Fluid Of Constant Viscosity Research Articles

Analytical solution for chemical fluid injection into linear heterogeneous porous media based on the method of characteristics

Injection of Newtonian fluids in porous media to displace linearly another Newtonian fluid is described by the Buckley-Leverett (BL) fractional flow theory derived for incompressible, constant viscosity fluids at constant injection rate in a homogeneous and isotropic reservoir. Injection of non-Newtonian fluids can be addressed by modifying the fractional flow equation by considering the dependence of injected fluid viscosity with velocity, changes of residual oil saturation and chemical adsorption on rock. In this work, the BL fractional flow model is extended to describe injection of non-Newtonian fluids into a 1D linear heterogeneous formation accounting for fluid adsorption, permeability reduction, inaccessible pore volume, and presence of a denuded water bank developed over time ahead of non-Newtonian zone. Results from derived analytical solutions demonstrate fluids saturation and pressure drop across the formation, fluids production rate and cumulative volumes to be in close agreement with the ones from a commercial reservoir simulator.

Extensional rheometry of mobile fluids. Part I: OUBER, an optimized uniaxial and biaxial extensional rheometer

Numerical optimization of a “six-arm cross-slot” device yields several three-dimensional shapes of fluidic channels that impose close approximations to an ideal uniaxial (biaxial) stagnation point extensional flow under the constraints of having four inlets and two outlets (two inlets and four outlets) and for Newtonian creeping flow. One of the numerically designed geometries is considered suitable for fabrication at the microscale, and numerical simulations with the Oldroyd-B and Phan-Thien and Tanner models confirm that the optimal flow fields are observed in the geometry for both constant viscosity and shear thinning viscoelastic fluids. The geometry, named the optimized uniaxial and biaxial extensional rheometer (OUBER), is microfabricated with high precision by selective laser-induced etching of a fused-silica substrate. Employing a refractive index-matched viscous Newtonian fluid, microtomographic-particle image velocimetry enables the measurement of the flow field in a substantial volume around the stagnation point. The flow velocimetry, performed at low Reynolds number (<0.1), confirms the accurate imposition of the desired and predicted flows, with a pure extensional flow at an essentially uniform deformation rate being applied over a wide region around the stagnation point. In Part II of this paper [Haward et al., J. Rheol. 67, 1011–1030 (2023)], pressure drop measurements in the OUBER geometry are used to assess the uniaxial and biaxial extensional rheometry of dilute polymeric solutions, in comparison to measurements made in planar extension using an optimized-shape cross-slot extensional rheometer [OSCER, Haward et al., Phys. Rev. Lett. 109, 128301 (2012)].

Resistive-force theory of slender bodies in viscosity gradients

In many natural settings, spatial variations in a fluid's viscosity arise due to changes in its local physico-chemical environment. We consider the low-Reynolds-number dynamics of slender bodies in fluids with a linearly varying viscosity. Assuming the spatial change in viscosity to be small compared to the spatial deviation of the body but large relative to the body's aspect ratio, we derive the modifications to resistive-force theory in a fluid due to a constant viscosity gradient. At leading order in the body slenderness, the results are identical to the classical theory in a constant-viscosity fluid but with the viscosity taking everywhere its local, non-constant value. At next order in slenderness, non-local terms arise due to the non-zero viscosity gradient. We use our results to predict the motion of straight and toroidal filaments settling under the action of gravity. We show that viscosity gradients induce rigid-body rotation of the filaments at a rate proportional to the components of the gradients along the filaments. This result contrasts with constant-viscosity fluids where the filaments do not rotate. We demonstrate further that if the viscosity gradient acts in the direction opposite to the gravitational field, then the filaments rotate towards a stable orientation, whose value depends on the ratio between the viscosity gradients parallel and perpendicular to the gravitational field; otherwise, the filaments align in the direction of the gravitational field. Our work shows that viscosity gradients can exert new forces on slender bodies, which could, for example, be used to control their orientation and drift.

