Flow Of Such Fluids Research Articles

On Fully Developed Flows of Fluids with a Pressure Dependent Viscosity in a Pipe

Stokes recognized that the viscosity of a fluid can depend on the normal stress and that in certain flows such as flows in a pipe or in channels under normal conditions, this dependence can be neglected. However, there are many other flows, which have technological significance, where the dependence of the viscosity on the pressure cannot be neglected. Numerous experimental studies have unequivocally shown that the viscosity depends on the pressure, and that this dependence can be quite strong, depending on the flow conditions. However, there have been few analytical studies that address the flows of such fluids despite their relevance to technological applications such as elastohydrodynamics. Here, we study the flow of such fluids in a pipe under sufficiently high pressures wherein the viscosity depends on the pressure, and establish an explicit exact solution for the problem. Unlike the classical Navier-Stokes solution, we find the solutions can exhibit a structure that varies all the way from a plug-like flow to a sharp profile that is essentially two intersecting lines (like a rotated V). We also show that unlike in the case of a Navier-Stokes fluid, the pressure depends both on the radial and the axial coordinates of the pipe, logarithmically in the radial coordinate and exponentially in the axial coordinate. Exact solutions such as those established in this paper serve a dual purpose, not only do they offer solutions that are transparent and provide the solution to a specific but simple boundary value problems, but they can be used also to test complex numerical schemes used to study technologically significant problems.

On steady flows of fluids with pressure– and shear–dependent viscosities

There are many technologically important problems such as elastohydrodynamics which involve the flows of a fluid over a wide range of pressures. While the density of the fluid remains essentially constant during these flows whereby the fluid can be approximated as being incompressible, the viscosity varies significantly by several orders of magnitude. It is also possible that the viscosity of such fluids depends on the shear rate. Here we consider the flows of a class of incompressible fluids with viscosity that depends on the pressure and shear rate. We establish the existence of weak solutions for the steady flows of such fluids subjected to homogeneous Dirichlet boundary conditions and to specific body forces that are not necessarily assumed to be small. A novel aspect of the study is the manner in which we treat the pressure that allows us to establish its compactness, as well as that of the velocity gradient. The method draws upon the physics of the problem, namely that the notion of incompressibility is an idealization that is attained by letting the compressibility of the fluid to tend to zero.

A note on the flow induced by a constantly accelerating edge in anOldroyd‐B fluid

Analytical expressions for the velocity field and the tangential stresses that are induced due to a constantly accelerating edge in an Oldroyd‐B fluid have been established for all values of material constants. The solutions that have been obtained satisfy the governing differential equations and all the imposed initial and boundary conditions. These solutions reduce to those for the Maxwell, second‐grade, and Navier‐Stokes fluid as limiting cases. Exact solutions such as those determined here for an unsteady problem serve a dual purpose. They have relevance to an interesting physical problem and the solutions can also be used to check the efficacy of the flows of such fluids in more complicated flow domains.

Flows of inelastic non-Newtonian fluids through arrays of aligned cylinders. Part 2. Inertial effects for square arrays

Numerical simulations are presented for flows of inelastic non-Newtonian fluids through periodic arrays of aligned cylinders. The truncated power-law fluid model is used for the relationship between the viscous stress and the rate-of-strain tensor. Results for the drag coefficient for creeping flows of such fluids have been presented in a companion paper [1]. In this second part the effects of finite fluid inertia are investigated for flows through square arrays. It is shown that the Reynolds-number dependence of the drag coefficient of a cylinder in the array is of the form C d≡ F/(ηU) = k 0 + k 2 Re2+ .. for small values of the Reynolds number Re ≡ ρaU/η, where F is the drag force, U is the averaged velocity in the array, η = K (U/a)n-1 is a viscosity scale with K and n the power-law coefficient and index and a the cylinder radius, and k 0 is the drag coefficient for creeping flows. The proportionality constant k 2 depends on the way the drag coefficient and the Reynolds number are defined. It is shown that the observed strong dependence of k 2 on n can almost be eliminated by using length scales different from a in the viscosity scales η used in the definition of Re and in the definition of the drag coefficient. Numerical simulation results are also presented for the velocity variance components. Results for flows at moderate Reynolds number, of order 100, are also presented; these are qualitatively similar to those for Newtonian fluids. The value of the Reynolds number beyond which the flow becomes unsteady was related to the Newtonian fluid case by rescaling. These results for moderate-Reynolds-number flow are compared against previously published experimental data.

