finite Volume Computational Analysis of the Heat Transfer Characteristic in a Double-Cylinder Counter-Flow Heat Exchanger with Viscoelastic Fluids

This work presents a computational analysis of the heat-exchange characteristics in a double-cylinder (also known as a double-pipe) geometrical arrangement. The heat-exchange is from a hotter viscoelastic fluid flowing in the core (inner) cylinder to a cooler Newtonian fluid flowing in the shell (outer) annulus. For optimal heat-exchange characteristics, the core and shell fluid flow in opposite directions, the so-called counter-flow arrangement.The mathematical modelling of the given problem reduces to a system of nonlinear coupled Partial Differential Equations (PDEs). Specifically, the rheological behaviour of the core fluid is governed by the Giesekus viscoelastic constitutive model. The governing system of coupled nonlinear PDEs is intractable to analytic treatment and hence is solved numerically using Finite Volume Methods (FVM). The FVM numerical methodology is implemented via the open-source software package OpenFOAM. The numerical methods are stabilized, specifically to address numerical instabilities arising from the High Weissenberg Number Problem (HWNP), via a combination of the Discrete Elastic Viscous Stress Splitting (DEVSS) technique and the Log-Conformation Reformulation (LCR) methodology. The DEVSS and LCR stabilization techniques are integrated into the relevant viscoelastic fluid solvers. The novelties of the study center around the simulation and analysis of the optimal heat-exchange characteristics between the heated Giesekus fluid and the coolant Newtonian fluid within a double-pipe counter-flow arrangement. Existing studies in the literature have either focused exclusively on Newtonian fluids and/or on rectangular geometries. The existing OpenFOAM solvers have also largely focused on non-isothermal viscoelastic flows. The relevant OpenFOAM solvers are modified for the present purposes by incorporating the energy equation for viscoelastic fluid flow. The flow characteristics are presented qualitatively (graphically) via the fluid pressure, temperature, velocity, and the polymer-stress components as well as the related normal stress differences. The results illustrate the required decrease in the core fluid temperature in the longitudinal direction due to the cooling effects of the shell fluid, whose temperature predictably increases in the counter-flow direction.

A numerical exploration of hyporheic zone solute transport behavior estimated from electrical resistivity inversions

An increasing number of studies have combined saline tracer injections with time-lapse electrical resistivity (ER) surveys to explore solute exchange within the hyporheic zone. The reliability and resolution of ER surveys of the hyporheic zone are rarely discussed. Here, we use a numerical modeling approach to assess how ER imaging resolves changes in saline tracer concentration within the hyporheic zone given different synthetic scenarios. We create a 3-D synthetic stream and surrounding hyporheic zone with an ER electrode transect and solve the coupled equations for porous fluid flow and solute transport. Then, we solve for the 3-D conduction of current between electrodes to create synthetic time-lapse ER surveys and invert the simulated resistances to obtain 2-D bulk electrical conductivity (σb) images perpendicular to the stream, which we compare with the known synthetic σb fields. The σb fields in the inversion generally capture the shape of the simulated σb fields, but with smoothing and artifacts as a function of the inversion process. Consequently, the calculated cross-sectional area of tracer plume is inaccurate. At inversion pixels beneath the stream, the accuracy of the inverted σb breakthrough curves when compared to the synthetic “truth” varies with stream size, pixel depth, and to a lesser extent injection time. The tails of these pixel breakthrough curves beneath the stream are consistently underestimated compared to the synthetic “truth,” i.e., σb in the inversions appear to return to background faster by one to six hours. The time series of average apparent bulk electrical conductivity requires no inversion and captures lingering saline tracer better than bulk EC breakthrough curves at individual pixels in the inversions.

Advanced two-phase subchannel method via non-linear iteration

A fast-running, robust two-phase flow, sub-channel model is presented based on non-linear solution of the steady-state subchannel fluid flow equations. The drift-flux model solves for conservation of liquid and vapor mass, mixture energy, and axial and transverse mixture momentum as part of an efficient planar marching scheme and nonlinear, nested outer and inner iteration. Models based on mechanistic subcooled boiling, two-phase turbulent void mixing, and drift are included. Solution verification and mesh convergence studies were performed for modern GE 10 × 10 fuel geometry and are shown to have excellent convergence behavior. Run time performance for a 50 axial mesh model showed 2.2 seconds on a single CPU core to tightly converge all 3D distributions (flow, void, pressure) for the GE 10 × 10 fuel geometry, supporting its efficient use within the Virtual Environment for Reactor Applications boiling water reactor framework.