Scaling of Transient Natural Convection Cooling in Side-Cooled Cavity: The Effect of Variable Viscosity

The transient natural convection of variable and constant viscosity fluids in a side cooled cubical cavity is studied experimentally. The convective flow in the cavity is visualized by the shadowgraph technique. The fluid was initially preheated and enough time is allowed to assure uniform temperature distribution. Temperature difference is introduced by a sudden cooling of the two side walls. The results indicate that the transient flow development is characterized by the following distinct processes: the initial growth of the vertical boundary layer and horizontal intrusions, the interaction of the horizontal intrusions and the stratification. New scaling relations are introduced to include the effect of the viscosity variation with temperature. The new suggested scalings showed good agreement with the measurements based on the shadowgraph images.

The effects of silica aerogel on the rheological behaviour and polymer matrix structure of Rigid Polyurethane foams

The incorporation of silica aerogel particles into the isocyanate used to prepare Rigid Polyurethane composite foams drastically modifies its rheological behaviour. The isocyanate changes from being a Newtonian fluid of constant viscosity (0.3 Pa s at 25 °C) to a non-linear power-law material with complex viscosity decreasing from 104 to 101 Pa s with a slope −1 versus frequency. The change in response is caused by the formation of a percolated network of particles at contents above 2.5 wt% of aerogel. Moreover, the addition of aerogel significantly slows down the modulus-build up kinetics and vitrification of the matrix by approximately 4.5 min with every 1 wt% of aerogel added. For the foam with 3.0 wt% of aerogel, the combination of high viscosity at early reaction times coupled with the slow modulus build-up impedes the drainage which leads to a decrease in the fraction of material in the struts of the matrix.

Study of Rayleigh-Taylor instability in viscosity-stratified fluid layers

The Rayleigh-Taylor instability in viscosity varying fluid layers is studied. A heavier fluid layer is superimposed on a lighter fluid layer. Both the fluid layers are bounded between two parallel isothermal horizontal walls. Out of the two walls, one wall is isothermally heated, while the other wall is isothermally cooled. The Rayleigh-Taylor instability is studied for axisymmetric configuration. Viscosities of both the fluids are considered to be varying with temperature. The effect of viscosity ratio, temperature ratio, heating location, Prandtl number and Weber number on the instability is studied. When the viscosity of fluid layers vary with temperature, the configuration becomes more unstable compared to that for constant viscosity fluid layers. For varying viscosity fluid layers the spike undergoes large deformations. At lower viscosity ratios, mushroom shaped spike is formed with elongated skirt structure. Whereas, at higher viscosity ratios, spike with fluid column structure is formed. It is found that increasing viscosity ratio shows stabilizing effect. Increasing temperature ratio is found to show destabilizing effect with complex spike structure formation at high temperature ratios. In top wall heating configuration, formation of mushroom shaped spike and skirt of the spike occurs earlier in a thinner and elongated form compared to that of bottom wall heating configuration. Prandtl number showed insignificant effect on the instability. For the parameter values of the study, when Weber is lower than 55 the configuration is stable and does not undergo instability. Overall, surface tension has shown stabilizing effect on the instability.

Numerical study of flow and heat transfer in a porous medium between two stretchable disks using quasi-linearization method

In this study, the flow as well as heat transfer of a classical Newtonian fluid of constant density and viscosity in a porous medium between two radially stretching disks is explored. The role of the porosity of the medium, the stretching of the disks, the viscous dissipation, and radiation on the flow and temperature fields is taken into account. The flow and heat equations are transformed into non-linear ODE by invoking the classical similarity transformations. These non-linear differential equations were linearized using Quasi linearization method. Further the linearized equations were discretized by employing the finite differences which were then solved numerically using the successive over relaxation parameter method. Some features of the flow and temperature are discussed in detail in the form of tables and graphs. The present study may be beneficial in lubricants and computational storage devices as well as fluid-flows and heat transmission in rotor-stator systems.

