Rarefied Poiseuille flow in elliptical and rectangular tubes

An isothermal steady rarefied gas flow in a long channel (tube) of elliptical or rectangular cross-section under the action of a given pressure gradient (Poiseuille flow) is studied on the basis of the Bhatnagar-Gross-Krook model. The solution is obtained using a conservative higher-order method. The velocity field in a channel cross-section is investigated as a function of the rarefaction degree and the cross-section geometry parameters. The main calculated function is the gas flow rate through the tube. The solutions obtained are compared with the available results.     

Onset of flow instability in a heated capillary tube

We study the stability of flow in a heated capillary tube with an evaporating meniscus. The behavior of the vapor/liquid system, which undergoes small perturbations, is analyzed by linear approximation, in the frame of a one-dimensional model of capillary flow, with a distinct interface. The effect of the physical properties of both phases, the wall heat flux and the capillary sizes, on the flow stability is studied. The velocity, pressure and temperature oscillations in a capillary tube with a constant wall heat flux or a constant wall temperature are determined. A scenario of a possible process at small and moderate Peclet numbers corresponding to the flow in capillaries is considered. The boundaries of stability, subdividing the domains of stable and unstable flows, are outlined, and the values of geometrical and operating parameters corresponding to the transition from stable to unstable flow are estimated. It is shown that the stable capillary flow occurs at relatively small wall heat fluxes, whereas at high ones, the flow is unstable, with continuously growing velocity, pressure and temperature oscillations.     

Numerical Analysis of the Influence of Thermal Stresses on the Nonlinear Thermomolecular Pressure Difference Effect

On the basis of a numerical analysis of the non-Navier-Stokes gas-dynamic equations for slow non-isothermal gas flows, the nonlinear thermomolecular pressure difference effect due to a large temperature gradient along the lateral surface of a capillary is investigated. It is shown that the magnitude of the effect is substantially different from the values calculated using the Navier-Stokes equations. For two models of molecular interaction (Maxwell molecules and hard spheres), the possibility of a quasi-one-dimensional interpretation of the effect for experimental estimation purposes is demonstrated. The solutions of the relaxation kinetic equation for flow in a plane capillary at small Knudsen numbers and the gas-dynamic equations for slow non-isothermal flows are compared and the range of their applicability is estimated.     

Formation of the finely dispersed gas phase in upward and downward fluid flows

The influence of the fluid and gas flow rate parameters on the diameter of bubbles separating from a single capillary immersed in a fluid flow in a channel of 1×1 cm cross-section is investigated. The dependences of the mean bubble diameter on the fluid velocity and the gas flow rate through the capillary are obtained. The frequency of the bubble detachment from the capillary is estimated. It is shown that the histogram of the bubble size distribution changes qualitatively with the excess of a certain critical bubble detachment frequency, which is associated with the transition from the discrete to jet mode of detachment.     

Stokes flow in a cylindrical tube containing a line of spheroidal particles

Viscous flow in a circular cylindrical tube containing an infinite line of rigid spheroidal particles equally spaced along the axis of the tube is considered for (a) uniform axial translation of the spheroids (b) flow past a line of stationary spheriods and (c) flow of the suspending fluid and spheroids under an imposed pressure gradient. The fluid is assumed to be incompressible and Newtonian. The Reynolds number is assumed to be small and the equations of creeping flow are used. Two types of solutions are developed: (i) an exact solution in the form of an infinite series which is valid for ratios of the spheroid diameter to the tube diameter up to 0.80, (ii) an approximate solution using lubrication theory which is valid for spheroids which nearly fill the tube. The drag on each spheroid and the pressure drop are computed for all cases. Both prolate and oblate spheroids are considered. The results show that the drag and pressure drop depend on the spheroidal diameter perpendicular to the axis of tube primarily and the effects of the spheroidal thickness and spacing are secondary. The results are of interest in connection with mechanics of capillary blood flow, sedimentation, fluidized beds, and fluid-solid transport.     

End effects in rarefied gas outflow into vacuum through a long tube

The problem of steady outflow of rarefied gas from a reservoir into vacuum through a long cylindrical tube of circular cross-section at a constant temperature is considered on the basis of the kinetic S-model. The kinetic equation is solved numerically using a conservative second-order method. The basic calculated quantity is the gas flow rate through the channel; the flowfields are also studied. The solutions obtained are compared with the known results.     

