Stokes flow inside and around a fluid drop distorted by the presence of a tube wall

Using a quadratic optimisation process to satisfy the boundary conditions, the drag coefficient and the flow patterns inside and outside a fluid drop translating axially in a tube have been determined accurately even when the drop is clearly elongated. Convincing comparisons with experimental results (flow visualizations, velocity and drag measurements) are presented. The effects of the viscosity and density ratios are examined.     

Deformation and breakup of a non-Newtonian slender drop in an extensional flow

The deformation and breakup of a non-Newtonian slender drop in a Newtonian liquid in a simple extensional and creeping flow has been theoretically studied. The power-law was chosen for the fluid inside the drop, and the deformation of the drop is described by a single ordinary differential equation, which was numerically solved. Asymptotic analytical expressions for the local radius were derived near the center and close to the end of the drop. The results for the shape of the drop and the breakup criterion are presented as a function of the capillary number, the viscosity ratio and type of non-Newtonian fluid inside the drop. An approximate analytical solution is also suggested which is in good agreement with the numerical results.     

Viscous flow of a suspension of liquid drops in a cylindrical tube

Viscous flow in a circular cylindrical tube containing an infinite line of viscous liquid drops equally spaced along the tube axis is considered under the assumption that a surface tension, sufficiently large, holds the drops in a nearly spherical shape. Three cases are considered: (1) axial translation of the drops, (2) flow of the external fluid past a line of stationary drops, and (3) flow of external fluid and liquid drops under an imposed pressure gradient. Both fluids are taken to be Newtonian and incompressible, and the linearized equations of creeping flow are used.The results show that both drag and pressure drop per sphere increase as the spacing increases at fixed radius and also increase as the radius of the drop increases. The presence of the internal motion reduces the drag and pressure gradients in all cases compared to rigid spheres, particularly for drops approaching the size of the tube.     

Gravity-induced spreading of a drop of a viscous fluid over a surface

The unsteady-state nonlinear problem of spreading of a drop of a viscous fluid on the horizontal surface of a solid under the action of gravity and capillary forces is considered for small Reynolds numbers. The method of asymptotic matching is applied to solve the axisymmetrical problem of spreading when the gravity exerts a significant effect on the dynamics of the drop. The flow structure in the drop is determined at large times in the neighborhood of a self-similar solution. The ranges of applicability of the quasiequilibrium model of drop spreading with a dynamic edge angle and a self-similar solution are found. It is shown that the transition from one flow model to another occurs at very large Bond numbers. Institute of Mechanics of Multiphase Systems, Siberian Division, Russian Academy of Sciences, Tyumen’ 625000. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 3, pp. 59–67, May–June, 1999.     

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.     

A boundary‐only approach to the deformation of a shear‐thinning drop in extensional Newtonian flow

The influence of shear thinning on drop deformation is examined through a numerical simulation. A two‐dimensional formulation within the scope of the boundary element method (BEM) is proposed for a drop driven by the ambient flow inside a channel of a general shape, with emphasis on a convergent–divergent channel. The drop is assumed to be shear thinning, obeying the Carreau–Bird model and the suspending fluid is Newtonian. The viscosity of the drop at any time is estimated on the basis of a rate‐of‐strain averaged over the region occupied by the drop. The viscosity thus changes from one time step to the next, and it is strongly influenced by drop deformation. It is found that small drops, flowing on the axis, elongate in the convergent part of the channel, then regain their spherical form in the divergent part; thus confirming experimental observations. Newtonian drops placed off‐axis are found to rotate during the flow with the period related to the initial extension, i.e. to the drop aspect ratio. This rotation is strongly prohibited by shear thinning. The formulation is validated by monitoring the local change of viscosity along the interface between the drop and the suspending fluid. It is found that the viscosity averaged over the drop compares, generally to within a few per cent, with the exact viscosity along the interface.     