Flow models of natural debris flows originating from overconsolidated clay materials

In this paper, slides and debris flows in overconsolidated clay materials are simulated numerically. As a case study, the models are applied to the Storegga slide in the Norwegian Sea and in particular to the sub-region called Ormen Lange, where the information available is the most precise for a subaqueous debris flow. Three different models for the rheology of clay are used: a viscoplastic (Bingham) fluid model, a viscoplastic fluid with interspersed solid blocks, and a viscoplastic model with yield strength increasing with depth. The small-scale debris flows in the Ormen Lange area can be reasonably well understood in terms of a pure Bingham model without granular effects and blocks. The presence of intact blocks in the region, however, indicates that at least the top layer of the sliding sediments was not destroyed by the flow. It suggests that the flow occurred mainly at high shear rate in a lubricating layer of mud deriving partly from the disintegration of the block's own material, and possibly from the entrainment of hemipelagic sediments along the flow path while the top part was left unsheared. The failure of the model with blocks probably stems from the use of the Coulomb friction law to represent the interaction between the block and the seabed. The Bingham model works better because during the flow of such fluids an unsheared plug region is formed naturally, even in unconsolidated materials. Combining the simulations with these three models, a possible scenario for the Ormen Lange debris flows is deduced according to which the lubricating layer supporting the blocks has a yield strength of about 10–15 kPa.

Global Analysis of the Flows of Fluids with Pressure-Dependent Viscosities

To describe the flows of fluids over a wide range of pressures, it is necessary to take into account the fact that the viscosity of the fluid depends on the pressure. That the viscosity depends on the pressure has been verified by numerous careful experiments. While the existence of solutions local-in-time to the equations governing the flows of such fluids are available for small, special data and rather unrealistic dependence of the viscosity on the pressure, no global existence results are in place. Our interest here is to establish the existence of weak solutions for spatially periodic three-dimensional flows that are global in time, for a large class of physically meaningful viscosity-pressure relationships.

Numerical simulations and global existence of solutions of two-dimensional flows of fluids with pressure- and shear-dependent viscosities

There is a considerable amount of experimental evidence that unequivocally shows that there are fluids whose viscosity depends on both the mean normal stress (pressure) and the shear rate. Recently, global existence of solutions for the flow of such fluids for the three-dimensional case was established by Málek, Nečas and Rajagopal. Here, we present a proof for the global existence of solutions for such fluids for the two-dimensional case. After establishing the global-in-time existence, we discretize the equations via the finite element method, outline the Newton type iterative method to solve the non-linear algebraic equations and provide numerical computations of the steady flow of such fluids in geometries that have technological significance.

Towards a reliable method for predicting the rheological properties of multiphase fluids

Measurement of the constitutive behavior of multiphase fluids is one of the most interesting and difficult problems in rheology. The boundary element method is particularly suited to modeling the flow of such fluids because of the relative ease of representing their complex geometry accurately without excessive meshing difficulties. Many papers have been published on the subject, but to the author's knowledge, the literature on the actual implementation of such methods is scarce. Here, a parallel implementation of a code based on the completed double layer boundary integral equation, with Barnes–Hut multipole acceleration, is described briefly. After determining the level of mesh refinement required to obtain convergent results, the code is used to predict the rheological behavior of suspensions of rigid particles of different shapes. The effect of concentration and spatial distributions on the rheological behavior of the suspensions is investigated. It is found that the properties of suspensions of rigid particles are correctly predicted for concentrations up to 30% solids, while at higher concentrations accurate predictions require excessively fine discretization. A new hybrid numerical-asymptotic technique is proposed, which produces a significant improvement in accuracy accompanied by a reduction in computational cost.

