scholarly journals Finite Volume Simulations of Particle-laden Viscoelastic Fluid Flows: Application To Hydraulic Fracture Processes

C. Fernandes ◽  
S. A. Faroughi ◽  
R. Ribeiro ◽  
A. Isabel ◽  
G. H. McKinley

Abstract Accurately resolving the coupled momentum transfer between the liquid and solid phases of complex fluids is a fundamental problem in multiphase transport processes, such as hydraulic fracture operations. Specifically we need to characterize the dependence of the normalized average fluid-particle force < F > on the volume fraction of the dispersed solid phase and on the rheology of the complex fluid matrix, parameterized through the Weissenberg number Wi measuring the relative magnitude of elastic to viscous stresses in the fluid. Here we use direct numerical simulations (DNS) to study the creeping flow (Re << 1) of viscoelastic fluids through static random arrays of monodisperse spherical particles using a finite volume Navier-Stokes/Cauchy momentum solver. The numerical study consists of N = 150 different systems, in which the normalized average fluid-particle force <F> is obtained as a function of the volume fraction φ (0 < φ ≤ 0.2) of the dispersed solid phase and the Weissenberg number Wi (0 ≤ Wi ≤ 4). From these predictions a closure law < F >( Wi,φ ) for the drag force is derived for the quasi-linear Oldroyd-B viscoelastic fluid model (with fixed retardation ratio β = 0.5) which is, on average, within 5.7% of the DNS results. Additionally, a flow solver able to couple Eulerian and Lagrangian phases (in which the particulate phase is modeled by the discrete particle method (DPM)) is developed, which incorporates the viscoelastic nature of the continuum phase and the closed-form drag law. Two case studies were simulated using this solver, in order to assess the accuracy and robustness of the newly-developed approach for handling particle-laden viscoelastic flow configurations with O (10 5 − 10 6 ) rigid spheres that are representative of hydraulic fracture operations. Three-dimensional settling processes in a Newtonian fluid and in a quasi-linear Oldroyd-B viscoelastic fluid are both investigated using a rectangular channel and an annular pipe domain. Good agreement is obtained for the particle distribution measured in a Newtonian fluid, when comparing numerical results with experimental data. For the cases in which the continuous fluid phase is viscoelastic we compute the evolution in the velocity fields and predicted particle distributions are presented at different elasticity numbers 0 ≤ El ≤ 30 (where El = Wi/Re ) and for different suspension particle volume fractions.

2011 ◽  
Vol 677 ◽  
pp. 272-304 ◽  
A. M. AFONSO ◽  
F. T. PINHO ◽  

High-elasticity simulations of flows through a two-dimensional (2D) 4 : 1 abrupt contraction and a 4 : 1 three-dimensional square–square abrupt contraction were performed with a finite-volume method implementing the log-conformation formulation, proposed by Fattal & Kupferman (J. Non-Newtonian Fluid Mech., vol. 123, 2004, p. 281) to alleviate the high-Weissenberg-number problem. For the 2D simulations of Boger fluids, modelled by the Oldroyd-B constitutive equation, local flow unsteadiness appears at a relatively low Deborah number (De) of 2.5. Predictions at higher De were possible only with the log-conformation technique and showed that the periodic unsteadiness grows with De leading to an asymmetric flow with alternate back-shedding of vorticity from pulsating upstream recirculating eddies. This is accompanied by a frequency doubling mechanism deteriorating to a chaotic regime at high De. The log-conformation technique provides solutions of accuracy similar to the thoroughly tested standard finite-volume method under steady flow conditions and the onset of a time-dependent solution occurred approximately at the same Deborah number for both formulations. Nevertheless, for Deborah numbers higher than the critical Deborah number, and for which the standard iterative technique diverges, the log-conformation technique continues to provide stable solutions up to quite (impressively) high Deborah numbers, demonstrating its advantages relative to the standard methodology. For the 3D contraction, calculations were restricted to steady flows of Oldroyd-B and Phan-Thien–Tanner (PTT) fluids and very high De were attained (De ≈ 20 for PTT with ϵ = 0.02 and De ≈ 10000 for PTT with ϵ = 0.25), with prediction of strong vortex enhancement. For the Boger fluid calculations, there was inversion of the secondary flow at high De, as observed experimentally by Sousa et al. (J. Non-Newtonian Fluid Mech., vol. 160, 2009, p. 122).

