A Two-Layer Mathematical Model of Blood Flow in Porous Constricted Blood Vessels

In this paper, we discussed a mathematical model for two-layered non-Newtonian blood flow through porous constricted blood vessels. The core region of blood flow contains the suspension of erythrocytes as non-Newtonian Casson fluid and the peripheral region contains the plasma flow as Newtonian fluid. The wall of porous constricted blood vessel configured as thin transition Brinkman layer over layered by Darcy region. The boundary of fluid layer is defined as stress jump condition of Ocha-Tapiya and Beavers–Joseph. In this paper, we obtained an analytic expression for velocity, flow rate, wall shear stress. The effect of permeability, plasma layer thickness, yield stress and shape of the constriction on velocity in core & peripheral region, wall shear stress and flow rate is discussed graphically. This is found throughout the discussion that permeability and plasma layer thickness have accountable effect on various flow parameters which gives an important observation for diseased blood vessels.     

Experimental and numerical study of the rotation and the erosion of fillers suspended in viscoelastic fluids under simple shear flow

When a porous agglomerate immersed in a fluid is submitted to a shear flow, hydrodynamic stresses acting on its surface may cause a size reduction if they exceed the cohesive stress of the agglomerate. The aggregates forming the agglomerate are slowly removed from the agglomerate surface. Such a behaviour is known when the suspending fluid is Newtonian but unknown if the fluid is viscoelastic. By using rheo-optical tools, model fluids, carbon black agglomerates and particles of various shapes, we found that the particles had a rotational motion around the vorticity axis with a period which is independent on shape (flat particles not considered), but which is exponentially increasing with the elasticity of the medium expressed by the Weissenberg number (We). Spherical particles are always rotating for We up to 2.6 (largest investigated We in this study) but elongated particles stop rotating for We>0.9 while orienting along the flow direction. Erosion is strongly reduced by elasticity. Since finite element numerical simulation shows that elasticity increases the local stress around a particle, the origin of the erosion reduction is interpreted as an increase of cohesiveness of the porous agglomerate due to the infiltration of a viscoelastic fluid.     

A model for extreme plasticity

We present a mathematical model for elastoplasticity in the regime where the applied stress greatly exceeds the yield stress. This scenario is typically found in violent impact testing, where millimetre thick metal samples are subjected to pressures on the order of 10–10² GPa, while the yield stress can be as low as 10⁻² GPa. In such regimes the metal can be treated as a barotropic compressible fluid in which the strength, measured by the ratio of the yield stress to the applied stress, is negligible to lowest order. Our approach is to exploit the smallness of this ratio by treating the effects of strength as a small perturbation to a leading order barotropic model. We find that for uniaxial deformations, these additional effects give rise to features in the response of the material which differ significantly from the predictions of barotropic flow.     

Flow of a fluid-solid mixture: Normal stress differences and slip boundary condition

In this paper, using mixture theory we study the flow of a dense suspension, composed of solid particles and a fluid; the emphasis is on the influence of the slip boundary condition and the effect of normal stress differences. Very little work has been done considering both the slip at the walls and the normal stress effects in the frame of a two-component flow. In this paper, the stress tensor for the solid component is modeled as a nonlinear fluid which not only includes the viscous effects but also the normal stress effects; the fluid constituent is modeled as a viscous fluid. We look at the flow between two flat plates.     

Granulation and bistability in non-Brownian suspensions

In granulation, a dense colloidal suspension is converted into pasty lumps by application of flow. Often, such lumps are bistable: each can exist either as a fluid droplet (with a shiny surface) or as a jammed granule, whose rough surface creates a bulk stress via capillary action. Such bistability can be explained if the bulk steady-state flow curve is sufficiently nonmonotonic that, above some threshold of stress, flow ceases entirely. This is a stronger condition than the one required to see discontinuous shear thickening, but closely related to it. For instance, inertia can play a role in shear thickening, but not in static bistability. Suitable flow curves were previously found in a phenomenological model of the colloidal glass transition, in which Brownian motion is arrested at high stresses. However, granulation often involves particles too large for Brownian motion to be significant, so that another nonmonotonicity mechanism is needed. A very recent theory, in which the proportion of frictional rather than lubricated contacts increases with stress, provides just such a mechanism, and we use it here to give a simple explanation of granular bistability in non-Brownian suspensions, which requires knowledge only of the steady-state flow curve. However, jamming is in general a history-dependent phenomenon which allows bistability to arise under broader conditions than those just described, possibly including systems that do not shear-thicken at all. In this paper, we focus on explanations of granular bistability based on steady-state shear-thickening, but also discuss alternative explanations based on flow history effects.     

