Deformation of a two-dimensional drop at non-zero Reynolds number in time-periodic extensional flows: numerical simulation

2001 ◽  
Vol 436 ◽  
pp. 177-206 ◽  
Author(s):  
KAUSIK SARKAR ◽  
WILLIAM R. SCHOWALTER
Keyword(s):  
Reynolds Number ◽  
Interfacial Tension ◽  
Vortex Flow ◽  
Extensional Flow ◽  
Reynolds Numbers ◽  
Time Dependent ◽  
Drop Deformation ◽  
Two Dimensional ◽  
Extensional Flows ◽  
Potential Vortex

The shape of a two-dimensional viscous drop deforming in several time-dependent flow fields, including that due to a potential vortex, has been studied. Vortex flow was approximated by linearizing the induced velocity field at the drop centre, giving rise to an extensional flow with rotating axes of stretching. A generalization of the potential vortex, a flow we have called rotating extensional flow, occurs when the frequency of revolution of the flow is varied independently of the shear rate. Drops subjected to this forcing flow exhibit an interesting resonance phenomenon. Finally we have studied drop deformation in an oscillatory extensional flow.Calculations were performed at small but non-zero Reynolds numbers using an ADI front-tracking/finite difference method. We investigate the effects of interfacial tension, periodicity, viscosity ratio, and Reynolds number on the drop dynamics. The simulation reveals interesting behaviour for steady stretching flows, as well as time-dependent flows. For a steady extensional flow, the drop deformation is found to be non-monotonic with time in its approach to an equilibrium value. At sufficiently high Reynolds numbers, the drop experiences multiple growth–collapse cycles, with possible axes reversal, before reaching a final shape. For a vortex flow, the long-time deformation reaches a steady value, and the drop attains a revolving steady elliptic shape. For rotating extensional flows as well as oscillatory extensional flows, the maximum value of deformation displays resonance with variation in parameters, first increasing and then decreasing with increasing interfacial tension or forcing frequency. A simple ODE model with proper forcing is offered to explain the observed phenomena.

scholarly journals Interfacial Tension Measurements in Microfluidic Quasi-Static Extensional Flows

Micromachines ◽  
2021 ◽  
Vol 12 (3) ◽  
pp. 272
Author(s):  
Doojin Lee ◽  
Amy Q. Shen
Keyword(s):  
Interfacial Tension ◽  
Microfluidic Device ◽  
Extensional Flow ◽  
Immiscible Fluids ◽  
Droplet Microfluidics ◽  
Channel Height ◽  
Deformation Model ◽  
Droplet Deformation ◽  
Extensional Flows ◽  
2D Model

Droplet microfluidics provides a versatile tool for measuring interfacial tensions between two immiscible fluids owing to its abilities of fast response, enhanced throughput, portability and easy manipulations of fluid compositions, comparing to conventional techniques. Purely homogeneous extension in the microfluidic device is desirable to measure the interfacial tension because the flow field enables symmetric droplet deformation along the outflow direction. To do so, we designed a microfluidic device consisting of a droplet production region to first generate emulsion droplets at a flow-focusing area. The droplets are then trapped at a stagnation point in the cross junction area, subsequently being stretched along the outflow direction under the extensional flow. These droplets in the device are either confined or unconfined in the channel walls depending on the channel height, which yields different droplet deformations. To calculate the interfacial tension for confined and unconfined droplet cases, quasi-static 2D Darcy approximation model and quasi-static 3D small deformation model are used. For the confined droplet case under the extensional flow, an effective viscosity of the two immiscible fluids, accounting for the viscosity ratio of continuous and dispersed phases, captures the droplet deformation well. However, the 2D model is limited to the case where the droplet is confined in the channel walls and deforms two-dimensionally. For the unconfined droplet case, the 3D model provides more robust estimates than the 2D model. We demonstrate that both 2D and 3D models provide good interfacial tension measurements under quasi-static extensional flows in comparison with the conventional pendant drop method.