Axisymmetric squirmers in Stokes fluid with nonuniform viscosity

The ciliary locomotion and feeding of an axisymmetric micro-swimmer in a complex fluid whose viscosity depends on nutrient concentration are investigated numerically. The micro-swimmer is modeled as having spheroidal geometry, and ciliary beating is modeled by a slip velocity; i.e. a squirmer is adapted. Looking at the coupling between swimming and feeding of spheroidal squirmers, it is found that swimming speed and feeding are most affected by a non-uniform viscosity environment when the ratio of advection-forces to diffusion-transport, characterized by the nondimensional Pecl\'et number, is moderate ($\text{Pe}\approx 10$). These changes are correlated to significant increases in the pressure force on the surface of the squirmer. The swimming and feeding changes are found to be more significant in oblate spheroids than prolate spheroids. Most interestingly, nontrivial symmetric slip boundary conditions on the surface of the squirmer, which results in zero net motion in a constant viscosity fluid, can yield non-zero propulsion when paired with asymmetric nutrient boundary conditions. The fact that symmetric fluid boundary conditions can result in asymmetric propulsion has implications for the design of artificial micro-swimmers and other low-$\text{Re}$ squirmers in complex fluids.

Numerical computing for axisymmetric transport phenomenon in Carreau liquid using variable conductance models

Conservation laws and variable conductance (viscosity, thermal conductivity and mass diffusion coefficient) models are used to develop mathematical problems describing transport mechanism in Carreau fluid over a non-uniformly moving surface. Boundary conditions are developed by no-slip theory. Mathematical models are transformed into suitable residual integrals which are approximated by Galerkin approximations. The obtained residuals are used for solving problems numerically using finite element method. Numerical investigation of variable viscosity, thermal conductivity and mass diffusion coefficients is carried out to examine the impact of parameters. The heat dissipated as a result of friction among the particles of the fluid of constant viscosity is greater than the heat dissipated due to friction force in the fluid of temperature-dependent viscosity. It is also found that ohmic phenomenon in the fluid of variable viscosity is prominent than that in the fluid of constant viscosity. Therefore, this fact must be in mind while using the fluids of variable viscosity in engineering applications. The transport of mass in fluid of constant viscosity is greater than that in fluid of variable viscosity. The Lorentz force is observed to oppose the flow. Hence, flow is decelerated by an increase in the intensity of magnetic field. The rate of transportation of mass has shown increasing trend when mass diffusion coefficient increases due to temperature.

Combined effects of partial slip and variable diffusion coefficient on mass and heat transfer subjected to chemical reaction

This article is a generalization of already published work and investigates the combined effects of chemical reaction, partial slip, temperature and concentration, jumps on the transport of heat energy and solute of variable mass conductance in a fluid of temperature dependent viscosity and thermal conductivity. Publish benchmark (Hayat et al 2015 Partial slip effects in flow over nonlinear stretching surface Appl. Math. Mech. 36 1513–26) is the special case of present work. It is also important to highlight that partial slip effects in the presence of variable diffusion coefficients are investigated first time. It is found that the distractive chemical reaction accelerates the rate of diffusion of solute. However, diffusion rate is decelerated by constructive chemical reaction. The concentration of solute in the liquid regime increases as the mass diffusion increases due to the rise in temperature. It is also observed that the diffusion of momentum of wall in fluid of constant viscosity is faster than the diffusion of momentum fluid of viscosity varying under rise in temperature. Further, an increase in thermal conductivity due to rise in temperature causes an enhancement in rate of heat conduction from wall into the fluid. Eventually, temperature of fluid rises.

Some Contradictions in the Multi-Layer Hele-Shaw Flow

An important problem concerning the Hele-Shaw displacements is to minimize the Saffman - Taylor instability. To this end, some constant viscosity fluid layers can be introduced in an intermediate region (I.R.) between the displacing fluids. However, we prove that very small (positive) values of the growth rates can be obtained only for a very large (unrealistic) I.R. On the contrary, when the I.R. length is constrained by certain conditions (for instance, geological), then the maximum value of the growth constants can not fall below a certain value, not depending on the number of layers. This maximum value is not so small.