Towards a reliable method for predicting the rheological properties of multiphase fluids

Measurement of the constitutive behavior of multiphase fluids is one of the most interesting and difficult problems in rheology. The boundary element method is particularly suited to modeling the flow of such fluids because of the relative ease of representing their complex geometry accurately without excessive meshing difficulties. Many papers have been published on the subject, but to the author's knowledge, the literature on the actual implementation of such methods is scarce. Here, a parallel implementation of a code based on the completed double layer boundary integral equation, with Barnes–Hut multipole acceleration, is described briefly. After determining the level of mesh refinement required to obtain convergent results, the code is used to predict the rheological behavior of suspensions of rigid particles of different shapes. The effect of concentration and spatial distributions on the rheological behavior of the suspensions is investigated. It is found that the properties of suspensions of rigid particles are correctly predicted for concentrations up to 30% solids, while at higher concentrations accurate predictions require excessively fine discretization. A new hybrid numerical-asymptotic technique is proposed, which produces a significant improvement in accuracy accompanied by a reduction in computational cost.

Lattice-Gas Simulations of Ternary Amphiphilic Fluid Flow in Porous Media

We develop our existing two-dimensional lattice-gas model to simulate the flow of single phase, binary immiscible and ternary amphiphilic fluids. This involves the inclusion of fixed obstacles on the lattice, together with the inclusion of "no-slip" boundary conditions. Here we report on preliminary applications of this model to the flow of such fluids within model porous media. We also construct fluid invasion boundary conditions, and the effects of invading aqueous solutions of surfactant on oil-saturated rock during imbibition and drainage are described.

Status of Computational Fluid Dynamics and Its Application to Materials Manufacturing

Computational fluid dynamics was born principally in the aerospace field as a method for fluid flow and heat transfer research methods following experimental and analytical approaches. Along with progress in the cost performance of computers, computational fluid dynamics is now establishing itself as a tool to improve production processes and product quality in the steel, nonferrous metals, glass, plastics, and composite materials industries.Materials manufacturers use computational fluid dynamics for diverse purposes:1. Reduction in experimental conditions and costs;2. Detailed analysis of mechanisms with multifaceted information unobtainable through experimentation;3. Universal tool for scale-up; and4. Evaluation of novel processes.It can be readily imagined that accuracy, flexibility, and other requirements of computational fluid dynamics should vary with specific applications.Fluids generally observed in materials manufacturing processes are molten materials such as metal, glass, and plastics, and gases for stirring and refining. In the flow of such fluids, materials quality and process characteristics are governed by the following:1. Transport phenomena in the bulk region (where fluid flow is normally turbulent);2. Chemical reaction at interfaces;3. Transport phenomena in boundary layers near the interfaces; and4. Complex coupled phenomena (heat transfer, diffusion, chemical reaction, phase transformation like solidification, free surface, electromagnetic force, and bubble flow).

Utilization of nonlinear diffusion solutions forn-diffusion and power-law materials

The equation ofn-diffusion, as originally formulated by J. R. Philip for various problems involving unsteady turbulent flows, applies directly to the one-dimensional flow of a non-Newtonian power-law fluid as well as indirectly to the laminar boundary layer flow of such fluids over a flat plate. The purpose of this note is to utilize a simple relation between one-dimensionaln-diffusion and nonlinear diffusion with power-law diffusivity such that ifp(x, t) denotes ann-diffusion pressure thenc(x, t)=|ϖp/ϖx| satisfies the nonlinear diffusion equation with power law diffusivity. This means in particular that the large number of solutions presently known for nonlinear diffusion can be utilized in the context ofn-diffusion. Known solutions ofn-diffusion are obtained via this procedure as well as newn-diffusion solutions, including the source solution and a solution of the problem of fluid withdrawal at a constant flow rate for a non-Newtonian fluid in a porous medium of infinite extent. Solutions arising from other exact nonlinear diffusion profiles are also investigated as well as the limiting case ofn-diffusion forn tending to infinity and the results are displayed graphically.

Approximation of the vector potential for viscous incompressible flow via the constant stress finite element

A convergence analysis for a piecewise quadratic finite element method to solve vector biharmonic problems inR 3 in primal variables is presented. The application to the equations of the vector potential for the flow of incompressible viscous fluids is focused. For simplicity, only the particular case of stationary stokesian flows is treated in detail, but it is showed that the method applies as well to more complex flows of such fluids.

