Axisymmetric Straining Flow Research Articles

Dynamics of a gas bubble in a straining flow: Deformation, oscillations, self-propulsion

The dynamics of a gas bubble suspended in an axisymmetric straining flow is examined with modern tools of global stability analysis. The complete bifurcation diagram of a bubble constrained to remain trapped at the stagnation point is determined. It is made of a stable branch, along which the bubble oscillates, and an unstable branch along which it develops non-oscillating waisted shapes prefiguring breakup. Two unstable non-oscillating modes also allow a free bubble to drift away from the stagnation point, not only along the elongation axis but also within the compressional plane thanks to a specific self-propulsion mechanism.

Mono-emulsion droplet stretching under direct current electric field.

We study the mechanism of stretching and breaking of a mono-emulsion droplet under a direct current electric field using theoretical and experimental approaches aided by numerical simulation. Axisymmetric straining flow driven by an electric field results in the equilibrium deformation of the droplet along the direction of the electric field if the electric capillary number Cae that is the ratio of electric stresses to capillary stresses, is less than a critical value (Cae)crit. At (Cae)crit, the droplet breaks either before showing the slow deformation stage or rapidly. Furthermore, we developed a theoretical model to understand the mechanism of the transition from equilibrium deformation to non-equilibrium breaking. The Cae that can induce Taylor's deformation D = (α - β)/(α + β) ≈ 0.295; where α and β are the lengths of semi-major and semi-minor axes of the droplet, corresponds to (Cae)crit. At this stage, the maximum flow velocity shifts to the outside of the droplet along the electric field direction and large electric stresses are mainly concentrated at the droplet's side apex causing daughter droplet ejection. Finally, we compare the values of (Cae)crit obtained from the theoretical model ((Cae)crit ≈ 0.25) for which the conductivity ratio (R) between the droplet and ambient liquid i.e., R ∼ O(10) with our experimental results ((Cae)crit ≈ 0.245) and realize that (Cae)crit decreases as R increases. We also observe that even though the viscosity ratio can alter the emulsion breakup modes, it has no effect on (Cae)crit for the onset of breaking.

Two touching spherical drops in a uniaxial compressional flow: The effect of interfacial slip

This study presents a semi-analytical solution for the problem of two touching drops with slipping interfaces pushed against each other in a uniaxial compressional flow at low capillary and Reynolds numbers. The jump in the tangential velocity at the liquid-liquid interface is modeled using the Navier slip condition. Analytical solutions of the contact force, the drop-scale stresses, and the drop-scale pressure are provided as functions of the slip coefficient α, the viscosity ratio κ, and the drop size ratio k. Since unequal drop sizes are considered, two problems are solved in the tangent sphere co-ordinate system to determine the steady state position: a pair of touching drops with its contact point at the origin of an axisymmetric straining flow, and two touching drops placed in a uniform flow parallel to the axis of symmetry of the drops. A general observation is that the effect of slip is manifested most strongly for drops whose viscosity is much greater than the suspending fluid κ≫1. For highly viscous drops, the flow and stress fields transition from those corresponding to solid particles for ακ ≪ 1, to those for inviscid drops in the limit ακ ≫ 1. The analytical expressions provided here for the contact force and the stress distributions will serve to provide the restrictions that complete the definition of the lubrication flow problem in the thin film between the two colliding drops. While the contact force that drains fluid out of the thin film is relatively unaffected by slip, the tangential stress and pressure in the near-contact region are mitigated significantly for ακ ≫ 1. The latter is expected to assist coalescence at high capillary numbers.

Force on a Small Particle Attached to a Plane Wall in a Hiemenz Straining Flow

The force acting on a spherical particle fixed to a wall and immersed in an axisymmetric straining flow is examined for small Reynolds numbers. The steady, incompressible flow field is computed using an axisymmetric finite-volume method over conditions spanning five decades in the Reynolds number. The flow is characterized by the formation of a vortex ring structure in the wedge region formed between the particle lower surface and the plane wall. A power law expression for the dimensionless particle force is obtained as a function of the Reynolds number, which is found to hold with excellent accuracy for Reynolds numbers below about 0.1.