Pore-Level Modeling of Gas and Condensate Flow in Two- and Three-Dimensional Pore Networks: Pore Size Distribution Effects on the Relative Permeability of Gas and Condensate

We present a mechanistic model of retrograde condensation processes in two- and three-dimensional capillary tube networks under gravitational forces. Condensate filling-emptying cycles in pore segments and gas connection–isolation cycles are included. With the pore-level distribution of gas and condensate in hand, we determine their corresponding relative permeabilities. Details of pore space and displacement are subsumed in pore conductances. Solving for the pressure field in each phase, we find a single effective conductance for each phase as a function of condensate saturation. Along with the effective conductance for the saturated network, the relative permeability for each phase is calculated. Our model porous media are two- and three-dimensional regular networks of pore segments with distributed size and square cross-section. With a Monte Carlo sampling we find the optimum network size to avoid size effects and then we investigate the effect of network dimensionality and pore size distribution on the relative permeabilities of gas and condensate.     

Structure of the nonisothermal swirling gas-droplet flow behind an abrupt tube expansion

Flow structure and heat and mass transfer in a swirling two-phase stream is numerically modeled using the Reynolds stress transport model. The gas phase is described by the 3DRANS system of equations with account for the inverse influence of particles on the transport processes in the gas. The gas phase turbulence is calculated using the Reynolds stress transport model with account for the presence of disperse particles. The two-phase nonswirling flow behind an abrupt tube expansion contains a secondary corner vortex which is absent from the swirling flow. The disperse phase is redistributed over the tube cross-section. Large particles are concentrated in the wall region of the channel under the action of the centrifugal forces, while the smaller particles are in the central zone of the chamber.     

用管轴流速确定动脉中血液振荡流的速度剖面

本文通过求解圆管内血液振荡流的基本方程，求得圆管内血液流的速度与压力梯度之间的关系式，文章提出一种利用管轴外流速计算管内压力梯度，进而确定血液振荡流动速度分布的方法，该方法用于检测活体血管内血液振荡流的速度剖面，具有操作简单，精度较高的优点，最后，以人体颈动脉为例，讨论血液周期振荡流的速度分布特征，发现在任意时刻，除了邻近管壁速度迅速降为零之外，沿管截面速度分布相当均匀，呈现出与定常流不同的速度分布特征。     

Origin and quantification of coupling between relative permeabilities for two-phase flows in porous media

An extended formulation of Darcy's two-phase law is developed on the basis of Stokes' equations. It leads, through results borrowed from the thermodynamics of irreversible processes, to a matrix of relative permeabilities. Nondiagonal coefficients of this matrix are due to the viscous coupling exerted between fluid phases, while diagonal coefficients represent the contribution of both fluid phases to the total flow, as if they were alone. The coefficients of this matrix, contrary to standard relative permeabilities, do not depend on the boundary conditions imposed on two-phase flow in porous media, such as the flow rate. This formalism is validated by comparison with experimental results from tests of two-phase flow in a square cross-section capillary tube and in porous media. Coupling terms of the matrix are found to be nonnegligible compared to diagonal terms. Relationships between standard relative permeabilities and matrix coefficients are studied and lead to an experimental way to determine the new terms for two-phase flow in porous media.     

The effect of longitudinal vibration on poiseuille flow in a non-circular pipe

The effect is discussed of a superposed longitudinal sinusoidal vibration on the flow, under a constant pressure gradient, of a slightly non-Newtonian fluid in a straight pipe of non-circular cross-section. It is found that a flow in transverse planes is produced. This is the superposition of steady and sinusoidal flows. The stream-function corresponding to the steady flow is obtained in the case when the pipe has a rectangular cross-section. The discussion is based on a Rivlin-Ericksen constitutive equation in which only“second-order” non-Newtonian terms are present.     