Generalization of the Hagen–Poiseuille velocity profile to non-Newtonian fluids and measurement of their viscosity

By using the constitutive equation for the stress tensor that yields a viscosity formula similar to the Eyring formula for non-Newtonian fluids and solving the hydrodynamic equation for a tube flow of such fluids, a generalization is obtained of the Hagen–Poiseuille formula for velocity profile in tube flow under pressure. The volume flow rate is calculated with the velocity profile, and a method of measuring zero shear rate viscosity is suggested. As the pressure difference increases, the calculated velocity profiles become increasingly flatter in the midportion of the tube than that predicted by the Hagen–Poiseuille formula, which remains parabolic for all values of the pressure difference.

Unsteady flow of an elastico-viscous fluid in tubes of uniform cross-section

In this paper, the uniqueness of solution for internal bounded unsteady flows of a shortmemory fluid is first established. Closed-form solutions are then obtained for the equations characterizing flows of such fluids in circular and rectangular tubes of uniform cross-section under an arbitrary pressure gradient. Special cases including the oscillatory flow between two parallel plates are discussed.

Fluids and thrusting

Recent studies of the structure of the continental crust by COCORP and BIRPS have shown that major thrust structures in the lower crust are ubiquitous. Plate-tectonic processes of continental crust destruction or construction at sites where subduction or collision occurs (the Himalayas or the Andes) involve thrusting of continental materials or ocean crust beneath continents, and local overthrusting may accompany transcurrent motion. In all such cases massive fluid flow must occur. As first stressed by M.K. Hubbert and W.W. Rubey, thrusting mechanisms require that fluid pressures are close to, or greater than, lithostatic pressures. When thrusting occurs, compaction of the underplate must occur rapidly, expelling pore water. Crustal thickening leads to gradual heating of the underplate with prograde metamorphism near the top and even melting at the base. If the overthrust plate is thick and hot at the base, retrograde metamorphism will occur at the base of the overthrust plate and rising fluids will encounter inverted thermal gradients. In such a region veins need not occur but leaching phenomena may dominate. If an underthrust plate is hydrated, and/or contains sedimentary aquifers, large fluid volumes are expelled through shear zones into the overthrust plate. The scale of fluid-release processes can be large. Thus, in thin-skinned tectonics where the overthrust plate is 10–15 km thick, induced fluid release can easily reach 4 · 10 9 g m −2 of thrust surface. The flow of such fluids will be controlled by the thrust surface and lithology, both of which will influence hydraulic fracture mechanisms and spacing of fractures. The chemistry of thrust-derived fluids may be highly variable depending on lithology and the time constants of thrusting. Vertical thermal and redox environments will be similarly dependent. In the subduction process, impressive quantities of fluids must pass back up the thrust surface, and such fluids have recently been directly observed during drilling. Dewatering of spilites, serpentinites and sediment layers of the underthrust oceanic lithosphere must produce massive fluid flow back to the surface but some of these fluids may form hydrated minerals in the overlying mantle and, at great depth, flux mantle melting. Extreme metasomatism in blueschist belts must result from such fluids. A case of particular interest involves the cessation of subduction when a high-level slab reaches thermal equilibrium. In general, flow regimes in the thrust and fault zones associated with collisions follow a sequence from conditions of high T- P with locally derived fluids at low water/rock ratios during initiation of the structures, to high fluxes of reduced metamorphic fluids along conduits as the structures propagate and intersect hydrothermal reservoirs. Later in the tectonic evolution, and at shallower crustal levels, there may be incursion of oxidizing near-surface fluid reservoirs into the faults. These fluids may have extremely low δ 18O-values, where mountain ranges form on rebound faults, and high-altitude depleted fluids penetrate down the rebound structure. Extensive mineralization may be associated with such thrust-derived fluids. Examples which will be discussed include the Mother Lode Au deposits of California, U.S.A., and the large U deposits of Lagoa Real, Brazil.

