Hydrodynamic instability of a suspension of spherical particles through a branching network of circular tubes

A numerical method is presented for computing the unsteady flow of a monodisperse suspension of spherical particles through a branching network of circular tubes. The particle motion and interparticle spacing in each tube are computed by integrating in time a one-dimensional convection equation using a finite-difference method. The particle fraction entering a descendent tube at a divergent bifurcation is related to the local and instantaneous flow rates through a partitioning law proposed by Klitzman and Johnson involving a dimensionless exponent, q ≥ 1. When q = 1, the particle stream is divided in proportion to the flow rate; as q → ∞, the particles are channeled into the tube with the highest flow rate. The simulations reveal that when the network involves two or more generations, a supercritical Hopf bifurcation occurs at a critical value of q, yielding spontaneous, self-sustained oscillations in the segment flow rates, pressure drop across the network, and particle spacing in each tube. A phase diagram is presented to establish conditions for unsteady flow. As found recently for blood flow in a capillary network, oscillations can be induced for a given network tree order by decreasing the ratio of the tube diameter from one generation to the next or by decreasing the diameter of the terminal segments. The instability is more prominent for rigid than deformable particles, such as drops, bubbles, and cells, due to strong lubrication forces between the tightly fitting particles and tube walls. Variations in the local particle spacing, therefore, have a more significant effect on the effective viscosity of the suspension in each tube and pressure drop required to drive a specified flow rate.     

Oscillating two-phase flow in a prestressed thick elastic tube

Summary The pulsating flow of a fluid with dusty particles in a prestressed thick walled elastic tube has been studied. The tube, subjected to a static inner pressure P_i and an axial stretch λ, is taken to be an incompressible, isotropic, elastic material. The fluid with particles is treated as incompressible Newtonian. Employing the theory of small deformation superimposed on large initial deformations, for an axially symmetric perturbed motion the governing equations are obtained in cylindrical polar coordinates. The analytical solutions of the equations of motion for the dust and the fluid have been obtained. Because of the variable character of the coefficients of the resulting equations for the solid body they are solved numerically. The dispersion relation is obtained as a function of the stretch, the thickness ratio and the parameters for dusty particles.     

A momentum integral solution for pulsatile flow in a rigid tube with and without longitudinal vibration

A momentum integral solution is obtained for fully developed pulsatile flow in a circular, rigid tube of infinite length. A fourth-order polynomial with unknown coefficients is used to represent the radial variation of axial velocity across the tube. Boundary conditions applied at the tube wall and the centerline give the velocity profile in terms of centerline velocity. The momentum integral equation then gives a differential equation for the centerline velocity; and complete solutions, including mass flow rate, are obtained for a sinusoidal pressure gradient, with and without a sinusoidal longitudinal wall velocity. Excellent agreement is found with Womersley's results for a stationary tube. For wall motion at the same frequency as the pressure gradient, both an increase in mass flow magnitude and an exact cancellation are found, depending on phasing. These results are used to discuss the application of momentum integral methods to pulsatile flows and possible fluid-dynamical aspects of cardiovascular behavior during whole-body vibration.     

Three-dimensional flow with heat transfer of a viscoelastic fluid over a stretching surface with a magnetic field

The present research study deals with the steady flow and heat transfer of a viscoelastic fluid over a stretching surface in two lateral directions with a magnetic field applied normal to the surface. The fluid far away from the surface is ambient and the motion in the flow field is caused by stretching surface in two directions. This result is a three-dimensional flow instead of two-dimensional as considered by many authors. Self-similar solutions are obtained numerically. For some particular cases, closed form analytical solutions are also obtained. The numerical calculations show that the skin friction coefficients in x- and y-directions and the heat transfer coefficient decrease with the increasing elastic parameter, but they increase with the stretching parameter. The heat transfer coefficient for the constant heat flux case is higher than that of the constant wall temperature case.     