Elementary implications of viscometry on the free surface profile for the problems of climbing and channel flow

Manifestations of normal stress effects in general fluid mechanics account for the major significant phenomenological differences between the so-called non-Newtonian fluids and fluids of the Navier-Stoke type. The climbing phenomenon, which was put into evidence by the experiments of Weissenberg, is one of the most interesting examples to illustrate this point. A second example, perhaps not as dramatic as the climbing phenomenon, is the bulging of the free surface in the flow down a titled trough. Previously, both phenomena have been analyzed using perturbation techniques which contain rather intricate calculations. In the present work we give a simplified approximate analysis of the free surface profile for a homogeneous, incompressible, simple fluid with the inclusion of surface tension for both phenomena. The present analysis is rather elementary and does not involve any intricate mathematical calculations. It uses the fact that both flows when slow, are nearly viscometric.     

Is Weibull distribution the correct model for predicting probability of failure initiated by non-interacting flaws?

The utility of the Weibull distribution has been traditionally justified with the belief that it is the mathematical expression of the weakest-link concept in the case of flaws locally initiating failure in a stressed volume. This paper challenges the Weibull distribution as a mathematical formulation of the weakest-link concept and its suitability for predicting probability of failure locally initiated by flaws. The paper shows that the Weibull distribution predicts correctly the probability of failure locally initiated by flaws if and only if the probability that a flaw will be critical is a power law or can be approximated by a power law of the applied stress.Contrary to the common belief, on the basis of a theoretical analysis and Monte Carlo simulations we show that in general, for non-interacting flaws randomly located in a stressed volume, the distribution of the minimum failure stress is not necessarily a Weibull distribution. For the simple cases of a single group of identical flaws or two flaw size groups each of which contains identical flaws, for example, the Weibull distribution fails to predict correctly the probability of failure. Furthermore, if in a particular load range, no new critical flaws are created by increasing the applied stress, the Weibull distribution also fails to predict correctly the probability of failure of the component. In all these cases however, the probability of failure is correctly predicted by the suggested alternative equation. This equation is the correct mathematical formulation of the weakest-link concept related to random flaws in a stressed volume. The equation does not require any assumption concerning the physical nature of the flaws and the physical mechanism of failure and can be applied in cases of locally initiated failure by non-interacting entities.     

A variational approach for the deformation of a saturated porous solid. A second-gradient theory extending Terzaghi's effective stress principle

Summary The principle of virtual power is used to derive the equilibrium field equations of a porous solid saturated with a fluid, including second density-gradient effects; the intention is the elucidation and extension of the effective stress principle of Terzaghi and Fillunger. In the context of a first density-gradient theory for a saturated solid we interpret the porewater pressure as a Lagrange multiplier in the expression for the deformation energy, assuring that the saturation constraint is verified. We prove that this saturation pressure is distributed among the constituents according to their respective volume fraction (Delesse law) only if they are both true density-preserving. We generalize the Delesse law to the case of compressible constituents. If a material-dependent effective stress contribution is to arise, it is, in general, nonvanishing simultaneously in both the solid and fluid constituents. Moreover, saturation pressure, effective stresses and compressibility constitutive equations determine the exchange volume forces. In a theoretical formulation without non-isotropic strain measures, second density-gradient effects must be incorporated, not only to accommodate for the equilibrium-solid-shear stress and the interaction among neighboring solid-matrix pores, but also to describe internal capillarity effects. The earlier are accounted for by a dependence of the thermodynamic energy upon the density-gradient of the solid, while the latter derives from a mixed density-gradient dependence. Examples illustrate the necessity of these higher gradient effects for properly posed boundary value problems describing the mechanical behaviour of the disturbed rock zone surrounding salt caverns. In particular, we show that solid second-gradient strains allow for the definition of the concept of static permeability, which is distinct from the dynamic Darcy permeability. Received 1 February 1999; accepted for publication 9 March 1999     

