Displacement of a two-dimensional immiscible droplet adhering to a wall in shear and pressure-driven flows

1999 ◽  
Vol 383 ◽  
pp. 29-54 ◽  
Author(s):  
ANTHONY D. SCHLEIZER ◽  
ROGER T. BONNECAZE
Keyword(s):  
Capillary Number ◽  
Contact Angles ◽  
Fluid Viscosity ◽  
Diffusive Transport ◽  
Two Dimensional ◽  
Parallel Plates ◽  
Contact Lines ◽  
Critical Capillary Number ◽  
Pressure Driven Flow ◽  
Pressure Driven

The dynamic behaviour and stability of a two-dimensional immiscible droplet subject to shear or pressure-driven flow between parallel plates is studied under conditions of negligible inertial and gravitational forces. The droplet is attached to the lower plate and forms two contact lines that are either fixed or mobile. The boundary-integral method is used to numerically determine the flow along and dynamics of the free surface. For surfactant-free interfaces with fixed contact lines, the deformation of the interface is determined for a range of capillary numbers, droplet to displacing fluid viscosity ratios, droplet sizes and flow type. It is shown that as the capillary number or viscosity ratio or size of the droplet increases, the deformation of the interface increases and above critical values of the capillary number no steady shape exists. For small droplets, and at low capillary numbers, shear and pressure-driven flows are shown to yield similar steady droplet shapes. The effect of surfactants is studied assuming a fixed amount of surfactant that is subject to convective–diffusive transport along the interface and no transport to or from the bulk fluids. Increasing the surface Péclet number, the ratio of convective to diffusive transport, leads to an accumulation of surfactant at the downstream end of the droplet and creates Marangoni stresses that immobilize the interface and reduce deformation. The no-slip boundary condition is then relaxed and an integral form of the Navier-slip model is used to examine the effects of allowing the droplet to slip along the solid surface in a pressure-driven flow. For contact angles less than or equal to 90°, a stable droplet spreads along the wall until a steady shape is reached, when the droplet translates across the wall at a constant velocity. The critical capillary number is larger for these droplets compared to those with pinned contact lines. For contact angles greater than 90°, the wetted area between a stable droplet and the wall decreases until a steady shape is reached. The critical capillary number for these droplets is less than that for pinned droplets. Above the critical capillary number the droplet completely detaches for a contact angle of 120°, or part of it is pinched off leaving behind a smaller attached droplet for contact angles less than or equal to 90°.

scholarly journals Boundary conditions for free surface inlet and outlet problems

2012 ◽  
Vol 708 ◽  
pp. 100-110 ◽  
Author(s):  
M. Taroni ◽  
C. J. W. Breward ◽  
P. D. Howell ◽  
J. M. Oliver
Keyword(s):  
Free Surface ◽  
Boundary Conditions ◽  
Capillary Number ◽  
A Priori ◽  
Contact Angles ◽  
Local Behaviour ◽  
Coating Flows ◽  
Film Forming ◽  
Critical Capillary Number ◽  
Field Behaviour

AbstractWe investigate and compare the boundary conditions that are to be applied to free-surface problems involving inlet and outlets of Newtonian fluid, typically found in coating processes. The flux of fluid is a priori known at an inlet, but unknown at an outlet, where it is governed by the local behaviour near the film-forming meniscus. In the limit of vanishing capillary number $\mathit{Ca}$ it is well known that the flux scales with ${\mathit{Ca}}^{2/ 3} $, but this classical result is non-uniform as the contact angle approaches $\lrm{\pi} $. By examining this limit we find a solution that is uniformly valid for all contact angles. Furthermore, by considering the far-field behaviour of the free surface we show that there exists a critical capillary number above which the problem at an inlet becomes over-determined. The implications of this result for the modelling of coating flows are discussed.