Micromachines ◽  
2019 ◽  
Vol 10 (4) ◽  
pp. 262 ◽  
Zhang ◽  
Zhang ◽  
Wu ◽  
Shen ◽  
Chen ◽  

Fluid mixing plays an essential role in many microfluidic applications. Here, we compare the mixing in time pulsing flows for both a Newtonian fluid and a viscoelastic fluid at different pulsing frequencies. In general, the mixing degree in the viscoelastic fluid is higher than that in the Newtonian fluid. Particularly, the mixing in Newtonian fluid with time pulsing is decreased when the Reynolds number Re is between 0.002 and 0.01, while it is enhanced when Re is between 0.1 and 0.2 compared with that at a constant flow rate. In the viscoelastic fluid, on the other hand, the time pulsing does not change the mixing degree when the Weissenberg number Wi ≤ 20, while a larger mixing degree is realized at a higher pulsing frequency when Wi = 50.

2021 ◽  
Vol 36 (3) ◽  
pp. 165-176
Kirill Nikitin ◽  
Yuri Vassilevski ◽  
Ruslan Yanbarisov

Abstract This work presents a new approach to modelling of free surface non-Newtonian (viscoplastic or viscoelastic) fluid flows on dynamically adapted octree grids. The numerical model is based on the implicit formulation and the staggered location of governing variables. We verify our model by comparing simulations with experimental and numerical results known from the literature.

Symmetry ◽  
2020 ◽  
Vol 12 (9) ◽  
pp. 1475
Humaira Yasmin ◽  
Naveed Iqbal ◽  
Aiesha Hussain

The peristaltic flow of Johnson–Segalman fluid in a symmetric curved channel with convective conditions and flexible walls is addressed in this article. The channel walls are considered to be compliant. The main objective of this article is to discuss the effects of curvilinear of the channel and heat/mass convection through boundary conditions. The constitutive equations for Johnson–Segalman fluid are modeled and analyzed under lubrication approach. The stream function, temperature, and concentration profiles are derived. The analytical solutions are obtained by using regular perturbation method for significant number, named as Weissenberg number. The influence of the parameter values on the physical level of interest is outlined and discussed. Comparison is made between Jhonson-Segalman and Newtonian fluid. It is concluded that the axial velocity of Jhonson-Segalman fluid is substantially higher than that of Newtonian fluid.

2021 ◽  
Bertrand Rollin ◽  
Frederick Ouellet ◽  
Bradford Durant ◽  
Rahul Babu Koneru ◽  
S. Balachandar

Abstract We study the interaction of a planar air shock with a perturbed, monodispersed, particle curtain using point-particle simulations. In this Eulerian-Lagrangian approach, equations of motion are solved to track the position, momentum, and energy of the computational particles while the carrier fluid flow is computed in the Eulerian frame of reference. In contrast with many Shock-Driven Multiphase Instability (SDMI) studies, we investigate a configuration with an initially high particle volume fraction, which produces a strongly two-way coupled flow in the early moments following the shock-solid phase interaction. In the present study, the curtain is about 4 mm in thickness and has a peak volume fraction of about 26%. It is composed of spherical particles of d = 115μm in diameter and a density of 2500 kg.m−3, thus replicating glass particles commonly used in multiphase shock tube experiments or multiphase explosive experiments. We characterize both the evolution of the perturbed particle curtain and the gas initially trapped inside the particle curtain in our planar three-dimensional numerical shock tube. Control parameters such as the shock strength, the particle curtain perturbation wavelength and particle volume fraction peak-to-trough amplitude are varied to quantify their influence on the evolution of the particle cloud and the initially trapped gas. We also analyze the vortical motion in the flow field. Our results indicate that the shock strength is the primary contributor to the cloud particle width. Also, a classic Richtmyer-Meshkov instability mixes the gas initially trapped in the particle curtain and the surrounding gas. Finally, we observe that the particle cloud contribute to the formation of longitudinal vortices in the downstream flow.

2006 ◽  
Shriram Pillapakkam ◽  
Pushpendra Singh ◽  
Denis L. Blackmore ◽  
Nadine Aubry

A finite element code based on the level set method is developed for performing two and three dimensional direct numerical simulations (DNS) of viscoelastic two-phase flow problems. The Oldroyd-B constitutive equation is used to model the viscoelastic liquid and both transient and steady state shapes of bubbles in viscoelastic buoyancy driven flows are studied. The influence of the governing dimensionless parameters, namely the Capillary number (Ca), the Deborah Number (De) and the polymer concentration parameter c, on the deformation of the bubble is also analyzed. Our simulations demonstrate that the rise velocity oscillates before reaching a steady value. The shape of the bubble, the magnitude of velocity overshoot and the amount of damping depend mainly on the parameter c and the bubble radius. Simulations also show that there is a critical bubble volume at which there is a sharp increase in the bubble terminal velocity as the increasing bubble volume increases, similar to the behavior observed in experiments. The structure of the wake of a bubble rising in a Newtonian fluid is strikingly different from that of a bubble rising in a viscoelastic fluid. In addition to the two recirculation zones at the equator of the bubble rising in a Newtonian fluid, two more recirculation zones exist in the wake of a bubble rising in viscoelastic fluids which influence the shape of a rising bubble. Interestingly, the direction of motion of the fluid a short distance below the trailing edge of a bubble rising in a viscoelastic fluid is in the opposite direction to the direction of the motion of the bubble, thus creating a “negative wake”. In this paper, the velocity field in the wake of the bubble, the effect of the parameters on the velocity field and their influence on the shape of the bubble are also investigated.