A New Non-Darcy Flow Model for Low-Velocity Multiphase Flow in Tight Reservoirs

The pore and pore-throat sizes of shale and tight rock formations are on the order of tens of nanometers. The fluid flow in such small pores is significantly affected by walls of pores and pore-throats. This boundary layer effect on fluid flow in tight rocks has been investigated through laboratory work on capillary tubes. It is observed that low permeability is associated with large boundary layer effect on fluid flow. The experimental results from a single capillary tube are extended to a bundle of tubes and finally to porous media of tight formations. A physics-based, non-Darcy low-velocity flow equation is derived to account for the boundary layer effect of tight reservoirs by adding a non-Darcy coefficient term. This non-Darcy equation describes the fluid flow more accurately for tight oil reservoir with low production rate and low pressure gradient. Both analytical and numerical solutions are obtained for the new non-Darcy flow model. First, a Buckley–Leverett-type analytical solution is derived with this non-Darcy flow equation. Then, a numerical model has been developed for implementing this non-Darcy flow model for accurate simulation of multidimensional porous and fractured tight oil reservoirs. Finally, the numerical studies on an actual field example in China demonstrate the non-negligible effect of boundary layer on fluid flow in tight formations.     

Study of Dispersion in Porous Media by Pulsed Field Gradient NMR: Influence of the Fluid Rheology

Pulsed field gradient nuclear magnetic resonance (PFG-NMR) is used to measure the molecular displacements for the flow of a fluid through a capillary tube and a packed bed made of monodisperse PMMA beads. The molecules average displacement is studied using both the formalism of propagators and the cumulant method. In the Poiseuille case, the dispersion coefficients determined by the cumulant method compare satisfactorily with the theoretical values obtained. The technique is then extended to study the flow through a porous medium. We thus analyze Newtonian (water) and non-Newtonian (Xanthan) flows and put a particular emphasis on comparing the dispersion mechanisms between Newtonian and non-Newtonian fluids.     

颗粒毛细效应影响因素的离散元分析

颗粒毛细效应是指将一根细管插入填充有颗粒物质的容器中并对管施加竖直振动时颗粒在管内上升并最终达到一个稳定的高度的现象, 该现象为颗粒物料的逆重力输运提供了一种潜在的技术途径. 为探究颗粒毛细效应的影响因素, 采用离散元方法, 模拟再现了颗粒毛细效应过程,展示了不同管径下颗粒竖直方向速度演变特性, 考察了不同容器宽度和振动条件下颗粒最终毛细上升高度随管径的演变规律. 结果表明, 在容器宽度与粒径比为40、管振幅与粒径比为14.33、管振动频率为12 Hz情况下, 管径与粒径比$D/d = 3.33$时, 管内颗粒堵塞严重, 使得颗粒上升缓慢,并造成颗粒柱中断; $D/d = 8.33$时, 起初毛细上升高度增加迅速, 随后毛细上升高度的增大逐渐减缓, 管内颗粒在管径方向几乎不存在速度梯度; $D/d =15$时, 随着颗粒毛细上升高度的增大, 管内颗粒柱分离为速度截然不同的两层, 上层颗粒在管径方向几乎不存在速度梯度, 而下层颗粒存在明显的速度梯度.研究还发现, 在毛细效应能够发生的管径范围内, 存在一个对应于颗粒最终毛细上升高度最大值的临界管径, 当管径小于临界管径时, 颗粒最终毛细上升高度随管径的增大而增大, 当管径大于临界管径时, 颗粒最终毛细上升高度随管径的增大而趋于减小; 增大容器宽度,临界管径有所增大; 增大振幅、适当提高频率能够有效促进临界管径的增大.      

Speed measurements in the developing region of an electrohydrodynamic spray using laser diagnostics

Experiments were conducted to assess the ability of a particle-counter-sizer velocimeter (PCSV) to measure droplet speeds in the developing region of an electrohydrodynamic spray. The working fluid, ethanol, was pumped through a stainless steel capillary tube at a constant mass flow rate. The capillary tube was held at various positive potential voltages with respect to an externally grounded funnel located below the tube. Data were acquired at two axial stations, directly under the capillary tip, in the developing region of the subject spray. The results indicated that the PCSV, with an experimental modification to the speed reduction process, performed well in this region. Speeds were measured at the near station within an average uncertainty of 2.2% and a corresponding maximum uncertainty of 3.0%; speeds at the far station were measured within an average uncertainty of 4.6% and a corresponding maximum uncertainty of 6.0%.     