Splitting of a two-dimensional liquid plug at an airway bifurcation

2016 ◽  
Vol 793 ◽  
pp. 1-20 ◽  
Author(s):  
Benjamin L. Vaughan ◽  
James B. Grotberg
Keyword(s):  
Gravitational Field ◽  
Capillary Number ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Medical Treatments ◽  
Liquid Plug ◽  
Splitting Ratio ◽  
Critical Capillary Number ◽  
Airway Walls ◽  
Airway Bifurcation

Certain medical treatments involve the introduction of exogenous liquids in the lungs. These liquids can form plugs within the airways. The plugs propagate throughout the branching network in the lungs being forced by airflow. They leave a deposited film on the airway walls and split at bifurcations. Understanding the resulting distribution of liquid throughout the lungs is important for the effective administration of the prescribed treatments. In this paper, we investigate numerically the splitting of a liquid plug by a two-dimensional pulmonary bifurcation under the influence of a transverse gravitational field. The splitting is characterized by the splitting ratio, which is the ratio of volume of the liquid plug in the daughter channels and depends on the capillary number and the orientation of the bifurcation plane with respect to a three-dimensional gravitational field. It is observed that gravity induces asymmetry in the splitting, causing the splitting ratio to be reduced. This effect is mitigated as the capillary number is increased. It is also observed that there exists a critical capillary number where the plug will not split and will instead propagate entirely into the gravitationally favoured daughter channel. We also compute the wall stresses on the bifurcation walls and observe the locations where stresses and their gradients are the highest in magnitude.

scholarly journals Analytical Solution of Mixed Electroosmotic/Pressure Driven Flow of Viscoelastic Fluids between a Parallel Flat Plates Micro-Channel: The Maxwell Model Using the Oldroyd and Jaumann Time Derivatives

Micromachines ◽  
2020 ◽  
Vol 11 (11) ◽  
pp. 986
Author(s):  
Laura Casas ◽  
José A. Ortega ◽  
Aldo Gómez ◽  
Juan Escandón ◽  
René O. Vargas
Keyword(s):  
Flow Behavior ◽  
Viscoelastic Fluids ◽  
Volumetric Flow Rate ◽  
Maxwell Model ◽  
Parallel Plates ◽  
Flat Plates ◽  
Practical Applications ◽  
Low Viscosity ◽  
Pressure Driven Flow ◽  
Pressure Driven

In the present work, an analytical approximate solution of mixed electroosmotic/pressure driven flow of viscoelastic fluids between a parallel plates microchannel is reported. Inserting the Oldroyd, Jaumann, or both time derivatives into the Maxwell model, important differences in the velocity profiles were found. The presence of the shear and normal stresses is only close to the wall. This model can be used as a tool to understand the flow behavior of low viscosity fluids, as most of them experiment on translation, deformation and rotation of the flow. For practical applications, the volumetric flow rate can be controlled with two parameters, namely the gradient pressure and the electrokinetic parameter, once the fluid has been rheologically characterized.

Multiscale Simulations of Polymer Flow Between Two Parallel Plates

2021 ◽  
Vol 143 (4) ◽  
Author(s):  
Hong-Ji Yan ◽  
Zhen-Hua Wan ◽  
Feng-Hua Qin ◽  
De-Jun Sun
Keyword(s):  
Molecular Dynamics ◽  
Molecular Dynamics Simulations ◽  
Chain Length ◽  
Multiscale Method ◽  
Time Averaging ◽  
Parallel Plates ◽  
Large Spatial Scale ◽  
Dynamics Simulations ◽  
Pressure Driven Flow ◽  
Pressure Driven

Abstract A modified multiscale method without constitutive equation is proposed to investigate the microscopic information and macroscopic flow properties of polymeric fluid with the memory effect between parallel plates. In this method, the domain is entirely described by macromodel with isolated molecular dynamics simulations applied to calculate the necessary local stresses. The present method is first verified by the creep-recovery motion and pressure-driven flow, and all results are in excellent agreement with the available numerical solutions in literature. Then, the method is extended to simulate two typical problems of relatively large spatial scale in general beyond the capability of molecular dynamics simulations. In the planar Couette flow, the relationship between macroscopic properties and the time evolution of local molecular information is investigated in detail without long time averaging. All results that are consistent with nonequilibrium molecular dynamics and literature qualitatively or quantitatively demonstrate the validity of present multiscale method in simulating transient viscoelastic flows and the capacity to obtain the polymer information. In the pressure-driven flow, a general monotonically decreasing relationship between the maximum or average velocities and the polymer concentrations has been found regardless of the polymer chain length. Particularly, the reference concentration that satisfies a power law with chain length is closely related to the overlap concentration, and the reference velocity is exactly the relevant velocity of Newtonian fluid with corresponding zero shear rate viscosity.