2009 ◽  
Vol 132 (1) ◽  
Arunn Narasimhan ◽  
B. V. K. Reddy

Bidisperse porous medium (BDPM) consists of a macroporous medium whose solid phase is replaced with a microporous medium. This study investigates using numerical simulations, steady natural convection inside a square BDPM enclosure made from uniformly spaced, disconnected square porous blocks that form the microporous medium. The side walls are subjected to differential heating, while the top and bottom ones are kept adiabatic. The bidispersion effect is generated by varying the number of blocks (N2), macropore volume fraction (ϕE), and internal Darcy number (DaI) for several enclosure Rayleigh numbers (Ra). Their effect on the BDPM heat transfer (Nu) is investigated. When Ra is fixed, the Nu increases with an increase in both DaI and DaE. At low Ra values, Nu is strongly affected by both DaI and ϕE. When N2 is fixed, at high Ra values, the porous blocks in the core region have negligible effect on the Nu. A correlation is proposed to evaluate the heat transfer from the BDPM enclosure, Nu, as a function of Raϕ, DaE, DaI, and N2. It predicts the numerical results of Nu within ±15% and ±9% in two successive ranges of modified Rayleigh number, RaϕDaE.

Kaushik Das ◽  
Debashis Basu ◽  
Todd Mintz

The present study makes a comparative assessment of different turbulence models in simulating the flow-assisted corrosion (FAC) process for pipes with noncircular cross sections and bends, features regularly encountered in heat exchangers and other pipeline networks. The case study investigates material damage due to corrosion caused by dissolved oxygen (O2) in a stainless steel pipe carrying an aqueous solution. A discrete solid phase is also present in the solution, but the transport of the solid particles is not explicitly modeled. It is assumed that the volume fraction of the solid phase is low, so it does not affect the continuous phase. Traditional two-equation models are compared, such as isotropic eddy viscosity, standard k-ε and k-ω models, shear stress transport (SST) k-ω models, and the anisotropic Reynolds Stress Model (RSM). Computed axial and radial velocities, and turbulent kinetic energy profiles predicted by the turbulence models are compared with available experimental data. Results show that all the turbulence models provide comparable results, though the RSM model provided better predictions in certain locations. The convective and diffusive motion of dissolved O2 is calculated by solving the species transport equations. The study assumes that solid particle impingement on the pipe wall will completely remove the protective film formed by corrosion products. It is also assumed that the rate of corrosion is controlled by diffusion of O2 through the mass transfer boundary layer. Based on these assumptions, corrosion rate is calculated at the internal pipe walls. Results indicate that the predicted O2 corrosion rate along the walls varies for different turbulence models but show the same general trend and pattern.

2018 ◽  
Vol 140 (9) ◽  
Teng Zhou ◽  
Yongbo Deng ◽  
Hongwei Zhao ◽  
Xianman Zhang ◽  
Liuyong Shi ◽  

Viscoelastic solution is encountered extensively in microfluidics. In this work, the particle movement of the viscoelastic flow in the contraction–expansion channel is demonstrated. The fluid is described by the Oldroyd-B model, and the particle is driven by dielectrophoretic (DEP) forces induced by the applied electric field. A time-dependent multiphysics numerical model with the thin electric double layer (EDL) assumption was developed, in which the Oldroyd-B viscoelastic fluid flow field, the electric field, and the movement of finite-size particles are solved simultaneously by an arbitrary Lagrangian–Eulerian (ALE) numerical method. By the numerically validated ALE method, the trajectories of particle with different sizes were obtained for the fluid with the Weissenberg number (Wi) of 1 and 0, which can be regarded as the Newtonian fluid. The trajectory in the Oldroyd-B flow with Wi = 1 is compared with that in the Newtonian fluid. Also, trajectories for different particles with different particle sizes moving in the flow with Wi = 1 are compared, which proves that the contraction–expansion channel can also be used for particle separation in the viscoelastic flow. The above results for this work provide the physical insight into the particle movement in the flow of viscous and elastic features.

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