Flow of Non-Newtonian Power-Law Fluids Through Porous Media

Abstract The partial differential equation that describes the flow, of non-Newtonian, power-law, slightly compressible fluids in porous media is derived. An approximate solution, in closed form, is developed for the unsteady-state flow behavior and verified by. two different methods. Using the unsteady-state solution, a method for analyzing injection test data is formulated and used to analyze four injection tests. Theoretical results were used to derive steady-state equations of flow, equivalent transient drainage radius, and a method for analyzing isochronal test data. The theoretical fundamentals of the flow, of non-Newtonian power-law fluids in porous media are established. Introduction Non-Newtonian power-law fluids are those that obey the relation = constant. Here, is the viscosity, e is the shear rate at which the viscosity is measured, and n is a constant. Examples of such fluids are polymers. This paper establishes the theoretical foundation of the flow of such fluids in porous media. The partial differential equation describing this flow is derived and solved for unsteady-state flow. In addition, a method for interpreting isochronal tests and an equation for calculating the equivalent transient drainage radius are presented. The unsteady-state flow solution provides a method for interpreting flow tests (such as injection tests).Non-Newtonian power-law fluids are injected into the porous media for mobility control, necessitating a basic porous media for mobility control, necessitating a basic understanding of the flow behavior of such fluids in porous media. Several authors have studied the porous media. Several authors have studied the rheological properties of these fluids using linear flow experiments and standard viscometers. Van Poollen and Jargon presented a theoretical study of these fluids. They described the flow by the partial differential equation used for Newtonian fluids and accounted for the effect of shear rate on viscosity by varying the viscosity as a function of space. They solved the equation numerically using finite difference. The numerical results showed that the pressure behavior vs time differed from that for Newtonian fluids. However, no methods for analyzing flow-test data (such as injection tests) were offered. This probably was because of the lack of analytic solution normally required to understand the relationship among the variables.Recently, injectivity tests were conducted using a polysaccharide polymer (biopolymer). The data showed polysaccharide polymer (biopolymer). The data showed anomalies when analyzed using methods derived for Newtonian fluids. Some of these anomalies appeared to be fractures. However, when the methods of analysis developed here were applied, the anomalies disappeared. Field data for four injectivity tests are reported and used to illustrate our analysis methods. Theoretical Consideration General Consideration The partial differential equation describing the flow of a non-Newtonian, slightly compressible power-law fluid in porous media derived in Appendix A is ..........(1) where the symbols are defined in the nomenclature. JPT P. 155

Variational aspects of relativistic field theories, with application to perfect fluids

By investigating perturbations of classical field theories based on variational principles we develop a variety of relations of interest in several fields, general relativity, stellar structure, fluid dynamics, and superfluid theory. The simplest and most familiar variational principles are those in which the field variations are unconstrained. Working at first in this context we introduce the Noether operator, a fully covariant generalization of the socalled canonical stress energy tensor, and prove its equivalence to the symmetric tensor T μν . By perturbing the Noether operator's definition we establish our fundamental theorem, that any two of the following imply the third (a) the fields satisfy their field equations, (b) the fields are stationary, (c) the total energy of the fields is an extremum against all perturbations. Conversely, a field theory which violates this theorem cannot be derived from an unconstrained principle. In particular both Maxwell's equations for F μν and Euler's equations for the perfect fluid have stationary solutions which are not extrema of the total energy [(a) + (b) (c)]. General relativity is a theory which does have an unconstrained variational principle but the definition of Noether operator is more ambiguous than for other fields. We define a pseudotensorial operator which includes the Einstein and Landau-Lifschitz complexes as special cases and satisfies a certain criterion on the asymptotic behavior. Then our extremal theorem leads to a proof of the uniqueness of Minkowski space: It is the only asymptotically flat, stationary, vacuum solution to Einstein's equations having R 4 global topology and a maximal spacelike hypersurface. We next consider perfect fluid dynamics. The failure of the extremal-energy theorem elucidates why constraints have always been used in variational principles that lead to Euler's equations. We discuss their meaning and give what we consider to be the “minimally constrained” principle. A discussion of one constraint, “preservation of particle identity,” from the point of view of path-integral quantum mechanics leads to the conclusion that it is inapplicable to degenerate Bose fluids, and this gives immediately the well-known irrotational flow of such fluids. Finally, we develop a restricted extremal theorem for the case of perfect fluids with self-gravitation, which has the same form as before except that certain perturbations are forbidden in (c). We show that it is a generalization of the Bardeen-Hartle-Sharp variational principle for relativistic stellar structure. It may be useful in constructing nonaxisymmetric stellar models (generalized Dedekind ellipsoids). We also give the Newtonian versions of the main results here, and we show to what extent the extremal theorems extend to fields that may not even have a variational principle.