A Quasi-linear Viscoelastic Constitutive Equation for the Brain: Application to Hydrocephalus

Hydrocephalus is a condition which occurs when an excessive accumulation of cerebrospinal fluid in the brain causes enlargement of the ventricular cavities. Modern treatments of shunt implantation are effective, but have an unacceptably high rate of failure in most reported series. One of the common factors causing shunt failure is the misplacement of the proximal catheter's tip, which can be remedied if the healed configuration of the ventricular space can be predicted. In a recent study we have shown that this is accomplished by a mathematical model which requires as input the knowledge of the speed at which the ventricular walls move inwardly. In this paper we report on a theoretical method of calculating this speed and show that it will become of great practical usefulness as soon as more experimental results become available.     

Weak formulation for a two-phase nonlinear flow in an undeformable porous medium

In this paper we present a mathematical model for the two-phase flow of a mono-component fluid in an undeformable porous medium. The main practical application is the problem of gas extraction in a geothermal reservoir for which the model can be used for predicting the extinction time of a specific phase in the reservoir. The system is modeled assuming that temperature is not evolving and that the driving mechanism in the case of co-existence of the two phases is capillarity. We also assume that the fluid can be found in liquid and gaseous phase and that there can be regions where this two phases co-exist. The various phases are separated by evolving boundaries (the mathematical formulation turns out to be a free boundary problem) which are determined imposing mass balance relations. We give an integral formulation for the so-called overall density, which is the sum of the densities of each phase weighted by saturation. Finally we present some numerical simulations to investigate the dependence of the solution on the physical parameters and on the boundary conditions involved in the system.     

Interstitial fluid flow: simulation of mechanical environment of cells in the interosseous membrane

In vitro experiments have shown that subtle fluid flow environment plays a significant role in living biological tissues,while there is no in vivo practical dynamical measurement of the interstitial fluid flow velocity.On the basis of a new finding that capillaries and collagen fibrils in the interosseous membrane form a parallel array,we set up a porous media model simulating the flow field with FLUENT software,studied the shear stress on interstitial cells’ surface due to the interstitial fluid flow,and analyzed the effect of flow on protein space distribution around the cells.The numerical simulation results show that the parallel nature of capillaries could lead to directional interstitial fluid flow in the direction of capillaries.Interstitial fluid flow would induce shear stress on the membrane of interstitial cells,up to 30 Pa or so,which reaches or exceeds the threshold values of cells’ biological response observed in vitro.Interstitial fluid flow would induce nonuniform spacial distribution of secretion protein of mast cells.Shear tress on cells could be affected by capillary parameters such as the distance between the adjacent capillaries,blood pressure and the permeability coefficient of capillary’s wall.The interstitial pressure and the interstitial porosity could also affect the shear stress on cells.In conclusion,numerical simulation provides an effective way for in vivo dynamic interstitial velocity research,helps to set up the vivid subtle interstitial flow environment of cells,and is beneficial to understanding the physiological functions of interstitial fluid flow.     

Thermo-Osmosis Effect in Saturated Porous Medium

In this paper, the thermo-poroelasticity theory is used to investigate the quasi-static response of temperatures, pore pressure, stress, displacement, and fluid flux around a cylindrical borehole subjected to impact thermal and mechanical loadings in an infinite saturated poroelastic medium. It has been reported in literatures that coupled flow known as thermo-osmosis by which flux is driven by temperature gradient, can significantly change the fluid flux in clay, argillaceous and many other porous materials whose permeability coefficients are very small. This study presents a mathematical model to investigate the coupled effect of thermo-osmosis in saturated porous medium. The energy balance equations presented here fulfill local thermal non-equilibrium condition (LTNE) which is different from the local thermal equilibrium transfer theory, accounting for that temperatures of solid and fluid phases are not the same and governed by different heat transfer equations. Analytical solutions of temperatures, pore pressure, stress, displacement, and fluid flux are obtained in Laplace transform space. Numerical results for a typical clay are used to investigate the effect of thermo-osmosis. The effects of LTNE on temperatures, pore pressure, and stress are also studied in this paper.     