Migration of DNAPL in Saturated Porous Media: Validation of High-Resolution Shock-Capturing Numerical Simulations through a Sandbox Experiment

This paper shows a comparison between experiments carried out in a laboratory-scale sandbox where the migration of a dense nonaqueous phase liquid (DNAPL), hydrofluoroether (HFE-7100), in a saturated porous medium was investigated, and validation was performed using high-resolution shock-capturing numerical simulations to resolve the nonlinear governing coupled partial differential equations of a three-phase immiscible fluid flow. The contaminant was released using a colored fluid as a tracer for a fixed time and pressures different from the atmospheric one into the saturated zone, first by using a column laboratory experiment, and then a sandbox-scale example with a hydraulic gradient. A digital image analysis procedure was used to determine the saturation distribution of the contaminant during its migration. These results are compared with the values determined for a DNAPL migration in a similar porous media through a numerical simulation. They show good agreement with the experimental results and also show that CactusHydro can follow the migration of a plume evolution very precisely and can also be used to evaluate the effects and environmental impacts deriving from leaks of DNAPL in saturated zones.

Theoretical Approach to Predicting the Diffusion Radius of Fracture Grouting in Soil–Rock Mixtures

Previously conducted studies have established that the soil–rock mixture in the Chongqing area has the characteristics of loose structure, poor stability, strong permeability, and so on. When building a tunnel in a soil–rock mixture stratum, it is necessary to reinforce the surface rock mass and surrounding rock by grouting to improve the safety of tunnel excavation. To study the diffusion mechanism of cement slurry (Bingham fluid) in soil–rock mixtures, based on the Bingham fluid flow equation and slurry diffusion model, the Bingham fluid fracture diffusion formula was derived, and field grouting tests and indoor model tests were carried out with soil–rock mixtures in the Chongqing area as the research object. The fracture grouting diffusion formula was verified and analyzed using the test data. The research results show that the theoretical calculation results of various working conditions are close to the actual test results (the error of indoor model test results is less than 3%, and the error of field test results is less than 5%). A Bingham fluid fracture diffusion formula has been developed that applies to various working conditions of fracture grouting of soil–rock mixtures and has a good prediction effect on the value of the fracture diffusion radius.

Three‐dimensional numerical simulation of a laminar fluid flow in an enhanced microchannel with circular protrusions

In this research, thermal analysis is performed numerically to investigate the performance of microchannel with different protrusions. Four different configurations of circular protrusions are considered to simulate by computational fluid dynamics. The deionized water has been utilized as working fluid in this research, so that its thermal‐physical properties are dependent on temperature. The finite volume approach is employed for solving the governing equations of the laminar fluid flow in steady state condition. The outcomes of the numerical simulation show a very good agreement compared to experimental data. It can be found that the employing protrusions on the bottom wall impresses the thermal criteria in microchannels. The influence of protrusions on the heat transfer coefficient and the ratio of Nusselt number changes compared to the smooth channel are evaluated at various velocity of the fluid. An important result is that the microchannel with higher radius of protrusions has a better heat transfer rate.

Melting behavior of an organic phase change material in a square thermal energy storage capsule with an array of wire electrodes

A numerical analysis of melting of an organic phase change material (PCM) in a square thermal energy storage (TES) capsule with an array of high voltage wire electrodes has been performed. Fully coupled set of governing equations for fluid flow, heat transfer, phase change, electric field and charge transport are solved using the opensource finite-volume framework of OpenFOAM. The melt interface is tracked using a fixed grid approach based on enthalpy-porosity method. The main objective of the study is to investigate the effect of direction of electric field on the melting process under different configurations of grounded walls. The transient evolution of the melting process in the presence of electric field is mapped in terms of liquid fraction, mean velocity and mean Coulomb force. The melt flow morphology, temperature field, velocity and electric field distribution at different grounded electrode configurations are visualized. Electric field leads to generation and coalescence of small vortices that influence the melt interface, velocity distribution and the total melting performance. 12 cases with different grounded wall configurations that lead to different electric field directions are considered, herein. Grounding the left and right walls of the square TES capsule notably influence the melting performance. Case 4 with grounded left wall led to the shortest melting time of 2.68 h with 97 % reduction, as compared to the case without electric field (5.28 h). Case 5 with grounded right wall has a negative impact on the total melting time (5.55 h), which is 4.86 % higher than the case without electric field (5.28 h). In general, application of electric field led to an increase in total achievable melt fraction for a given period of 6 h. Case 2 with all the four walls grounded led to a maximum melt fraction of 0.97 which is 9.39 % higher than the case without electric field (0.88). Case 4 with grounded left wall showed the maximum power storage capacity of 17.2 kW which is 49.3 % higher than the case without electric field (8.7 kW). Results presented in this study provide deeper insights on the mechanisms of electric field on melting process and will serve as a reference for design of a TES module assisted with electrohydrodynamic (EHD) flow induced by an array of wire electrodes.