Generalized Couette flow in channels of irregular cross‐section

Abstract

The flow behavior of non‐Newtonian power‐law fluids in channels of irregular cross‐section is examined. The driving force of the flow may be a constant pressure gradient (Poiseuille flow), a moving boundary (Couette flow) or the combination of the two (generalized Couette flow). There are three factors that influence the fluid motion in a channel, namely, the power‐law index n, the channel geometry and a dimensionless quantity E which can be viewed as the ratio of drag flow to pressure flow. The effects of these variables on velocity distributions and volumetric flow rates for various channel geometries are analyzed. The direct application of the numerical results on extruder design and operation is discussed.     

Three dimensional numerical and asymptotic solutions for the peristaltic transport of a heat-conducting fluid

Summary Using the Oberbeck-Boussinesq (O-B) equations as a mathematical model, asymptotic solutions in closed form and numerical solutions are obtained for the peristaltic transport of a heat-conducting fluid in a three-dimensional flexible tube. The results show that the relation between mass flux and pressure drop remains almost linear and the efficiency of the transport depends mainly on the ratio of the wave amplitudeh and the average radius of the tubed. However, the 3-D flow is much different from the 2-D flow in the following ways: (i) The 3-D flow is much more sensitive to the change of the volume expansion coefficient _r ; (ii) Trapping and backflow are much more common in 3-D case; (iii) The longwave asymptotic approximation in 3-D case is not as good as in 2-D case, especially when _r is not small; (iv) The 3-D flow is more sensitive to Reynolds number change.     

A Study of the Flow through Capillary Tubes Tuned for a Cooling Circuit with Saturated Fluorocarbon Refrigerants

Capillary tube expansion devices are widely used in refrigeration equipment; nevertheless, the mechanism of the flow is still not fully described and understood, so the experimental verification of most predictions is still necessary. A modified numerical model of capillary flow has been developed both for standard refrigerants and with emphasis for saturated fluorocarbon (C₂F_2n+2) refrigerants. These refrigerants have several unique properties (high dielectric performance, chemical stability, and radiation resistance). Therefore, they can be used in some special applications, where other common fluids cannot be applied. The main aim of this study was to prepare a practical capillary flow model, which would improve the procedure of predicting the behavior of capillary tubes for cooling circuits of particle detectors being built at the international CERN laboratory in Geneva. The generated numerical model was verified through available data from the literature and also via measurements performed in a real cooling circuit with pure, oil-free octafluoropropane (C₃F₈) refrigerant. Paper presented at the Seventeenth European Conference on Thermophysical Properties, September 5–8, 2005, Bratislava, Slovak Republic.     

Shear flow over a particulate or fibrous plate

Simple shear flow over a porous plate consisting of a planar array of particles is studied as a model of flow over a membrane. The main objective is to compute the slip velocity defined with reference to the velocity profile far above the plate, and the drift velocity induced by the shear flow underneath the plate. The difference between these two velocities is shown to be proportional to the thickness of the plate. When the geometry of the particle array is anisotropic, the directions of the slip and drift velocity are generally different from the direction of the overpassing shear flow. An integral formulation is developed to describe flow over a plate consisting of a periodic lattice of particles with arbitrary shape, and integral representations for the velocity and pressure are developed in terms of the doubly-periodic Green's function of three-dimensional Stokes flow. Based on the integral representation, asymptotic expressions for the slip and drift velocity are derived to describe the limit where the particle size is small compared to the inter-particle separation, and numerical results are presented for spherical and spheroidal particles of arbitrary size. The asymptotic results are found to be accurate over an extended range of particle sizes. To study the limit of small plate porosity, the available solution for shear flow over a plane wall with a circular orifice is used to describe flow over a plate with a homogeneous distribution of circular perforations, and expressions for the slip and drift velocity are derived. Corresponding results are presented for axial and transverse shear now over a periodic array of cylinders arranged distributed in a plane. Streamline pattern illustrations confirm that a negative drift velocity is due to the onset of eddies between closely-spaced particles.     

