The Efficient Finite Element Methods for Time-Fractional Oldroyd-B Fluid Model Involving Two Caputo Derivatives

In this paper,we consider the numerical schemes for a timefractionalOldroyd-B fluidmodel involving the Caputo derivative.We propose two efficient finite element methods by applying the convolution quadrature in time generated by the backward Euler and the second-order backward difference methods.Error estimates in terms of data regularity are established for both the semidiscrete and fully discrete schemes.Numerical examples for two-dimensional problems further confirmthe robustness of the schemes with first-and second-order accurate in time.

Analytical study of Cattaneo–Christov heat flux model for a boundary layer flow of Oldroyd-B fluid

We investigate the Cattaneo-Christov heat flux model for a two-dimensional laminar boundary layer flow of an incompressible Oldroyd-B fluid over a linearly stretching sheet. Mathematical formulation of the boundary layer problems is given. The nonlinear partial differential equations are converted into the ordinary differential equations using similarity transformatioris. The dimensionless velocity and temperature profiles are obtained through optimal homotopy analysis method （OHAM）. The influences of the physical parameters on the velocity and the temperature are pointed out. The results show that the temperature and the thermal boundary layer thickness are smaller in the Cattaneo-Christov heat flux model than those in the Fourier＇s law of heat conduction.

Boundary layer flow of Oldroyd-B fluid by exponentially stretching sheet

The present paper investigates the steady flow of an Oldroyd-B fluid. The fluid flow is induced by an exponentially stretched surface. Suitable transformations reduce a system of nonlinear partial differential equations to a system of ordinary dif- ferential equations. Convergence of series solution is discussed explicitly by a homotopy analysis method （HAM）. Velocity, temperature and heat transfer rates are examined for different involved parameters through graphs. It is revealed that for a larger retardation time constant, the velocity is enhanced and the temperature is lowered. It is noted that relaxation time constant and the Prandtl number enhance the heat transfer rate.

RHEOLOGY OF SEMICONCENTRATED FIBRE SUSPENSION IN THE OLDROYD-B FLUID

The constitutive equation for a semiconcentrated fibre suspension in the Oldroyd-B fluid has been derived from a statistical model of such a suspension by employing the molecular theory for polymeric liquids.To circumvent theoretical difficulties in viscoelastic fluid mechanics,several simplified models are used to account for the interactions of fibres and polymer molecules.Some of material functions are calculated in terms of the constitutive equation.

Melting heat and thermal radiation effects in stretched flow of an Oldroyd-B fluid

The incompressible flow of a non-Newtonian fluid with mixed convection along a stretching sheet is analyzed. The heat transfer phenomenon is discussed through thermal radiation. The effects of the melting heat transfer and heat generation/absorption are also taken. Suitable transformations are utilized to attain the nonlinear ordinary differential expressions. The convergent series solutions are presented. The fluid flow, temperature, and surface heat transfer rate are examined graphically. It is observed that the velocity decreases when the relaxation time increases while increases when the retardation time is constant. The results also reveal that the temperature distribution reduces when the radiation parameter increases.

Flow of Oldroyd-B fluid with nanoparticles and thermal radiation

The two-dimensional boundary layer flow of an Oldroyd-B fluid in the presence of nanoparticles is investigated. Convective heat and mass conditions are considered in the presence of thermal radiation and heat generation. The Brownian motion and thermophoresis effects are retained. The nonlinear partial differential equations are reduced into the ordinary differential equation （ODE） systems. The resulting ODE systems are solved for the series solutions. The results are analyzed for various physical parameters of interest. Numerical values of the local Nusselt and Sherwood numbers are also computed and analyzed.