Tertiary solutions and their stability in rotating plane Couette flow

1998 ◽  
Vol 358 ◽  
pp. 357-378 ◽  
Author(s):  
M. NAGATA
Keyword(s):  
Reynolds Number ◽  
Couette Flow ◽  
Stability Boundary ◽  
Streamwise Vortex ◽  
Reynolds Numbers ◽  
Time Dependent ◽  
Plane Couette Flow ◽  
Streamwise Direction ◽  
The Stability ◽  
Oscillatory Instabilities

The stability of nonlinear tertiary solutions in rotating plane Couette flow is examined numerically. It is found that the tertiary flows, which bifurcate from two-dimensional streamwise vortex flows, are stable within a certain range of the rotation rate when the Reynolds number is relatively small. The stability boundary is determined by perturbations which are subharmonic in the streamwise direction. As the Reynolds number is increased, the rotation range for the stable tertiary motions is destroyed gradually by oscillatory instabilities. We expect that the tertiary flow is overtaken by time-dependent motions for large Reynolds numbers. The results are compared with the recent experimental observation by Tillmark & Alfredsson (1996).

Bifurcation Characteristics of Flows in Rectangular Sudden Expansion Channels

2005 ◽  
Author(s):  
Francine Battaglia ◽  
George Papadopoulos
Keyword(s):  
Reynolds Number ◽  
Laminar Flow ◽  
Critical Reynolds Number ◽  
Three Dimensional ◽  
Reynolds Numbers ◽  
Expansion Ratio ◽  
Sudden Expansion ◽  
Two Dimensional ◽  
Effective Expansion ◽  
Three Dimensional Simulations

The effect of three-dimensionality on low Reynolds number flows past a symmetric sudden expansion in a channel was investigated. The geometric expansion ratio of in the current study was 2:1 and the aspect ratio was 6:1. Both experimental velocity measurements and two- and three-dimensional simulations for the flow along the centerplane of the rectangular duct are presented for Reynolds numbers in the range of 150 to 600. Comparison of the two-dimensional simulations with the experiments revealed that the simulations fail to capture completely the total expansion effect on the flow, which couples both geometric and hydrodynamic effects. To properly do so requires the definition of an effective expansion ratio, which is the ratio of the downstream and upstream hydraulic diameters and is therefore a function of both the expansion and aspect ratios. When the two-dimensional geometry was consistent with the effective expansion ratio, the new results agreed well with the three-dimensional simulations and the experiments. Furthermore, in the range of Reynolds numbers investigated, the laminar flow through the expansion underwent a symmetry-breaking bifurcation. The critical Reynolds number evaluated from the experiments and the simulations was compared to other values reported in the literature. Overall, side-wall proximity was found to enhance flow stability, helping to sustain laminar flow symmetry to higher Reynolds numbers in comparison to nominally two-dimensional double-expansion geometries. Lastly, and most importantly, when the logarithm of the critical Reynolds number from all these studies was plotted against the reciprocal of the effective expansion ratio, a linear trend emerged that uniquely captured the bifurcation dynamics of all symmetric double-sided planar expansions.

Rheological Aspects of Drops Deforming in Finite Reynolds Number Oscillatory Extensional Flows

2004 ◽  
Author(s):  
Xiaoyi Li ◽  
Kausik Sarkar
Keyword(s):  
Numerical Simulation ◽  
Reynolds Number ◽  
Extensional Flow ◽  
Three Dimensional ◽  
Industrial Applications ◽  
Drop Deformation ◽  
Interface Morphology ◽  
Drop Dynamics ◽  
Variation Of Parameters ◽  
Broad Implication

The evolving morphology of droplets in a flowing emulsion determines its rheological properties. A two-way interaction between drops and the flow governs the rheological stresses arising from drop deformation. In this paper, the rheology of droplet emulsions under oscillatory extensional flow is investigated using direct numerical simulation (DNS). The deformation of a three dimensional drop is simulated. The rheological responses are related with the interface morphology using Bachelor’s stress formulation [6]. Detailed investigation of the variation of parameters such as interfacial tension, flow frequency and inertia displayed complex non-Newtonian response of the emulsion that will have broad implication in industrial applications. The results are explained and discussed with a simple model for the drop dynamics.