Motion of a drop along the centreline of a capillary in a pressure-driven flow

2009 ◽  
Vol 640 ◽  
pp. 27-54 ◽  
Author(s):  
ETIENNE LAC ◽  
J. D. SHERWOOD
Keyword(s):  
Capillary Number ◽  
Streaming Potential ◽  
Companion Paper ◽  
Boundary Integral ◽  
Viscous Drops ◽  
Low Viscosity ◽  
Gravity Effects ◽  
Circular Capillary ◽  
Pressure Driven Flow ◽  
Pressure Driven

The deformation of a drop as it flows along the axis of a circular capillary in low Reynolds number pressure-driven flow is investigated numerically by means of boundary integral computations. If gravity effects are negligible, the drop motion is determined by three independent parameters: the size a of the undeformed drop relative to the radius R of the capillary, the viscosity ratio λ between the drop phase and the wetting phase and the capillary number Ca, which measures the relative importance of viscous and capillary forces. We investigate the drop behaviour in the parameter space (a/R, λ, Ca), at capillary numbers higher than those considered previously. If the fluid flow rate is maintained, the presence of the drop causes a change in the pressure difference between the ends of the capillary, and this too is investigated. Estimates for the drop deformation at high capillary number are based on a simple model for annular flow and, in most cases, agree well with full numerical results if λ ≥ 1/2, in which case the drop elongation increases without limit as Ca increases. If λ < 1/2, the drop elongates towards a limiting non-zero cylindrical radius. Low-viscosity drops (λ < 1) break up owing to a re-entrant jet at the rear, whereas a travelling capillary wave instability eventually develops on more viscous drops (λ > 1). A companion paper (Lac & Sherwood, J. Fluid Mech., doi:10.1017/S002211200999156X) uses these results in order to predict the change in electrical streaming potential caused by the presence of the drop when the capillary wall is charged.

scholarly journals Color-gradient lattice Boltzmann modeling of immiscible two-phase flows on partially wetting surfaces

2017 ◽  
Vol 232 (3) ◽  
pp. 416-430 ◽  
Author(s):  
Yuan Yu ◽  
Haihu Liu ◽  
Yonghao Zhang ◽  
Dong Liang
Keyword(s):  
Lattice Boltzmann ◽  
Capillary Number ◽  
Viscosity Ratio ◽  
High Viscosity ◽  
Contact Angles ◽  
Displacement Velocity ◽  
Two Dimensional ◽  
Change Rate ◽  
Finger Length ◽  
Color Gradient

A zero-interfacial-force condition is derived and implemented to improve the wetting boundary scheme for a lattice Boltzmann color-gradient model. This new wetting boundary scheme is validated by two static problems, i.e. a droplet resting on a flat surface and a cylindrical surface, and one dynamic problem, i.e. the capillary filling in a two-dimensional channel. In these simulations, we observe that non-physical mass transfer is suppressed and spurious velocities become smaller. Meanwhile, accurate results including dynamic contact line movement are achieved for a broad range of contact angles. The model is then applied to study the displacement of immiscible fluids in a two-dimensional channel. Both the displacement velocity and the change rate of finger length are found to exhibit a linear dependence on the contact angle at the viscosity ratio of unity. The displacement velocity decreases but the change rate of finger length increases with increasing capillary number, while the displacement velocity tends to be constant, i.e. two-thirds of the maximum inlet velocity, at high viscosity ratios or low capillary numbers. In contrast to the displacement velocity, the change rate of finger length is negligible at high viscosity ratios or low capillary numbers, where the finger length is in an equilibrium state, while the equilibrium finger length itself is smaller at a higher viscosity ratio or a lower capillary number.

The Study of Combined Electroosmotic and Pressure Driven Flow in Wavy Nanochannels

2010 ◽  
Author(s):  
Naga Siva Kumar Gunda ◽  
Suman Chakraborty ◽  
Sushanta Kumar Mitra
Keyword(s):  
Flow Pattern ◽  
Stokes Equations ◽  
Flow Direction ◽  
Debye Length ◽  
Surface Profile ◽  
Base Flow ◽  
Navier Stokes ◽  
Parallel Plates ◽  
Pressure Driven Flow ◽  
Pressure Driven