刚性圆管中血液周期振荡流的切应力分布

本文通过求解圆管内血液振荡流的基本方程，求得圆管内血液流的压力梯度与切应力之间的关系式。在此基础上，详细讲座了圆管中轴向流速和切变率谐波的变化规律，指出流速谐波和切变率谐波的幅值都将随着谐波次数的增大而逐渐减小。为了使所得结果便于应用。文章通过管轴向中心线流速与压力梯度之间的关系式，进一步给出一种利用管轴向中心线流速计算管内切应力分布的简便方法。该方法用于检测活体血管内血液振荡流的切应力分布，具有操作简单，精度较高的优点。最后，以人体颈动脉为例，讨论血液周期振荡流的切应力的分布特性。发现在任意时刻，除了邻近管壁处切应力急剧增大到一定数值之外，沿管截面切应力分布相当均匀且接近于零，呈现出与定常流不同的切应力分布特征。     

Effect of elastic wall motion on oscillatory flow of micropolar fluid in an annular tube

Summary Oscillatory flow of a micropolar fluid in an annular tube is investigated. The outer wall of the tube is taken to be elastic and the variation in the diameter of the elastic wall due to pulsatile nature of pressure gradient is assumed to be small. The wall motion is governed by a tube law. The nonlinear equations governing the fluid flow and the tube law are solved using perturbation analysis. The steady-streaming phenomenon due to the interaction of convected inertia with viscous effects is studied. The analysis, is carried out for zero mean flow rate. It presents the effects of the elastic nature of the wall combined with micropolar fluid parameters on the mean pressure gradient and wall shear stress for different catheter sizes and frequency parameters. It is found that the effect of micropolarity is of considerable importance for small steady-streaming Reynolds number. Also, it is observed that the relationship between mean pressure gradient and the flow rate depends on the amplitude of the diameter variation, flow rate waveforms and the phase difference between them.     

Newtonian flow in a triangular duct with slip at the wall

We consider the Newtonian Poiseuille flow in a tube whose cross-section is an equilateral triangle. It is assumed that boundary slip occurs only above a critical value of the wall shear stress, namely the slip yield stress. It turns out that there are three flow regimes defined by two critical values of the pressure gradient. Below the first critical value, the fluid sticks everywhere and the classical no-slip solution is recovered. In an intermediate regime the fluid slips only around the middle of each boundary side and the flow problem is not amenable to analytical solution. Above the second critical pressure gradient non-uniform slip occurs everywhere at the wall. An analytical solution is derived for this case and the results are discussed.     

The effect of the diameter on air-water annular and churn flow in vertical pipes with and without surfactants

In this work, we present results of flow visualisation, pressure gradient measurements, and liquid holdup measurements for air-water flow without and with surfactants in vertical pipes with diameters of 34 mm, 50 mm, and 80 mm. The surfactants cause the formation of foam. This foam has a larger volume and a smaller density than the liquid. The larger volume results in a larger pressure gradient at large gas flow rates. At small gas flow rates, the lower density of the foam causes the transition between the regular annular flow regime and the irregular churn flow regime to shift to lower gas flow rates. As a result foam reduces the pressure gradient and the liquid holdup at small gas flow rates. Surfactants reduce the pressure gradient more effectively for thinner liquid films at the wall; therefore, they are more effective for small pipe diameters and small liquid flow rates.     

Measurement of the liquid film thickness in micro tube slug flow

Slug flow is one of the representative flow regimes of two-phase flow in micro tubes. It is well known that the thin liquid film formed between the tube wall and the vapor bubble plays an important role in micro tube heat transfer. In the present study, experiments are carried out to clarify the effects of parameters that affect the formation of the thin liquid film in micro tube two-phase flow. Laser focus displacement meter is used to measure the thickness of the thin liquid film. Air, ethanol, water and FC-40 are used as working fluids. Circular tubes with five different diameters, D = 0.3, 0.5, 0.7, 1.0 and 1.3 mm, are used. It is confirmed that the liquid film thickness is determined only by capillary number and the effect of inertia force is negligible at small capillary numbers. However, the effect of inertia force cannot be neglected as capillary number increases. At relatively high capillary numbers, liquid film thickness takes a minimum value against Reynolds number. The effects of bubble length, liquid slug length and gravity on the liquid film thickness are also investigated. Experimental correlation for the initial liquid film thickness based on capillary number, Reynolds number and Weber number is proposed.