A Stability Investigation of Two-Dimensional Surface Waves on Evaporating, Isothermal or Condensing Liquid Films: Part II — A Universal Linear Stability Formulation

2004 ◽  
Author(s):  
Xuemin Ye ◽  
Weiping Yan ◽  
Chunxi Li
Keyword(s):  
Reynolds Number ◽  
Surface Waves ◽  
Isothermal Condition ◽  
Reynolds Numbers ◽  
Liquid Films ◽  
Neutral Stability ◽  
Two Dimensional ◽  
Lower Reynolds Number ◽  
Liquid Property ◽  
Dimensional Surface

When liquid film is under evaporating or condensing conditions, the flow stability is clearly different to that under isothermal condition due to thermal non-equilibrium effect at interface, especially under lower Reynolds number. The universal linear temporal and spatial stability formulations of the two-dimensional surface waves on evaporating or isothermal or condensing liquid films are established in present paper with the collocation method based on the boundary layer theory and complete boundary conditions. The models include the effects of Reynolds number, thermocapillarity, inclination angle, liquid property, evaporation, isothermal or condensation. The effects of above factors are investigated with the neutral stability curves at different Reynolds numbers, and stabilities characteristics are fully indicated in theory for evaporating or condensing films.

Three-dimensional instabilities in the boundary-layer flow over a long rectangular plate

2011 ◽  
Vol 681 ◽  
pp. 411-433 ◽  
Author(s):  
HEMANT K. CHAURASIA ◽  
MARK C. THOMPSON
Keyword(s):  
Reynolds Number ◽  
Shear Layer ◽  
Three Dimensional ◽  
Reynolds Numbers ◽  
Vortical Flow ◽  
Forced Flow ◽  
Recirculation Region ◽  
Two Dimensional ◽  
Optimal Perturbation ◽  
Dimensional Instability

A detailed numerical study of the separating and reattaching flow over a square leading-edge plate is presented, examining the instability modes governing transition from two- to three-dimensional flow. Under the influence of background noise, experiments show that the transition scenario typically is incompletely described by either global stability analysis or the transient growth of dominant optimal perturbation modes. Instead two-dimensional transition effectively can be triggered by the convective Kelvin–Helmholtz (KH) shear-layer instability; although it may be possible that this could be described alternatively in terms of higher-order optimal perturbation modes. At least in some experiments, observed transition occurs by either: (i) KH vortices shedding downstream directly and then almost immediately undergoing three-dimensional transition or (ii) at higher Reynolds numbers, larger vortical structures are shed that are also three-dimensionally unstable. These two paths lead to distinctly different three-dimensional arrangements of vortical flow structures. This paper focuses on the mechanisms underlying these three-dimensional transitions. Floquet analysis of weakly periodically forced flow, mimicking the observed two-dimensional quasi-periodic base flow, indicates that the two-dimensional vortex rollers shed from the recirculation region become globally three-dimensionally unstable at a Reynolds number of approximately 380. This transition Reynolds number and the predicted wavelength and flow symmetries match well with those of the experiments. The instability appears to be elliptical in nature with the perturbation field mainly restricted to the cores of the shed rollers and showing the spatial vorticity distribution expected for that instability type. Indeed an estimate of the theoretical predicted wavelength is also a good match to the prediction from Floquet analysis and theoretical estimates indicate the growth rate is positive. Fully three-dimensional simulations are also undertaken to explore the nonlinear development of the three-dimensional instability. These show the development of the characteristic upright hairpins observed in the experimental dye visualisations. The three-dimensional instability that manifests at lower Reynolds numbers is shown to be consistent with an elliptic instability of the KH shear-layer vortices in both symmetry and spanwise wavelength.

Two-Dimensional Unsteady Viscous Flows

2017 ◽  
Author(s):  
Jian-Jun Shu
Keyword(s):  
Circular Cylinder ◽  
Flow Field ◽  
Hydrodynamic Force ◽  
Viscous Flows ◽  
Reynolds Numbers ◽  
Fundamental Solutions ◽  
Time Dependent ◽  
Two Dimensional ◽  
Low Reynolds Numbers ◽  
Rotational Motions

A number of new closed-form fundamental solutions for the two-dimensional generalized unsteady Oseen and Stokes flows associated with arbitrary time-dependent translational and rotational motions have been developed. As an example of application, the hydrodynamic force acting on a circular cylinder translating in an unsteady flow field at low Reynolds numbers is calculated using the new generalized fundamental solutions.