Flow induced on a salt waterbody due to the impingement of a freshwater drop or a water source

The particle image velocimetry (PIV) and planar laser-induced fluorescence (PLIF) techniques are used to study the flow induced on the surface of a body of saltwater when a drop impinges on its surface or when a source is present on the surface. The measurements show that the impingement of a fresh water drop causes a strong axisymmetric solutocapillary flow about the vertical line passing through the center of impact. The fluid directly below the center of impact rises upward, and near the surface it moves away from the center of impact. The flow, which develops within a fraction of second after the impact, persists for several seconds. In comparison, when a freshwater drop falls on a body of freshwater, the flow induced on the surface is much weaker and persists for a relatively shorter duration of time and the volume of water circulated is two orders of magnitude smaller. Similarly, when a fresh water source is present on a body of saltwater there is a solutocapillary flow which on the surface is away from the source and below the surface is towards the source.     

Modeling of a Dynamic Magneto-Fluid Seal in the Presence of a Pressure Drop

The ferrohydrodynamic problem, with a disconnected free surface, of the stability of an annular magneto-fluid seal under the action of centrifugal forces and a pressure drop is solved numerically. The effect of the shaft speed on the critical pressure drop is investigated in relation to the volume of the magneto-fluid plug, the magnetic field strength, the magnetic properties of the fluid, the gap width, and the shaft radius. The flow pattern and the thermal power released in the sealing layer are studied.     

Models for the deformation of a single ellipsoidal drop: a review

Dilute polymer blends and immiscible liquid emulsions are characterized by a globular morphology. The dynamics of a single drop subjected to an imposed flow field has been considered to be a valuable model system to get information on dilute blends. This problem has been studied either theoretically by developing exact theories for small drop deformations or by developing simplified models often based on phenomenological assumptions. In this paper, a critical overview of the available models for the dynamics of a single drop is presented, discussing four different systems, namely the Newtonian system, where a single Newtonian drop is immersed in an infinite Newtonian matrix; the non-Newtonian system, where at least one of the components, the drop fluid or the matrix one, is non-Newtonian; the confined Newtonian system, where the matrix is confined and wall effects alter the drop dynamics; and the confined non-Newtonian system.     

Modeling Non-Darcian Single- and Two-Phase Flow in Transparent Replicas of Rough-Walled Rock Fractures

While it is generally assumed that in the viscous flow regime, the two-phase flow relative permeabilities in fractured and porous media depend uniquely on the phase saturations, several studies have shown that for non-Darcian flows (i.e., where the inertial forces are not negligible compared with the viscous forces), the relative permeabilities not only depend on phase saturations but also on the flow regime. Experimental results on inertial single- and two-phase flows in two transparent replicas of real rough fractures are presented and modeled combining a generalization of the single-phase flow Darcy’s law with the apparent permeability concept. The experimental setup was designed to measure injected fluid flow rates, pressure drop within the fracture, and fluid saturation by image processing. For both fractures, single-phase flow experiments were modeled by means of the full cubic inertial law which allowed the determination of the intrinsic hydrodynamic parameters. Using these parameters, the apparent permeability of each fracture was calculated as a function of the Reynolds number, leading to an elegant means to compare the two fractures in terms of hydraulic behavior versus flow regime. Also, a method for determining the experimental transition flow rate between the weak inertia and the strong inertia flow regimes is proposed. Two-phase flow experiments consisted in measuring the pressure drop and the fluid saturation within the fractures, for various constant values of the liquid flow rate and for increasing values of the gas flow rate. Regardless of the explored flow regime, two-phase flow relative permeabilities were calculated as the ratio of the single phase flow pressure drop per unit length divided by the two-phase flow pressure drop per unit length, and were plotted versus the measured fluid saturation. Results confirm the dependence of the relative permeabilities on the flow regime. Also the proposed generalization of Darcy’s law shows that the relative permeabilities versus fluid saturation follow physical meaningful trends for different liquid and gas flow rates. The presented model fits correctly the liquid and gas experimental relative permeabilities as well as the fluid saturation.     