Solid surfaces of micro/nanochannels exhibit a certain degree of roughness that is incurred during fabrication and/or adsorption of macromolecules. The presence of such roughness changes the flow pattern in electroosmotic flows (EOF). The present study investigates the effect of surface waviness on combined EOF and pressure driven flow (PDF) of an electrolyte solution, in a nanochannel having charged walls. The surface profile of the top and bottom walls vary either in a varicose or in a sinuous mode. The problem is solved by using the Perturbation model, a modified linearized disturbance Navier-Stokes equations, by assuming two-dimensional combined EOF and PDF between two parallel plates as base flow. By discretizing the linearized disturbance equations using the Chebyshev collocation method in the wall normal direction and Fourier transformation in the flow direction, the perturbed velocity components are calculated. The effects of electric double layer (EDL) and amplitude of wavy surface on the flow pattern are studied. The effects of overlapped EDL are also studied as one of the limiting case. The formation of circulation regions is observed in the varicose mode channel when the EOF and PDF are flowing in the opposite direction. The decrease in the number of circulation regions is ob served for the decrease in the value of average half height of the channel to debye length ratio (κ). Serpentine or triangular type waviness in the streamline velocity is observed in sinuous mode type channel when the EOF and PDF are in opposite directions. The increase in the waviness of the streamline velocity is observed for decrease in the value of κ and increase in the amplitude a when both EOF and PDF are flowing in the same direction.

Thermal Analysis of Mixed Electroosmotic and Pressure Driven Flows in Two-Dimensional Straight Microchannels

Volume 4 ◽  
2004 ◽  
Author(s):  
Keisuke Horiuchi ◽  
Prashanta Dutta
Keyword(s):  
Heat Transfer ◽  
Thermal Analysis ◽  
Pressure Gradient ◽  
Two Dimensional ◽  
Transfer Coefficients ◽  
Nusselt Numbers ◽  
Micro Channels ◽  
Micro Flows ◽  
Pressure Driven Flow ◽  
Pressure Driven

Analytical solution for the temperature distributions, heat transfer coefficients, and Nusselt numbers of steady electroosmotic flows with an arbitrary pressure gradient are obtained for two-dimensional straight micro-channels. The thermal analysis considers interaction among inertial, diffusive and Joule heating terms in order to obtain the thermally developing behavior of mixed electroosmotic and pressure driven flows. In mixed flow cases, the governing equation for energy is not separable in general. Therefore, we introduced a new method that considers the extended Graetz problem. Heat transfer characteristics are presented for low Reynolds number micro-flows where the viscous and electric field terms are very dominant. Analytical results show that the heat transfer coefficient of mixed-electroosmotic and pressure driven flow is smaller than that of pure electroosmotic flow. For the parameter range studied here (Re&lt;0.7), the fully developed Nusselt number is independent of the thermal Peclet number and pressure gradient. Moreover, in mixed electroosmotic and pressure driven flows, the thermal entrance length increases with the imposed pressure gradient.

scholarly journals Semi-analytical solutions for two-dimensional elastic capsules in Stokes flow

2012 ◽  
Vol 468 (2146) ◽  
pp. 2915-2938 ◽  
Author(s):  
Michael Higley ◽  
Michael Siegel ◽  
Michael R. Booty
Keyword(s):  
Analytical Solutions ◽  
Stokes Flow ◽  
Finite Time ◽  
Capillary Number ◽  
Time Dependent ◽  
Two Dimensional ◽  
Elastic Interfaces ◽  
Critical Capillary Number ◽  
Field Velocity ◽  
Elastic Capsules

Elastic capsules occur in nature in the form of cells and vesicles and are manufactured for biomedical applications. They are widely modelled, but there are few analytical results. In this paper, complex variable techniques are used to derive semi-analytical solutions for the steady-state response and time-dependent evolution of two-dimensional elastic capsules with an inviscid interior in Stokes flow. This provides a complete picture of the steady response of initially circular capsules in linear strain and shear flows as a function of the capillary number Ca . The analysis is complemented by spectrally accurate numerical computations of the time-dependent evolution. An imposed nonlinear strain that models the far-field velocity in Taylor's four-roller mill is found to lead to cusped steady shapes at a critical capillary number Ca c for Hookean capsules. Numerical simulation of the time-dependent evolution for Ca > Ca c shows the development of finite-time cusp singularities. The dynamics immediately prior to cusp formation are asymptotically self-similar, and the similarity exponents are predicted analytically and confirmed numerically. This is compelling evidence of finite-time singularity formation in fluid flow with elastic interfaces.

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