Development of computationally efficient augmented Lagrangian SPH for incompressible flows and its quantitative comparison with WCSPH simulating flow past a circular cylinder

In Lagrangian particle-based methods such as smoothed particle hydrodynamics (SPH), computing totally divergence-free velocity field in a flow domain with the smallest error possible is the most critical issue, which might be achieved through solving pressure Poisson equation implicitly with higher particle resolutions. However, implicit solutions are computationally expensive and may be particularly challenging in the solution of multiphase flows with highly nonlinear deformations as well as fluid-structure interaction problems. Augmented Lagrangian SPH (ALSPH) method is a new alternative algorithm as a prevalent pressure solver where the divergence-free velocity field is achieved by iterative calculation of velocity and pressure fields. This study investigates the performance of the ALSPH technique by solving a challenging flow problem such as two-dimensional flow around a cylinder within the Reynolds number range of 50 to 500 in terms of improved robustness, accuracy, and computational efficiency. The same flow conditions are also simulated using the conventional weakly compressible SPH (WCSPH) method. The results of ALSPH and WCSPH solutions are not only compared in terms of numerical validation/ verification studies, but also rigorous investigations are performed for all related physical flow characteristics, namely, hydrodynamic coefficients, frequency domain analyses, and velocity divergence fields.     

Average pressure and velocity fields in non-uniform suspensions of spheres in Stokes flow

A widely used method for the approximate numerical simulation of the bulk behavior of particle suspensions consists in filling the entire space with copies of a fundamental cell in which N particles are arranged according to some probability distribution. Until now this method has only been used for suspensions that are spatially uniform in the mean. The case of spatially non-uniform systems, on the other hand, has not been considered. Here the average velocity and pressure fields for such a non-uniform suspension of identical rigid spheres in Stokes flow are calculated, and analytic solutions expressed in terms of multipole coefficients are presented. The results match and extend others obtained by the authors in parallel work using a completely different approach. In particular, the definition of a quantity to be identified with the mixture pressure is fully supported by the present results. An explicit result for the structure of the viscous stress in the suspension is also found. It is shown that, for spatially non-uniform systems, the stress contains a non-symmetric contribution analogous to a baroclinic source of vorticity.As a byproduct of the analysis, certain integrals of two periodic functions introduced by Hasimoto are calculated. These integrals would arise in similar problems, e.g. the electric field produced by electric multipoles in a periodic cubic structure.     

Effect of surface slip on Stokes flow past a spherical particle in infinite fluid and near a plane wall

The motion of a spherical particle in infinite linear flow and near a plane wall, subject to the slip boundary condition on both the particle surface and the wall, is studied in the limit of zero Reynolds number. In the case of infinite flow, an exact solution is derived using the singularity representation, and analytical expressions for the force, torque, and stresslet are derived in terms of slip coefficients generalizing the Stokes–Basset–Einstein law. The slip velocity reduces the drag force, torque, and the effective viscosity of a dilute suspension. In the case of wall-bounded flow, advantage is taken of the axial symmetry of the boundaries of the flow with respect to the axis that is normal to the wall and passes through the particle center to formulate the problem in terms of a system of one-dimensional integral equations for the first sine and cosine Fourier coefficients of the unknown traction and velocity along the boundary contour in a meridional plane. Numerical solutions furnish accurate predictions for (a) the force and torque exerted on a particle translating parallel to the wall in a quiescent fluid, (b) the force and torque exerted on a particle rotating about an axis that is parallel to the wall in a quiescent fluid, and (c) the translational and angular velocities of a freely suspended particle in simple shear flow parallel to the wall. For certain combinations of the wall and particle slip coefficients, a particle moving under the influence of a tangential force translates parallel to the wall without rotation, and a particle moving under the influence of a tangential torque rotates about an axis that is parallel to the wall without translation. For a particle convected in simple shear flow, minimum translational velocity is observed for no-slip surfaces. However, allowing for slip may either increase or decrease the particle angular velocity, and the dependence on the wall and particle slip coefficients is not necessarily monotonic.     