Improvements to a smooth particle hydrodynamics simulator for investigating submarine landslide generated waves

Submarine landslides can exhibit complex rheologies, including a finite yield stress and shear thinning, yet are often simulated numerically using a Newtonian fluid rheology and simplistic boundary conditions. Here we present improvements made to a Smoothed Particle Hydrodynamics simulator to allow the accurate simulation of submarine landslide generated waves. The improvements include the addition of Bingham and Herschel-Bulkley rheologies, which better simulate the behavior of submarine mudflows. The interaction between the base of the slide and the slope is represented more accurately through the use of a viscous stress boundary condition. This condition treats the interface between the seafloor and the slide as a fluid boundary layer with a user-defined viscosity and length scale. Modifications to the pressure and density calculations are described that improve their stability for landslide generated wave scenarios. An option for pressure decomposition is introduced to prevent particle locking under high pressure. This facilitates the application of this simulator to landslide scenarios beneath significant water depths. Additional modifications to the reaveraging and renormalization routines improve the stability of the free surface and fluid density. We present the mathematical formulations of these improvements alongside commentary on their performance and applicability to landslide generated wave modeling. The modifications are verified against analytical fluid flow solutions and a wave generation experiment.     

A finite strain model of stress, diffusion, plastic flow, and electrochemical reactions in a lithium-ion half-cell

We formulate the continuum field equations and constitutive equations that govern deformation, stress, and electric current flow in a Li-ion half-cell. The model considers mass transport through the system, deformation and stress in the anode and cathode, electrostatic fields, as well as the electrochemical reactions at the electrode/electrolyte interfaces. It extends existing analyses by accounting for the effects of finite strains and plastic flow in the electrodes, and by exploring in detail the role of stress in the electrochemical reactions at the electrode-electrolyte interfaces. In particular, we find that that stress directly influences the rest potential at the interface, so that a term involving stress must be added to the Nernst equation if the stress in the solid is significant. The model is used to predict the variation of stress and electric potential in a model 1-D half-cell, consisting of a thin film of Si on a rigid substrate, a fluid electrolyte layer, and a solid Li cathode. The predicted cycles of stress and potential are shown to be in good agreement with experimental observations.     

Large-scale vortical structure,supported by small-scale turbulent motions. Helicity as a cause for inverse energy cascade

In this article we demonstrate that turbulent stress contributions which depend on the rotation of the frame of reference (and therefore are system dependent) give rise to the inverse energy cascade, and thus introduce an ordering in the structure of turbulence.We first demonstrate that a non-rotating Boussinesq fluid subject to an artificial force that is not invariant under parity changes of the orthogonal group has a destabilizing effect in the B'enard problem. This destabilization is due to helicity and the stability regimes are divided into two regions: (1) If the helicitys is below a threshold values ^*, then long and very short wavelength disturbances at Rayleigh numbers Ra > Ra^crit are stable whereas those with intermediate wavelengths are unstable. (2) If the helicitys >s _* then all disturbances are unstable.For a rotating turbulent Boussinesq fluid we derive the most simple rotation dependent expression for the stress divergence and demonstrate that it leads quantitatively to a similar helicity dependent force. In the B6nard problem it gives rise to an analogous division of the stability/instability regime as obtained for non-rotating fluids subject to the artificial helicity dependent force.     