Stability of buoyancy-driven convection in an Oldroyd-B fluid-saturated anisotropic porous layer

The nonlinear stability of thermal convection in a layer of an Oldroyd-B fluid-saturated Darcy porous medium with anisotropic permeability and thermal diffu- sivity is investigated with the perturbation method. A modified Darcy-Oldroyd model is used to describe the flow in a layer of an anisotropic porous medium. The results of the linear instability theory are delineated. The thresholds for the stationary and oscillatory convection boundaries are established, and the crossover boundary between them is de- marcated by identifying a codimension-two point in the viscoelastic parameter plane. The stability of the stationary and oscillatory bifurcating solutions is analyzed by deriving the cubic Landau equations. It shows that these solutions always bifurcate supercritically. The heat transfer is estimated in terms of the Nusselt number for the stationary and oscillatory modes. The result shows that, when the ratio of the thermal to mechanical anisotropy parameters increases, the heat transfer decreases.

Nonlinear three-dimensional stretched flow of an Oldroyd-B fluid with convective condition, thermal radiation, and mixed convection

The effect of non-linear convection in a laminar three-dimensional Oldroyd-B fluid flow is addressed. The heat transfer phenomenon is explored by considering the non-linear thermal radiation and heat generation/absorption. The boundary layer as- sumptions are taken into account to govern the mathematical model of the flow analy- sis. Some suitable similarity variables are introduced to transform the partial differen- tial equations into ordinary differential systems. fifth-order techniques with the shooting method The Runge-Kutta-Fehlberg fourth- and are used to obtain the solutions of the dimensionless velocities and temperature. The effects of various physical parameters on the fluid velocities and temperature are plotted and examined. A comparison with the exact and homotopy perturbation solutions is made for the viscous fluid case, and an excellent match is noted. The numerical values of the wall shear stresses and the heat transfer rate at the wall are tabulated and investigated. The enhancement in the values of the Deborah number shows a reverse behavior on the liquid velocities. The results show that the temperature and the thermal boundary layer are reduced when the non- linear convection parameter increases. The values of the Nusselt number are higher in the non-linear radiation situation than those in the linear radiation situation.

Three-dimensional flow of Oldroyd-B fluid over surface with convective boundary conditions

The present study addresses the three-dimensional flow of an Oldroyd-B fluid over a stretching surface with convective boundary conditions. The problem formulation is presented using the conservation laws of mass, momentum, and energy. The solutions to the dimensionless problems are computed. The convergence of series solutions by the homotopy analysis method （HAM） is discussed graphically and numerically. The graphs are plotted for various parameters of the temperature profile. The series solutions are verified by providing a comparison in a limiting case. The numerical values of the local Nusselt number are analyzed.

Axial Couette flow of an Oldroyd-B fluid in an annulus

This paper establishes the velocity field and the adequate shear stress corresponding to the motion of an Oldroyd-B fluid between two infinite coaxial circular cylinders by means of finite Hankel transforms. The flow of the fluid is produced by the inner cylinder which applies a time-dependent longitudinal shear stress to the fluid. The exact analytical solutions, presented in series form in terms of Bessel functions, satisfy all imposed initial and boundary conditions. The general solutions can be easily specialized to give similar solutions for Maxwell, second grade and Newtonian fluids performing the same motion. Finally, some characteristics of the motion as well as the influence of the material parameters on the behavior of the fluid motion are graphically illustrated.

Unsteady Incompressible Flow of a Generalized Oldroyd-B Fluid between Two Oscillating Infinite Parallel Plates in Presence of a Transverse Magnetic Field

In this paper an attempt has been made to study the unsteady incompressible flow of a generalized Oldroyd-B fluid between two oscillating parallel plates in presence of a transverse magnetic field. An exact solution for the velocity field has been obtained by means of Laplace and finite Fourier sine transformations in series form in terms of Mittage-Leffler function. The dependence of the velocity field on fractional as well as material parameters has been illustrated graphically. The velocity fields for the classical Newtonian, generalized Maxwell, generalized second grade and ordinary Oldroyd-B fluids are recovered as limiting cases of the flow considered for the generalized Oldroyd-B fluid.