Effects of the Divergence Angle on the Onset of Asymmetric Flow Through a Planar Diffuser

2009 ◽  
Author(s):  
Majid Nabavi ◽  
Luc Mongeau
Keyword(s):  
Reynolds Number ◽  
Flow Regime ◽  
Critical Reynolds Number ◽  
Reynolds Numbers ◽  
Two Dimensional ◽  
Asymmetric Flow ◽  
Symmetric Flow ◽  
Turbulent Flow Regime ◽  
Flow Through ◽  
Gradual Expansion

In this study, two-dimensional laminar incompressible and turbulent compressible flow through the planar diffuser (gradual expansion) for different divergence half angles of the diffuser (θ), and different Reynolds numbers (Re) was numerically studied. The effects of θ on the critical Reynolds number at which the onset of asymmetric flow is observed, were investigated. In the laminar flow regime, it was observed that for every values of θ, there is a critical Re beyond which the flow is asymmetric. However, in the turbulent flow regime, for θ ≥ 20°, even at low Reynolds number the flow is asymmetric. Only for θ ≤ 10°, symmetric flow was observed below a critical Re.

Three-dimensional dynamics and transition to turbulence in the wake of bluff objects

1992 ◽  
Vol 238 ◽  
pp. 1-30 ◽  
Author(s):  
George Em Karniadakis ◽  
George S. Triantafyllou
Keyword(s):  
Reynolds Number ◽  
Three Dimensional ◽  
Period Doubling ◽  
Reynolds Numbers ◽  
Transition To Turbulence ◽  
Two Dimensional ◽  
Near Wake ◽  
Vortex Filaments ◽  
Average Flow ◽  
Time Average

The wakes of bluff objects and in particular of circular cylinders are known to undergo a ‘fast’ transition, from a laminar two-dimensional state at Reynolds number 200 to a turbulent state at Reynolds number 400. The process has been documented in several experimental investigations, but the underlying physical mechanisms have remained largely unknown so far. In this paper, the transition process is investigated numerically, through direct simulation of the Navier—Stokes equations at representative Reynolds numbers, up to 500. A high-order time-accurate, mixed spectral/spectral element technique is used. It is shown that the wake first becomes three-dimensional, as a result of a secondary instability of the two-dimensional vortex street. This secondary instability appears at a Reynolds number close to 200. For slightly supercritical Reynolds numbers, a harmonic state develops, in which the flow oscillates at its fundamental frequency (Strouhal number) around a spanwise modulated time-average flow. In the near wake the modulation wavelength of the time-average flow is half of the spanwise wavelength of the perturbation flow, consistently with linear instability theory. The vortex filaments have a spanwise wavy shape in the near wake, and form rib-like structures further downstream. At higher Reynolds numbers the three-dimensional flow oscillation undergoes a period-doubling bifurcation, in which the flow alternates between two different states. Phase-space analysis of the flow shows that the basic limit cycle has branched into two connected limit cycles. In physical space the period doubling appears as the shedding of two distinct types of vortex filaments.Further increases of the Reynolds number result in a cascade of period-doubling bifurcations, which create a chaotic state in the flow at a Reynolds number of about 500. The flow is characterized by broadband power spectra, and the appearance of intermittent phenomena. It is concluded that the wake undergoes transition to turbulence following the period-doubling route.

Computing internal viscous flow problems for the circle by integral methods

1977 ◽  
Vol 79 (3) ◽  
pp. 609-624 ◽  
Author(s):  
R. D. Mills
Keyword(s):  
Reynolds Number ◽  
Circular Cylinder ◽  
Integral Representation ◽  
Analytical Solutions ◽  
Reynolds Numbers ◽  
Cylinder Wall ◽  
Two Dimensional ◽  
Discontinuous Surface ◽  
Flow Problems ◽  
Integral Methods

Steady two-dimensional viscous motion within a circular cylinder generated by (a) the rotation of part of the cylinder wall and (b) fluid entering and leaving through slots in the wall is considered. Studied in particular are moving-surface problems with continuous and discontinuous surface speeds, simple inflow–outflow problems and the impinging-jet problem within a circle. The analytical solutions at zero Reynolds number are given for the last two types of problem. Numerical results are obtained at various Reynolds numbers from the integral representation of the solution. At zero Reynolds number this approach involves a quadrature around the circumference of the cylinder. At other Reynolds numbers it involves an iterative–integral technique based on the use of the ‘clamped-plate’ biharmonic Green's function.

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