Flow of wormlike micelle solutions past a confined circular cylinder

In this paper, we present the results of an investigation into the flow of a series of viscoelastic wormlike micelle solutions past a confined circular cylinder. Although this benchmark flow has been studied in great detail for polymer solutions, this paper reports the first experiments to use a viscoelastic wormlike micelle solution as the test fluid. The flow kinematics, stability and pressure drop were examined for two different wormlike micelle solutions over a wide range of Deborah numbers and cylinder to channel aspect ratios. A combination of particle image velocimetry and pressure drop measurements were used to characterize the flow kinematics, while flow-induced birefringence measurements were used to measure the micelle deformation and alignment in the flow. The pressure drop was found to decrease initially due to the shear thinning of the test fluid before increasing at higher flow rates as elastic effects begin to dominate the flow. Above a critical Deborah number, an elastic instability was observed for just one of the test fluids studied, the other remained stable for all Deborah number tested. Flow-induced birefringence and velocimetry measurements showed that observed instability originates in the extensional flow in the wake of the cylinder and appears not as periodic counter-rotating vortices as has been observed in the flow of polymer solutions past circular cylinders, but as a chaotic rupture event in the wake of the cylinder that propagates axially along the cylinder. Reducing the cylinder to channel aspect ratio and the degree of shearing introduced by the channel walls had a weak impact on the stability of the flow. These measurements, when taken in conjunction with previous work on flow of wormlike micelle solutions through a periodic array of cylinders, definitively show that the instability can be attributed to a breakdown of the wormlike micelle solutions in the extensional flow in the wake of the cylinder.     

Shear flow over a porous particle

Linear axisymmetric Stokes flow over a porous spherical particle is investigated. An exact analytic solution for the fluid velocity components and the pressure inside and outside the porous particle is obtained. The solution is generalized to include the cases of arbitrary three-dimensional linear shear flow as well as translational-shear Stokes flow. As the permeability of the particle tends to zero, the solutions obtained go over into the corresponding solutions for an impermeable particle. The problem of translational Stokes flow around a spherical drop (in the limit a gas bubble or an impermeable sphere) was considered, for example, in [1,2]. A solution of the problem of translational Stokes flow over a porous spherical particle was given in [3]. Linear shear-strain Stokes flow over a spherical drop was investigated in [2].Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 113–120, May–June, 1995.     

Hydrodynamics of a submerging drop: Lined structures on the crown surface

Macrophotography and high-speed videofilming are used to investigate the material transfer in a falling drop upon collision with the surface of a fluid at rest. In the experiments the drops of colored water, milk, mineral oil, and seed oil fell in pure or colored water. Emphasis was placed on recording the pattern of the drop material spreading over the surface of the receiving fluid. On the continuous surface of the primary cavity and the crown the drop material is concentrated in the form of thin fibers which form a regular streaky or netlike pattern in which triangular, quadrangular, and pentagonal cells are expressed. The cell rows are ordered in the form of layers on the lateral walls and the bottom of the cavity. The fiber dimensions and the degree of their expressiveness vary in the process of flow evolution. The upper row of structures on the crown surface is formed by vertical fibers.     

Fluid flow simulation in a random elliptical porous medium by using the lattice Boltzmann method

A fluid flow through an isotropic porous medium with randomly arranged elliptical particles is simulated by the lattice Boltzmann method. The dimensionless pressure drop and the dimensionless permeability are evaluated as functions of the Reynolds number. The effect of the aspect ratio of the major to minor semi-axis of the ellipse on the dimensionless permeability is considered for different values of porosity. The pressure drop is thoroughly investigated as a function of fluid viscosity for different values of the aspect ratio and porosity. The influence of various parameters of the problem on the mean tortuosity of the medium is considered.     