Optimised multi-stream microfluidic designs for controlled extensional deformation

In this study, we optimise two types of multi-stream configurations (a T-junction and a flow-focusing design) to generate a homogeneous extensional flow within a well-defined region. The former is used to generate a stagnation point flow allowing molecules to accumulate significant strain, which has been found very useful for performing elongational studies. The latter relies on the presence of opposing lateral streams to shape a main stream and generate a strong region of extension in which the shearing effects of fluid–wall interactions are reduced near the region of interest. The optimisations are performed in two (2D) and three dimensions (3D) under creeping flow conditions for Newtonian fluid flow. It is demonstrated that in contrast with the classical-shaped geometries, the optimised designs are able to generate a well-defined region of homogeneous extension. The operational limits of the obtained 3D optimised configurations are investigated in terms of Weissenberg number for both constant viscosity and shear-thinning viscoelastic fluids. Additionally, for the 3D optimised flow-focusing device, the operational limits are investigated in terms of increasing Reynolds number and for a range of velocity ratios between the opposing lateral streams and the main stream. For all obtained 3D optimised multi-stream configurations, we perform the experimental validation considering a Newtonian fluid flow. Our results show good agreement with the numerical study, reproducing the desired kinematics for which the designs are optimised.

Variable viscosity effects for the steady flow past a sphere

The effect of a spatially dependent viscosity in the unbounded flow around a rigid spherical particle that translates with constant velocity is investigated theoretically. The variable viscosity emulates the effect of a variable concentration of an additive material in a simple solvent. The analysis is performed utilizing a smooth function for the viscosity of the additive material, which can describe qualitatively both depletion and accumulation phenomena around the particle. Assuming steady state, creeping and isothermal conditions, and no external forces and torques, the momentum and mass balances are a generalization of the classical Stokes equations for which the linearity is preserved. Manipulating suitably the governing partial differential equations, a single ordinary integrodifferential equation for the radial part of the radial velocity component is derived. This equation is solved either numerically using a Chebyshev pseudospectral method or analytically using an asymptotic technique. A decrease in the total drag on the particle as the additive material increases is predicted in the depletion case. In the accumulation case, the total drag may increase or decrease in comparison to the simple Newtonian fluid. Analysis of the total drag to its individual contributions reveals that the friction drag (due to viscous forces) is affected substantially by the change of the viscosity, while the form drag (due to pressure) varies much smoother and milder. Finally, we investigate under which conditions the variable viscosity fluid can be approximated as a constant viscosity fluid with Navier type slip at the wall.

Influence of thermal properties on temperature of fluid with micro-structures

This work discusses the impact of variation of thermal properties (viscosity, thermal conductivity, vortex viscosity, dynamic viscosity, and spin gradient viscosity) with respect to the temperature of the micropolar fluid. Basic equations with the law of containing micro-structures conservation of angular moment are used to model the transport of heat in polymeric liquid, the micropolar fluid. Modeled set of coupled nonlinear problems are solved numerically using finite element method (FEM). The results are validated by comparing them with already published work. The macro-motion is impacted significantly by the vortex viscosity and spin gradient viscosity. The angular velocity is decreased when vortex viscosity is varied. The wall momentum diffuses in the micropolar fluid of temperature dependent viscosity more rapidly in comparison with wall momentum diffusion in the micropolar fluid of constant viscosity. However, the angular momentum of micropolar fluid with constant dynamic viscosity is greater than that of micropolar fluid with variable dynamic viscosity. The diffusion of heat in micropolar fluid speeds up as its thermal conductivity increases due to a rise in the temperature of the micropolar fluid.