Critical behavior of drops in linear flows. I. Phenomenological theory for drop dynamics near critical stationary states

The dynamics of viscous drops in linear creeping flows are investigated near the critical flow strength at which stationary drop shapes cease to exist. According to our theory the near-critical behavior of drops is dominated by a single slow mode evolving on a time scale that diverges at the critical point with exponent 1/2. The theory is based on the assumption that the system undergoes a saddle-node bifurcation. The predictions have been verified by numerical simulations for drops in axisymmetric straining flow and in two-dimensional flows with less vorticity than in shear flow. Application of our theory to the accurate determination of critical parameters is discussed.

On the mechanism of microbial cell disruption in high-pressure homogenisation

The tensions produced in the wall of a rigid, thin-walled, liquid-filled sphere as it moves with an axisymmetric straining flow are examined. This problem has not been previously addressed. A generalised correlation for the maximum wall tension, expressed in dimensionless form as a Weber number ( We), is developed in terms of the acceleration number ( Ac) and Reynolds number ( Re) of the straining flow. At low Reynolds number We is dominated by viscous forces, while inertial forces due to internal pressure gradients caused by sphere acceleration dominate at higher Re. The generalised correlation has been used to examine the case of a typical yeast cell (a thin-walled, liquid-filled sphere) passing through a typical high-pressure homogeniser (a straining-flow device). At 56 MPa homogenising pressure, a 6 μm yeast cell experiences tensions in the inertially dominated regime ( Re=100). The correlation gives We=0.206, corresponding to a maximum wall tension of 8 Nm -1. This is equivalent to an applied compressive force of 150 μN and compares favourably with the force required to break yeast cells under compressive micromanipulation (40–90 μN). Inertial forces may therefore be an important and previously unrecognised mechanism of microbial cell disruption during high-pressure homogenisation. Further work is required to examine the likelihood of cell deformation in the high-strain-rate short-residence-time environment of the homogeniser, and the effect that such deformation may have on the contribution of inertial forces to disruption.

Accelerated flows past a rigid sphere or a spherical bubble. Part 1. Steady straining flow

This work reports the first part of a series of numerical simulations carried out in order to improve knowledge of the forces acting on a sphere embedded in accelerated flows at finite Reynolds number, Re. Among these forces added mass and history effects are particularly important in order to determine accurately particle and bubble trajectories in real flows. To compute these hydrodynamic forces and more generally to study spatially or temporally accelerated flows around a sphere, the full Navier–Stokes equations expressed in velocity–pressure variables are solved by using a finite-volume approach. Computations are carried out over the range 0.1 ≤ Re ≤ 300 for flows around both a rigid sphere and an inviscid spherical bubble, and a systematic comparison of the flows around these two kinds of bodies is presented.Steady uniform flow is first considered in order to test the accuracy of the simulations and to serve as a reference case for comparing with accelerated situations. Axisymmetric straining flow which constitutes the simplest spatially accelerated flow in which a sphere can be embedded is then studied. It is shown that owing to the viscous boundary condition on the body as well as to vorticity transport properties, the presence of the strain modifies deeply the distribution of vorticity around the sphere. This modification has spectacular consequences in the case of a rigid sphere because it influences strongly the conditions under which separation occurs as well as the characteristics of the separated region. Another very original feature of the axisymmetric straining flow lies in the vortex-stretching mechanism existing in this situation. In a converging flow this mechanism acts to reduce vorticity in the wake of the sphere. In contrast when the flow is divergent, vorticity produced at the surface of the sphere tends to grow indefinitely as it is transported downstream. It is shown that in the case where such a diverging flow extends to infinity a Kelvin–Helmholtz instability may occur in the wake.Computations of the hydrodynamic force show that the effects of the strain increase rapidly with the Reynolds number. At high Reynolds numbers the total drag is dramatically modified and the evaluation of the pressure contribution shows that the sphere undergoes an added mass force whose coefficient remains the same as in inviscid flow or in creeping flow, i.e. CM = ½, whatever the Reynolds number. Changes found in vorticity distribution around the rigid sphere also affect the viscous drag, which is markedly increased (resp. decreased) in converging (resp. diverging) flows at high Reynolds numbers.

A spherical particle straddling a fluid/gas interface in an axisymmetric straining flow

Numerical solutions, obtained via the boundary-integral technique, are used to consider the effect of a linear axisymmetric straining flow on the existence of steady-state configurations in which a neutrally buoyant spherical particle straddles a gas–liquid interface. The problem is directly applicable to predictions of the stability of particle capture in flotation processes, and is also of interest in the context of contact angle and surface tension measurements. A primary goal of the present study is a determination of the critical capillary number, Cac, beyond which an initially captured particle is pulled from the interface by the flow, and the dependence of Cac on the equilibrium contact angle θc. We also present equilibrium configurations for a wide range of contact angles and subcritical capillary numbers.