The Oscillatory Motion of Oldroyd-B Fluid by Incorporating Some of the Mechanical Factors

The aim of this paper is to present a numerical study of oscillatory motion of Oldroyd-B fluid in a uniform magnetic field through a small circular pipe. First, we derive the orientation stress tensor by considering the Brownian force. Then, the orientation stress tensor is incorporated by taking Hookean dumbbells on Brownian configuration fields in the Oldroyd-B model. The Oldroyd-B model is then reformulated coupled with the momentum equation and the total stress tensor. Finally, we analyze the orientation stress tensor in the pipe by the numerical simulations of the model and showed that the effect of orientation stress tensor is considerable although the Brownian force is sufficiently small.

A Note on Hydromagnetic Flow of an Oldroyd-B Fluid near an Infinite Plate Induced by Half Rectified Sine Pulses

An initial value problem concerning the motion of an incompressible, electrically conducting, viscoelastic Oldroyd-B fluid bounded by an infinite rigid non-conducting plate is solved. The unsteady motion is generated impulsively from rest in the fluid due to half rectified sine pulses subjected on the plate in its own plane in presence of an external magnetic field. It is assumed that no external electric field is acting on the system and the magnetic Reynolds number is very small. The operational method is used to obtain exact solutions for the fluid velocity and the shear stress on the wall. Quantitative analysis of the results is presented with a view to disclose the simultaneous effects of the external magnetic field and the fluid elasticity on the flow and the wall shear stress for different periods of pulsation of the plate. It is also shown that the classical and hydromagnetic Rayleigh solutions appear as the limiting cases of the present analysis.

Oldroyd-B黏弹性液滴撞击弯曲壁面的数值研究

液滴撞击弯曲壁面现象广泛存在于冶金、化工和航空航天等领域,在某些场景下,液滴混入高分子聚合物后会表现黏弹性特性,为了进一步认识液滴黏弹性对撞击弯曲壁面的影响,对黏弹性液滴撞击弯曲壁面的过程进行数值模拟研究.研究基于相场方法和格子玻尔兹曼(LBM)方法,采用应力场分布函数求解Oldroyd-B本构方程,并施加适用于弯曲壁面的接触角模型,发展了固-液-气三相黏弹性流体模拟方法,对Oldroyd-B黏弹性液滴撞击弯曲壁面问题进行数值模拟,主要研究了黏度比β、韦伯数We和壁面接触角θ对撞击过程的影响.结果表明:撞击过程主要包括4个阶段:运动阶段、铺展阶段、拉伸阶段和撕裂阶段.黏度比β越低,液滴在铺展拉伸阶段动能衰减越慢,转化的表面能更多,但更早进入撕裂阶段,液滴撞击弯曲壁面后,更容易脱离壁面.韦伯数We较小时,液滴主要在壁面处附着或反弹;We较大时,液滴会撕裂并脱离壁面,We越大,液滴在铺展拉伸阶段动能衰减越快,进入撕裂阶段更慢.壁面的亲疏水性会影响液滴最终的状态,疏水性越高,对铺展阶段的阻碍作用越强,液滴越容易发生撕裂和脱离.

Effect of fracture fluid flowback on shale microfractures using CT scanning

The field data of shale fracturing demonstrate that the flowback performance of fracturing fluid is different from that of conventional reservoirs,where the flowback rate of shale fracturing fluid is lower than that of conventional reservoirs.At the early stage of flowback,there is no single-phase flow of the liquid phase in shale,but rather a gas-water two-phase flow,such that the single-phase flow model for tight oil and gas reservoirs is not applicable.In this study,pores and microfractures are extracted based on the experimental results of computed tomography(CT)scanning,and a spatial model of microfractures is established.Then,the influence of rough microfracture surfaces on the flow is corrected using the modified cubic law,which was modified by introducing the average deviation of the microfracture height as a roughness factor to consider the influence of microfracture surface roughness.The flow in the fracture network is simulated using the modified cubic law and the lattice Boltzmann method(LBM).The results obtained demonstrate that most of the fracturing fluid is retained in the shale microfractures,which explains the low fracturing fluid flowback rate in shale hydraulic fracturing.