Experimental and numerical study of erosion in a non-Newtonian hydraulic fracturing fluid

Erosion in hydraulic fracturing fluid is investigated utilizing Computational Fluid Dynamics (CFD) and with comparison to new experimental data for sand in shear thinning non-Newtonian fluids. A grid optimized CFD-based erosion prediction procedure is applied utilizing Dense Discrete Phase Model (DDPM) and one-way coupling model. The effect of different turbulence models on flow modeling, sand particle transport and impact characteristics are also examined. Constant viscosity fluid and power law constitutive equations are implemented and their capability in predicting erosion is examined by comparison to experimental data gathered in this investigation. This investigation has shown that significant error can arise from utilizing constant viscosity model to predict erosion resulting from particles entrained in non-Newtonian fluids. Thus, this investigation has resulted in selecting the best CFD approach and model combination for predicting erosion in this hydraulic fracturing non-Newtonian fluid.

Mechanism of shear thickening in suspensions of rigid spheres in Boger fluids. Part II: Suspensions at finite concentration

The steady shear rheology of nondilute suspensions of noncolloidal, rigid spheres in highly elastic, constant viscosity fluids is studied via numerical simulation using three-dimensional, finite-volume methods and via experiments using cone-and-plate measurements. In the numerical study, we use an immersed boundary method to simulate an ensemble of particles as a function of time until they achieve steady average bulk properties, and the simulations include fully resolved particle-scale hydrodynamics and fluid stresses. The simulations show that for low volume fraction, nondilute suspensions, the viscosity thickens at high shear and this effect can be fully determined by considering a single particle’s interactions with the suspending fluid. In fact, we show that experimentally measured viscosities of suspensions up to ϕ∼0.25, where ϕ is the volume fraction of particles, can be characterized by a shift factor that determines the zero-shear viscosity and a master curve that describes the viscosity thickening. The master curve is determined by single particle–fluid interactions that depend critically on the extensional properties of the suspending fluid. Thus, the “per particle” thickening contribution to the viscosity, at large values of the Weissenberg number, is not dependent on particle–particle interactions when plotted versus the suspension stress. This “master curve” for the thickening of the shear viscosity is not only demonstrated in the simulations, but also we show that it is consistent with all available experimental data including our own. We show that the first normal stress difference coefficient can be described similarly by a shift factor and a master curve. Our numerical simulations do not quantitatively reproduce the experimental data and in general underpredict the experimentally measured shear-thickening of the rheological properties. We suggest, and provide supporting evidence, that this can be explained by the inability of the constitutive model employed in the simulations to adequately describe the extensional properties of the suspending fluid.

Analysis of Sandface-Temperature-Transient Data for Slightly Compressible, Single-Phase Reservoirs

Summary This paper presents new semilog-straight-line and temperature-derivative methods for interpreting and analyzing sandface-temperature transient data from constant-rate drawdown and buildup tests conducted in infinite-acting reservoirs containing slightly compressible fluid of constant compressibility and viscosity. The procedures are dependent on the analytical solutions accounting for Joule-Thomson (J-T) heating/cooling, adiabatic-fluid expansion, and conduction and convection effects. The development of the analytical solutions is dependent on the fact that the effects of temperature changes on pressure-transient data can be neglected so that the pressure-diffusivity and thermal-energy-balance equations can be decoupled. The analytical solutions are verified by and are found in excellent agreement with the solutions of a commercial nonisothermal reservoir simulator. It is shown that drawdown and buildup sandface-temperature data may exhibit three infinite-acting radial-flow (IARF) periods (represented by semilog equations): one at early times reflecting the adiabatic expansion/compression effects, another at intermediate times reflecting the J-T expansion in the skin zone if skin exists, and the third at late times reflecting J-T expansion effects in the nonskin zone. Performing semilog analyses by use of these IARF regimes gives estimates of permeability of skin and nonskin zones as well as the radius of the skin zone, assuming that the J-T coefficient of the fluid and the viscosity are known. Parameters such as skin-zone permeability and radius are not readily accessible from conventional pressure-transient analysis (PTA) from which only the skin factor and nonskin-zone permeability can be obtained. The applicability of the proposed analysis procedure is demonstrated by considering synthetic and field-test data. The results indicate that the analysis procedure provides reliable estimates of skin-zone and nonskin-zone permeability and skin-zone radius from drawdown or buildup temperature data jointly with pressure data.