Harmonic waves in a prestressed thin elastic tube filled with a viscous fluid

In this work a theoretical analysis is presented for wave propagation ina thin-walled prestressed elastic tube filled with a viscous fluid. Thefluid is assumed to be incompressible and Newtonian, whereas the tubematerial is considered to be incompressible, isotropic and elastic.Considering the physiological conditions that the arteries experience, sucha tube is initially subjected to a mean pressure P_i and anaxial stretch _z. If it is assumed that in the course ofblood flow small incremental disturbances are superimposed on this initialfield, then the governing equations of this incremental motion are obtainedfor the fluid and the elastic tube. A harmonic-wave type of solution issought for these field equations and the dispersion relation is obtained.Some special cases, as well as the general case, are discussed and thepresent formulation is compared with some previous works on the samesubject.     

Finite element solutions for laminar flow and heat transfer around a square prism

Abstract

The finite element solutions of Navier‐Stokes and energy equations for steady laminar flow and heat transfer around square prisms, with attack angles of 0° and 45° have been obtained for a gas of Pr=0.7. The variations of surface shear stress, local pressure and Nusselt number are obtained over the entire prism surface including the zone beyond the point of separation. The predicted values of drag coefficients, the location of. separation, the average Nusselt number and the plots of velocity flow fields and isotherms are also presented. The trend of the present numerical results seems reasonable.     

Start-up flow in a porous medium channel

Summary The transient fluid motion in a porous medium channel is considered. Frictional resistance induced by the solid matrix and the channel walls is accounted for by a Darcian body force and a viscous shear stress, respectively. The adopted mathematical model leads to a one-parameter problem, in which the channel half-widthh, the porosity and the permeabilityK combine into a shape parameterA=(h ²/K)^1/2. Exact analytical solutions in terms of infinite series expansions are provided both for the start-up flow following the sudden imposition of a constant pressure gradient and for the transient motion induced by an instantaneously imposed flow rate. Time histories of the centerline velocity and the wall friction are presented, together with time-varying velocity profiles. It is observed that the start-up time required to reach a steady state is significantly reduced in the less porous channels, and this reduction is more pronounced when the start-up flow is driven by a pressure gradient than if the transient motion is forced by an imposed flow rate.     

Numerical exploration of flow topology and vortex stability in a curved duct

We performed incompressible flow simulation in a square duct with 90° bend and a curvature radius of 2.3 to extend our understanding of the vortical flow development in the bend. The solutions for the flow investigated at the Reynolds number of Re=790 are obtained in a tri-quadratic element system, where velocities stagger the pressure working variable, using the streamline-upwind finite element model and the BiCGSTAB iterative solver. The simulated results reveal that centrifugal force convects the quickly moving fluid particles towards the outer wall. The axial velocity, as a result, shows twin peaks in the curved channel. At about θ=66°, the secondary flow shows three complex pairs of vortices. Also noteworthy is the formation of a downstream spiralling flow motion. To better elucidate the dominating three-dimensional flow nature, the topological study of limiting streamlines was undertaken. Insight into the longitudinal flow instability is gained by tracking the formation and diminishing of limiting cycles. Copyright © 2006 John Wiley & Sons, Ltd.     

Finite element solution of a boundary integral equation for mode I embedded three-dimensional fractures

The singular integral equation governing the opening of a mode I embedded three-dimensional fracture in an infinite solid was solved by applying the finite element method. The strategy is to formulate the equation into weak form, and to transfer the differentiation from the singular term, 1/r, in the equation to the test function. A numerical algorithm was thus developed. The numerical solutions for circular and elliptical fractures under the action of polynomial pressure distributions were compared with the analytical solutions by Green and Sneddon,¹² Irwin,¹³ Shah and Kobayashi¹⁴ and Nishioka and Atluri.¹⁶ The results have demonstrated that the numerical method reported is accurate and efficient.     