Analytical solution of Bloch NMR fluid flow space–time-dependent equation using laplace transform and complex inversion integral

Nuclear magnetic resonance (NMR) is a phenomenon whereby magnetization is excited when static and time varying magnetic fields are applied simultaneously on a given medium such as human blood. The effect of the magnetization causes the protons of the medium to spinning. For several decades now, a set of three Bloch equations are used to describe the dynamics of the spinning protons. Exact solution of the Bloch equations has been the endeavors of many workers with partial success. In about a decade now, a milestone was the appearance of a single NMR fluid flow equation derived from the three set of Bloch equations. The single equation has been found insuperable up to now, defying all efforts to yield a closed form solution. Motivated by the exigency to achieve complete magnetization expression as a function of time and distance, for NMR signal calculations or experiments, we have actualized a closed form solution. We used the method of Laplace transforms and ultimately applied complex inversion theorem to obtain the inverse Laplace transforms. Our final expression is a labyrinth of several oscillatory systems that are characteristics of a nonlinear phenomenon. This is cognate with concepts of chaos.

A class of exact solutions of the Navier–Stokes equations in three and four dimensions

A few basic, intuitive, properties of the Navier–Stokes system of equations for incompressible fluid flows are discussed in this paper. We present a rephrased interpretation of the Navier–Stokes equation in a space having an arbitrary number of dimensions. We then derive spatially periodic solutions for the velocity and pressure fields that span an unbounded domain in three and four dimensions, given a smooth solenoidal initial velocity vector field. In these solutions all velocity components depend non-trivially on all coordinate directions.

Finite droplets vs long droplets: Discrepancy in release conditions in a microscopic constricted channel

Conditions of release of trapped droplets in constricted channels are of great significance in various domains, including microfluidic development and enhanced oil recovery. In our previous studies, a detailed and quantitative analysis of the threshold pressure needed to release a droplet from a constricted channel has been performed. However, droplets may exist in real applications as long droplets, which may exhibit different behavior than finite droplets. Therefore, in this study, direct numerical simulations, combining the fluid flow equations and the phase-field method, have been conducted on three-dimensional constrained channels to investigate discrepancies in release conditions of finite droplets and long droplets. The results have shown that for a finite droplet, the maximum pressure increases with the increase in the contact angle, whereas for a long droplet, the maximum pressure is almost the same both in the water-wet and neutral-wet conditions. Effects of droplet size on the release pressure have also been studied. For the finite droplet and at the water-wet condition (θ = 45°), the minimum release pressure increases linearly with the droplet length, while for the long droplet at similar conditions, the minimum release pressure does not change much as the length of the droplet increases. Furthermore, the release pressure decreases with the increased tapering angle.

Digital core: modeling the temperature field in the hollow space of the rock

This article considers the problem associated with conducting experiments in computational fluid dynamics, the solution of which is the application of the equations of incompressible fluid flow and the introduction of vortex flow functions.The aim of this work is to simulate the temperature viscous flow with a set Reynolds number inside a porous medium enclosed in the rock material to further describe the thermal front of motion in the form of a system of equations, with the choice of the most likely suitable system with differentiated equations for each segment of the thermal simulation (along the cube axes), the final cross-correlation, spread in the identified physical field of confusers/diffusers. Scientific novelty of the performed work is due to the original approach to the development and interpretation of the results of the algorithm for predicting the temperature field in the textural configuration of the confusers/diffusers of rocks.

Time‐dependent pressure‐driven flow in a horizontal cylinder filled with a bidisperse porous material

AbstractSome properties of time‐dependent that modify Brinkman equations for fluid flow in a cylindrical tube filled with Bidisperse Porous Material are discussed in this article. The fluid velocities through the fracture and porous phases of the Bidisperse Porous Medium (BDPM) resulting from the application of pressure gradient are described by two coupled second‐order partial differential equations. Laplace transform technique, D'Alembert and Riemann‐Sum Approximation Methods are used to obtain a semianalytical solution for the model. The choice of the D'Alembert is made to systematically decouple the coupled governing equations without altering their initial orders. The role of the coupling parameter: The coefficient of momentum transfer in the flow formation is considered. Accordingly, three cases are analyzed: (a) weak coupling which described the fluid flow in the absence of the coupling parameter, (b) the strong coupling resulting from a large value of the coupling parameter , and (c) fluid momentum for any arbitrary value of . It is observed that fluid stability is attained when and are decreased; a finding that agrees with the findings of Nield and Kuznetsov and Magyari. Also, the maximum velocity in the fracture phase of the BDPM is attained when the coefficient of momentum transfer is neglected while an opposing flow formation is demonstrated in the fracture and porous phases of BDPM as is increased.