Exploration of single wall carbon nanotubes for the peristaltic motion in a curved channel with variable viscosity

In the present paper, we have investigated peristaltic flow of single wall carbon nanotubes (SWCNTs) in a curved channel with variable viscosity. The present problem is modeled and exact solutions for non-dimensional differential equation are achieved under long wavelength and low Reynolds number approximation. The features of the peristaltic motion are examined by plotting graphs and discussed in detail. Velocity tables are also incorporated for different values of volume fraction of the nano particle material and for viscosity parameter. Results showed that addition of SWCNTs decreases nano particle temperature and velocity profile near the left wall. It is found that for pure blood case, velocity for the variable viscosity fluid is greater as compared to the constant viscosity fluid. It is also found that for the case of SWCNTs, velocity for the variable viscosity fluid near the left wall decreases whereas it gets opposite behavior near the right wall.

Extracting extensional properties through excess pressure drop estimation in axisymmetric contraction and expansion flows for constant shear viscosity, extension strain-hardening fluids

In this study, hyperbolic contraction-expansion flow (HCF) devices have been investigated with the specific aim of devising new experimental measuring systems for extensional rheological properties. To this end, a hyperbolic contraction-expansion configuration has been designed to minimize the influence of shear in the flow. Experiments have been conducted using well-characterized model fluids, alongside simulations using a viscoelastic White-Metzner/FENE-CR model and finite element/finite volume analysis. Here, the application of appropriate rheological models to reproduce quantitative pressure drop predictions for constant shear viscosity fluids has been investigated, in order to extract the relevant extensional properties for the various test fluids in question. Accordingly, experimental evaluation of the hyperbolic contraction-expansion configuration has shown rising corrected pressure drops with increasing elastic behaviour (D e=0 similar to 16), evidence which has been corroborated through numerical prediction. Moreover, theoretical to predicted solution correspondence has been established between extensional viscosity and first normal stress difference. This leads to a practical means to measure extensional viscosity for elastic fluids, obtained through the derived pressure drop data in these HCF devices.

HEAT TRANSFER IN JACKETED VESSEL EQUIPPED WITH DOWN-PUMPING PITCHED BLADE TURBINE

Heat transfer to a Newtonian fluid in a jacketed vessel equipped with a down-pumping pitched blade turbine (PBT) has been numerically investigated. The turbine has six blades at 45° angles and the turbine is placed concentrically in a cylindrical vessel with a flat top and bottom. An incompressible Newtonian fluid of constant density and viscosity has been considered. The cylindrical wall of the vessel is maintained at constant wall temperature with the help of an outer jacket. The governing momentum and energy equations were numerically solved to obtain the flow and heat transfer fields. The detailed flow and heat transfer fields have been explored for Reynolds number, Re = 7.2 × 104, and Prandtl number in the range 0.71 ≤ Pr ≤ 50. The obtained flow and heat transfer fields are presented as velocity contours, pressure contours, and isotherm profiles at various sections. It is observed that the velocity gradient is higher in the vicinity of the turbine as compared to the vessel wall. In the vertical plane, the velocity is found to be maximum just below the turbine because of the down-pumping of the fluid. The observation of isotherm profiles shows that the temperature gradient is very large near to the vessel wall and small in the vicinity of the turbine. An almost uniform temperature profile was obtained at the highest Prandtl number of 50. The degree of thermal mixing and heat transfer coefficient increases with increasing value of Prandtl number. The area-weighted average value of the Nusselt number shows positive dependence on the Prandtl number. A functional relationship within the average Nusselt number and Prandtl number has been proposed for the present configuration and the operating conditions.