Synthetic polymers:A review of applications in drilling fluids

With the growth of deep drilling and the complexity of the well profile,the requirements for a more complete and efficient exploitation of productive formations increase,which increases the risk of various complications.Currently,reagents based on modified natural polymers(which are naturally occurring compounds)and synthetic polymers(SPs)which are polymeric compounds created industrially,are widely used to prevent emerging complications in the drilling process.However,compared to modified natural polymers,SPs form a family of high-molecular-weight compounds that are fully synthesized by undergoing chemical polymerization reactions.SPs provide substantial flexibility in their design.Moreover,their size and chemical composition can be adjusted to provide properties for nearly all the functional objectives of drilling fluids.They can be classified based on chemical ingredients,type of reaction,and their responses to heating.However,some of SPs,due to their structural characteristics,have a high cost,a poor temperature and salt resistance in drilling fluids,and degradation begins when the temperature reaches 130℃.These drawbacks prevent SP use in some medium and deep wells.Thus,this review addresses the historical development,the characteristics,manufacturing methods,classification,and the applications of SPs in drilling fluids.The contributions of SPs as additives to drilling fluids to enhance rheology,filtrate generation,carrying of cuttings,fluid lubricity,and clay/shale stability are explained in detail.The mechanisms,impacts,and advances achieved when SPs are added to drilling fluids are also described.The typical challenges encountered by SPs when deployed in drilling fluids and their advantages and drawbacks are also discussed.Economic issues also impact the applications of SPs in drilling fluids.Consequently,the cost of the most relevant SPs,and the monomers used in their synthesis,are assessed.Environmental impacts of SPs when deployed in drilling fluids,and their manufacturing processes are identified,together with advances in SP-treatment methods aimed at reducing those impacts.Recommendations for required future research addressing SP property and performance gaps are provided.

Reservoir quality evaluation of the Narimba Formation in Bass Basin,Australia:Implications from petrophysical analysis,sedimentological features,capillary pressure and wetting fluid saturation

The evaluation of reservoir quality was accomplished on the Late Paleocene to Early Eocene Narimba Formation in Bass Basin,Australia.This study involved combination methods such as petrophysical analysis,petrography and sedimentological studies,reservoir quality and fluid flow units from derivative parameters,and capillary pressure and wetting fluid saturation relationship.Textural and diagenetic features are affecting the reservoir quality.Cementation,compaction,and presence of clay minerals such as kaolinite are found to reduce the quality while dissolution and secondary porosity are noticed to improve it.It is believed that the Narimba Formation is a potential reservoir with a wide range of porosity and permeability.Porosity ranges from 3.1%to 25.4%with a mean of 15.84%,while permeability ranges between 0.01 mD and 510 mD,with a mean of 31.05 mD.Based on the heterogenous lithology,the formation has been categorized into five groups based on permeability variations.Group I showed an excellent to good quality reservoir with coarse grains.The impacts of both textural and diagenetic features improve the reservoir and producing higher reservoir quality index(RQI)and flow zone indicators(FZI)as well as mostly mega pores.The non-wetting fluid migration has the higher possibility to flow in the formation while displacement pressure recorded as zero.Group II showed a fair quality reservoir with lower petrophysical properties in macro pores.The irreducible water saturation is increasing while the textural and digenetic properties are still enhancing the reservoir quality.Group III reflects lower quality reservoir with mostly macro pores and higher displacement pressure.It may indicate smaller grain size and increasing amount of cement and clay minerals.Group IV,and V are interpreted as a poor-quality reservoir that has lower RQI and FZI.The textural and digenetic features are negatively affecting the reservoir and are leading to smaller pore size and pore throat radii(r35)values to be within the range of macro,meso-,micro-,and nano pores.The capillary displacement pressure curves of the three groups show increases reaching the maximum value of 400 psia in group V.Agreement with the classification of permeability,r35 values,and pore type can be used in identifying the quality of reservoir.