A viscous line vortex in an imposed axisymmetric straining flow of an upper-convected Maxwell fluid

An exact solution is found which describes a viscous line vortex in an imposed axisymmetric straining flow of an upper-convected Maxwell fluid. The vorticity distribution becomes more concentrated around the symmetry axis as the relaxation time is increased.

Particle motion in axisymmetric stagnation flow toward an interface

AbstractNumerical solutions for the creeping motion of a spherical particle in a linear axisymmetric straining flow normal to a deformable interface are presented for a range of viscosity ratios, capillary numbers, and Bond numbers. The parameter ranges investigated have applications in areas of flotation (small interface deformation) and material processing (large interface deformation). The accuracy of previous solutions for flotation problems which neglect interface deformation is considered, along with the magnitude and form of interface deformation “defects” which may appear in material processing applications involving fluids containing bubbles or small particles.

Small-amplitude perturbations of shape for a nearly spherical bubble in an inviscid straining flow (steady shapes and oscillatory motion)

The method of domain perturbations is used to study the problem of a nearly spherical bubble in an inviscid, axisymmetric straining flow. Steady-state shapes and axisymmetric oscillatory motions are considered. The steady-state solutions suggest the existence of a limit point at a critical Weber number, beyond which no solution exists on the steady-state solution branch which includes the spherical equilibrium state in the absence of flow (e.g. the critical value of 1.73 is estimated from the third-order solution). In addition, the first-order steady-state shape exhibits a maximum radius at θ = ⅙π which clearly indicates the barrel-like shape that was found earlier via numerical finite-deformation theories for higher Weber numbers. The oscillatory motion of a nearly spherical bubble is considered in two different ways. First, a small perturbation to a spherical base state is studied with the ad hoc assumption that the steady-state shape is spherical for the complete Weber-number range of interest. This analysis shows that the frequency of oscillation decreases as Weber number increases, and that a spherical bubble shape is unstable if Weber number is larger than 4.62. Secondly, the correct steady-state shape up to O(W) is included to obtain a rigorous asymptotic formula for the frequency change at small Weber number. This asymptotic analysis also shows that the frequency decreases as Weber number increases; for example, in the case of the principal mode (n = 2), ω2 = ω00(1−0.31W), where ω0 is the oscillation frequency of a bubble in a quiescent fluid.

A simultaneous concentration of a vortex and a magnetic field in magnetohydrodynamic flows

A three-dimensional magnetodydrodynamic flow with a combined effect of convection and diffusion is reduced to a two-dimensional problem by a variable transformation. The original field is composed of a unidirectional vorticity and an aligned magnetic field superimposed on an axisymmetric straining flow in an incompressible, viscous, electrically conducting fluid. By restricting consideration to the axisymmetric vorticity and magnetic fields, an exact solution is presented, which shows time evolution from general initial profiles. In a particular case of constant straining, the solution simultaneously approaches the state of a concentrated cylindrical vortex and a magnetic tube of Gaussian profile sharing a common symmetry axis.

Numerical solution of free-boundary problems in fluid mechanics. Part 3. Bubble deformation in an axisymmetric straining flow

We consider the deformation of a bubble in a uniaxial extensional flow for Reynolds numbers in the range 0.1 [les ] R [les ] 100. The computations show that the bubble bursts at a relatively early stage of deformation for R [ges ] O(10), never reaching the highly elongated shapes observed and predicted at lower Reynolds numbers. We also compute the deformation of the bubble under the assumption of potential flow, and conclude that the potential-flow solution provides a good approximation to the real flow in this case for R [ges ] O(100).

The evolution of slender inviscid drops in an axisymmetric straining flow

The evolution of the shape of a slender inviscid drop in an axisymmetric straining motion is studied at low Reynolds numbers. It is found that the shape equation can be solved by polynommals with time-dependent coefficients. A global stability result can be used to show simply that only one possible equilibrium is stable. It is further shown that if the slender drop starts with a long-wavelength waist then it cannot go to this stable equilibrium and must either extend indefinitely or burst. In the class of trinomial shapes, it is shown that the drop either bursts or goes to the stable equilibrium, depending on whether or not the initial shape has a waist.