Nonhomogeneous Incompressible Herschel–Bulkley Fluid Flows Between Two Eccentric Cylinders

The equations for the nonhomogeneous incompressible Herschel–Bulkley fluid are considered and existence of a weak solution is proved for a boundary-value problem which describes three-dimensional flows between two eccentric cylinders when in each two-dimensional cross-section annulus the flow characteristics are the same. The rheology of such a fluid is defined by a yield stress τ* and a discontinuous stress-strain law. A fluid volume stiffens if its local stresses do not exceed τ*, and a fluid behaves like a nonlinear fluid otherwise. The flow equations are formulated in the stress–velocity–density–pressure setting. Our approach is different from that of Duvaut–Lions developed for the classical Bingham viscoplastic fluids. We do not apply the variational inequality but make use of an approximation of the generalized Bingham fluid by a non-Newtonian fluid with a continuous constitutive law.     

Cessation of Newtonian circular and plane Couette flows with wall slip and non-zero slip yield stress

We solve analytically the cessation flows of a Newtonian fluid in circular and plane Couette geometries assuming that wall slip occurs provided that the wall shear stress exceeds a critical threshold, the slip yield stress. In steady-state, slip occurs only beyond a critical value of the angular velocity of the rotating inner cylinder in circular Couette flow or of the speed of the moving upper plate in plane Couette flow. Hence, in cessation, the classical no-slip solution holds if the corresponding wall speed is below the critical value. Otherwise, slip occurs only initially along both walls. Beyond a first critical time, slip along the fixed wall ceases, and beyond a second critical time slip ceases also along the initially moving wall. Beyond this second critical time no slip is observed and the decay of the velocity is faster. The velocity decays exponentially in all regimes and the decay is reduced with slip. The effects of slip and the slip yield stress are discussed.     

Fractured water injection wells: Pressure transient analysis

In this paper we study the pressure drop in a hydraulic fracture after shut-in of a water injection well. The pressure transient behavior depends on fracture closure, lateral stress, rock elasticity and fracture fluid leak-off. Under the assumption that horizontal cross-sections of a vertical fracture do not depend on the vertical variable, we formulate a mathematical model which allows for determination of both pore pressure and elastic rock displacements jointly with the fracture aperture and fracture fluid pressure. An analytical consideration is performed for the case of an ideal very long fracture with the same aperture along its full length. In the general case, fracture closure is analyzed numerically.     

Rheology of a suspension of particles in viscoelastic fluids

In this paper we study the bulk stress of a suspension of rigid particles in viscoelastic fluids. We first apply the theoretical framework provided by Batchelor [J. Fluid Mech. 41 (1970) 545] to derive an analytical expression for the bulk stress of a suspension of rigid particles in a second-order fluid under the limit of dilute and creeping flow conditions. The application of the suspension balance model using this analytical expression leads to the prediction of the migration of particles towards the centerline of the channel in pressure-driven flows. This is in agreement with experimental observations. We next examine the effects of inertia (or flow Reynolds number) on the rheology of dilute suspensions in Oldroyd-B fluids by two-dimensional direct numerical simulations. Simulation results are verified by comparing them with the analytical expression in the creeping flow limit. It is seen that the particle contribution to the first normal stress difference is positive and increases with the elasticity of the fluid and the Reynolds number. The ratio of the first normal stress coefficient of the suspension and the suspending fluid decreases as the Reynolds number is increased. The effective viscosity of the suspension shows a shear-thinning behavior (in spite of a non-shear-thinning suspending fluid) which becomes more pronounced as the fluid elasticity increases.     

Mathematical Models for Waxy Crude Oils

In this paper we review a series of mathematical models formulated for the flow of waxy crude oils, that is, of mineral oils with a high content of paraffinic hydrocarbons (with the generic name of waxes) which may be dissolved or segregated as solid crystals at sufficiently low temperatures. The flow takes place in a laboratory test loop. The crystals have a tendency to form aggregates, producing a gel-like structure. The resulting product can be modeled as a Bingham fluid, but its rheological parameters (yield stress and viscosity) depend on the amount and state of the segregated phase, whose evolution is in turn influenced by the flow. Of course temperature plays a key role. Wax can form a solid deposit at the pipe wall, reducing the pipe radius and this phenomenon is also taken into account. The models presented have a different degree of complexity, depending on which phenomena they include. In presenting each of them we discuss their expected range of validity.