Effect of bubble morphology and behavior on power consumption in non-Newtonian fluids’aeration process

Due to a prolonged operation time and low mass transfer efficiency, the primary challenge in the aeration process of non-Newtonian fluids is the high energy consumption, which is closely related to the form and rate of impeller, ventilation, rheological properties and bubble morphology in the reactor. In this perspective, through optimal computational fluid dynamics models and experiments, the relationship between power consumption, volumetric mass transfer rate(kLa) and initial bubble size(d0) was constructed to establish an efficient operation mode for the aeration process of non-Newtonian fluids. It was found that reducing the d0could significantly increase the oxygen mass transfer rate, resulting in an obvious decrease in the ventilation volume and impeller speed. When d0was regulated within 2-5 mm,an optimal kLa could be achieved, and 21% of power consumption could be saved, compared to the case of bubbles with a diameter of 10 mm.

Microscopic characteristics of tight sandstone reservoirs and their effects on the imbibition efficiency of fracturing fluids:A case study of the Linxing area,Ordos Basin

The Linxing area within the Ordos Basin exhibits pronounced reservoir heterogeneity and intricate micro-pore structures,rendering it susceptible to water-blocking damage during imbibition extraction.This study delved into the traits of tight sandstone reservoirs in the 8th member of the Shihezi Formation(also referred to as the He 8 Member)in the study area,as well as their effects on fracturing fluid imbibition.Utilizing experimental techniques such as nuclear magnetic resonance(NMR),high-pressure mercury intrusion(HPMI),and gas adsorption,this study elucidated the reservoir characteristics and examined the factors affecting the imbibition through imbibition experiments.The findings reveal that:①The reservoir,with average porosity of 8.40%and average permeability of 0.642×10^(-3)μm^(2),consists principally of quartz,feldspar,and lithic fragments,with feldspathic litharenite serving as the primary rock type and illite as the chief clay mineral;②Nano-scale micro-pores and throats dominate the reservoir,with dissolution pores and intercrystalline pores serving as predominant pore types,exhibiting relatively high pore connectivity;③Imbibition efficiency is influenced by petrophysical properties,clay mineral content,and microscopic pore structure.Due to the heterogeneity of the tight sandstone reservoir,microscopic factors have a more significant impact on the imbibition efficiency of fracturing fluids;④A comparative analysis shows that average pore size correlates most strongly with imbibition efficiency,followed by petrophysical properties and clay mineral content.In contrast,the pore type has minimal impact.Micropores are vital in the imbibition process,while meso-pores and macro-pores offer primary spaces for imbibition.This study offers theoretical insights and guidance for enhancing the post-fracturing production of tight sandstone reservoirs by examining the effects of these factors on the imbibition efficiency of fracturing fluids in tight sandstones.

Endoscopic intramural cystogastrostomy for treatment of peripancreatic fluid collection: A viewpoint from a surgeon

Percutaneous or endoscopic drainage is the initial choice for the treatment of peripancreatic fluid collection in symptomatic patients.Endoscopic transgastric fenestration(ETGF)was first reported for the management of pancreatic pseu-docysts of 20 patients in 2008.From a surgeon’s viewpoint,ETGF is a similar procedure to cystogastrostomy in that they both produce a wide outlet orifice for the drainage of fluid and necrotic debris.ETGF can be performed at least 4 wk after the initial onset of acute pancreatitis and it has a high priority over the surgical approach.However,the surgical approach usually has a better success rate because surgical cystogastrostomy has a wider outlet(>6 cm vs 2 cm)than ETGF.However,percutaneous or endoscopic drainage,ETGF,and surgical approach offer various treatment options for peripancreatic fluid collection patients based on their conditions.