Stress Distributions in the Flow of Viscoelastic Fluid through Narrowed Projections

Many non-Newtonian visoous fluids are recently used in practice, and in consequence the chances of handling these fluids have increased in many fields. However, at present the rheological equation is not yet established for the viscoelastic non-Newtonian fluids. Experimental studies are required, therefore, of viscoelastic flow concerning its stress analysis. One of the most efficient experimental methods is considered to be the photoelastic procedure, in view of the fact that the viscoelastic fluids with velocity gradients have birefringence effect.In this paper, the results of photoelastic measurements of viscoelastic flows through narrowed portions are reported. The flow was produced in rectangular vessels as two-dimentional flow by applying pressure to the fluid, and the narrowed portions were formed by applying triangular or rectangular projections to the two opposite walls. As the viscoelastic fluid, taking the homogeneity, fluidity and transparency as well as the birefringence effect of the material into consideration, Epikote fluid was employed. An ordinary photoelastic apparatus was used, and clear isochromatic and isoclinic patterns of stressed viscoelastic fluid were obtained. The experimental results may be summarized as follows. In general, the shear stress plays an important role for the mechanical degradation of highpolymer fluids. It is significant, therefore, to predict the amount of shear stress in the flow of such fluids, In the viscoelastic flows through the narrowed portions experimented in this study, the maximum shear stress appeared at the corner of each projection. Moreover, the shear stresses along the flow direction midway between both the projections showed peak values at the points a short distance ahead or behind the corners of the projections.

Steady-State and Unsteady-State Flow of Non~Newtonian Fluids Through Porous Media

Abstract Non-Newtonian fluids may be injected into a reservoir during secondary recovery operations. The non-Newtonian fluid used in this work is a power-law type of fluid; that is, the viscosity of the fluid decreases as the flow rate or rate of shear increases. This paper presents equations for steady-state, linear and radial flow of such fluids, transient behavior results from a finite difference model of a radial system, and transient behavior results from a field test. The equations that describe the flow of a non-Newtonian fluid are non-linear and are solved numerically. Finite difference solutions are presented as curves of dimensionless pressure drop at the wellbore vs dimensionless time for a constant injection rate. Solutions were obtained for 5-percent, 10-percent and 100-percent PV of a non-Newtonian fluid for injection rates of 1, 10, 100 and 1000 cc/sec and for a 5 percent PV of non-Newtonian fluid located at r = rw, 3, 10, 20, 50 and 100 ft for a flow rate of 1 cc/sec. The buildup curves do not exhibit a straight-line portion as is the case for Newtonian flow through porous media. Correlations also are shown for the productivity index vs rate for the computer model study and the field tests. INTRODUCTION During various secondary recovery operations non-Newtonian fluids are injected. Such fluids, in general, include thickened water and gelled fluids. The term "non-Newtonian" implies that the viscosity is not only dependent upon temperature and pressure, but also on the rate of shear that is applied to the fluid. For example, water, which is a Newtonian fluid, will have essentially the same viscosity no matter what rate of shear is applied. A pseudoplastic fluid exhibits a decreasing viscosity when higher rates of shear are applied; a dilated fluid has an increasing viscosity with increasing rates of shear. The objective of studies performed and described in this report is to obtain relationships and mathematical and empirical descriptions of the flow of non- Newtonian fluids through porous media. Simple mathematical relationships, computer studies that include the unsteady- state behavior of such fluids, and field tests are used. PREVIOUS LABORATORY STUDIES Laboratory studies have been performed by several investigators. The fluids used in one investigation1 were surfactant-stabilized dispersions of water in hydrocarbons. Its Fig. 6 is reproduced as our Fig. 1. This effective-viscosity vs frontal-velocity diagram shows that this fluid is of the power-law type over a fairly large range of frontal velocities. Gogarty1 developed an equation for the effective viscosity as a function of shear rate that reduces to Eq. 2 for frontal velocities greater than about 2 ft/D. The fluid characteristics used in the present study are similar to those reported by Gogarty.