Peristaltic transport of a Herschel-Bulkley fluid in an inclined tube

Peristaltic flow of Herschel-Bulkley fluid in an inclined tube is analyzed. The velocity distribution, the stream function and the volume flow rate are obtained. Also, when the yield stress ratio τ→0, and when the inclination parameter α=0 and the fluid parameter n=1, the results agree with those of Jaffrin and Shapiro (Ann. Rev. Fluid Mech. 3 (1971) 13) for peristaltic transport of a Newtonian fluid in a horizontal tube. The effects of τ and n on the pressure drop and the mean flow are discussed through graphs. Furthermore, the results for the peristaltic transport of Bingham and power law fluids through a flexible tube are obtained and discussed. The results obtained for the flow characteristics reveal many interesting behaviors that warrant further study of the effects of Herschel-Bulkley fluid on the flow characteristics.     

Drop formation in liquid-liquid systems

Conclusion An experimental method for determining the momentum flux on a drop for a nozzle with an arbitrary fluid velocity profile at the exit has been described. I t was observed that the variation of drop volume with flow rate can differ some-what from that conceived in previous drop formation models, in that drop volume does not necessarily vary smoothly with flow rate. High-speed motion pictures of drop formation and detachment show that the drop oscillates as it grows. I t is believed that this oscillation is related to the observed unsteady variation of drop volume with flow rate. Presently with General Dynamics Space Systems Division, San Diego/CA, USA     

Experimental investigation of viscoelastic drop deformation in Newtonian matrix at high capillary number under simple shear flow

The steady state deformation of a viscoelastic drop (Boger fluid) in a Newtonian liquid at high capillary number under simple shear flow is investigated by direct visualization using a specially designed Couette apparatus which enables visualization from two perpendicular directions. Two drop deformation modes are found: (1) Mode I – drop deformation in the flow direction and (2) Mode II – drop deformation in the vorticity direction. The drop deformation mode depends on the relative strength of the elastic contribution to viscous contribution. If the elastic contribution is weak compared to the viscous contribution, the drop elongates in the flow direction via Mode I. If the elastic contribution is strong, the drop elongates in the vorticity direction via Mode II. The drop size also affects the drop deformation. At the same capillary number, bigger drops have larger deformations than smaller drops.     

Three‐dimensional boundary element analysis of drop deformation in confined flow for Newtonian and viscoelastic systems

An adaptive (Lagrangian) boundary element approach is proposed for the general three‐dimensional drop deformation in confined flow. The adaptive method is stable as it includes remeshing capabilities of the deforming interface between drop and suspending fluid, and thus can handle large deformations. Both drop and surrounding fluid are viscous incompressible and can be Newtonian or viscoelastic. A boundary‐only formulation is implemented for fluids obeying the linear Jeffrey's constitutive equation. Similarly to the formulation for two‐dimensional Newtonian fluids (Khayat RE, Luciani A, Utracki LA. Boundary element analysis of planar drop deformation in confined flow. Part I. Newtonian fluids. Engineering Analysis of Boundary Elements 1997; 19 : 279), the method requires the solution of two simultaneous integral equations on the interface between the two fluids and the confining solid boundary. Although the problem is formulated for any confining geometry, the method is illustrated for a deforming drop as it is driven by the ambient flow inside a cylindrical tube. The accuracy of the method is assessed by comparison with the analytical solution for two‐phase radial spherical flow, leading to good agreement. The influence of mesh refinement is examined for a drop in simple shear flow. Copyright © 2000 John Wiley & Sons, Ltd.     

Experimental and numerical study of a gas cyclone with a central filter

This paper studies a novel gas cyclone with a cylindrical filter face installed in the center from the vortex finder to the bottom hopper. The experimental results show that this composite cyclone has a higher collection efficiency and a lower pressure drop than the original cyclone. The mechanisms for the improvement are analyzed by both physical experiments and numerical simulations. By measuring dust samples collected at different places it is revealed that the center filter can prevent fine particles from entering the inner vortex and escaping, which accounts for the increase of the collection efficiency. In addition, the flow field of the composite cyclone is simulated by computational fluid dynamics and compared with that of the original cyclone. The analysis shows that with the filter layer installed, the swirling flow disappears in the vortex finder, which decreases the kinetic energy dissipation and hence lowers the pressure drop.