Sound Produced When a Vortex Pair is Swallowed by a Jet

An analysis is made of the sound generated during the high-Reynolds-number convection of a vortex pair in a jet of water exhausting from a large vessel through a slit aperture. The equations of motion are linearized about the classical free-streamline solution describing steady flow through the aperture. It is assumed that the vortex pair is swept through the aperture into the jet by the steady mean flow, with no account taken of the nonlinear influence on the motion of ‘images’ in the boundaries. Additional vorticity is shed from the edges of the aperture in order that the flow should remain smooth and continuous (the Kutta condition). This vorticity is convected away within a sheet of ‘bound’ vorticity on the mean free streamlines of the jet. A strong peak in the bound vorticity is established when the vortex pair enters the aperture. Both the incident and the shed vorticity generate sound, but their respective contributions to the acoustic pressure are of opposite phase. The dominant radiation in the water above the aperture is produced as the vortex enters the jet, and has the form of a pressure pulse of width ~h/M and monopole strength , where h is the width of the aperture, ρ _o the density of the water, v a typical flow velocity, and M is the jet Mach number.     

Theoretical and numerical results of a deterministic two-dimensional vortex method

The first part of the paper reviews results obtained in earlier work: (1) The outline of the derivation of an integral equation of Fredholm type with respect to vorticity from the Navier-Stokes equations, and (2) the analytical results for two deterministic vortex methods which are based on the corespreading model. The aim of the second part is to confirm the results of the previous analysis and to estimate the accuracy of these methods numerically. In the present paper, the model problem of a transient flow past an impulsively started circular cylinder is studied by several numerical methods. The numerical results show that (1) the Gaussian core-spreading methods are comparable with the random walk vortex method, (2) the numerical fluctuation of the deterministic methods is small, and (3) the number of panels is smaller than those in the random walk vortex method to obtain the flow with almost the same accuracy. The theoretical results in § 2–4 of the present paper are summarized from “Core-spreading vortex methods in two-dimensional viscous flows”,Computer methods in applied mechanics and engineering (in press), by Kida and Nakajima, with permission from Elsevier Science, The Boulevard, Langford Lane, Kidlington 0X5 1GB, UK. The experimental picture is transferred from “The early stage of development of the wake behind an impulsively started cylinder for 40<Re<⁴”, by R Bouard and M Coutanceau,J. Fluid Mech. (1980) 101: 583–607.     

Intensive parameter analysis of adiabatic capillary tube using approximate analytic solution

Parameter analysis is helpful to have an insight into the flow characteristics of capillary tubes. Based on the approximate analytic solutions developed by the author, influences of geometrical parameters (inner diameter and length) and inlet operating parameters (pressure, subcooling or quality) on the mass flow rate through an adiabatic capillary tube have been intensively studied in this work. Some simple theoretical relations have been developed. The relations show that the mass flow rate is the power function of the geometries. For the subcooled inlet, good generic linearity between (2−C₂) power of the mass flow rate and the inlet operating parameters was deduced. With further approximation, some well-known linear trends could be theoretically interpreted. For the low quality inlet, good linearity between (C₂−2) power of the mass flow rate and the inlet quality was proposed. Approximately, the reciprocal of the mass flow rate is linear with the inlet quality. Experimental data supplemented by numerical data for R22, R410A and R407C are employed to verify the relations.     

Steady viscous flow in constricted elastic tubes subjected to a uniform external pressure

Cardiovascular illness is most commonly caused by a constriction, called a stenosis. A non-linear mathematical model with a free moving boundary was introduced to study viscous flow in tapered elastic tubes with axisymmetric constrictions subject to a prescribed pressure drop and a uniform external pressure. An iterative numerical scheme using a boundary iteration method was developed to solve the model. Effects of stenosis severity and stiffness, pressure drop, external pressure and stiffness of the vessel wall on the flow and wall motion were evaluated. It was found that stenosis severity, pressure drop and external pressure played more dominant roles than tube wall stiffness and stenosis stiffness perturbation. Tubes with 71 and 78 per cent stenoses showed two areas of negative transmural pressure and complex contraction–expansion–contraction wall motion patterns. Two types of tube diameter contraction and negative transmural pressure were observed, one was just distal to the stenosis and the other was near the outlet of the tube. Experiments using stenotic silicone tubes were conducted to quantify the tube law and verify the predicted pressure–flow relationship. The agreement between the numerical results and experimental measurements is better than that from a previous model which assumed periodicity of the tube and imposed different pressure conditions. © 1998 John Wiley & Sons, Ltd.