Multiple finger propagation modes in Hele-Shaw channels of variable depth

Journal of Fluid Mechanics ◽

10.1017/jfm.2014.100 ◽

2014 ◽

Vol 746 ◽

pp. 123-164 ◽

Cited By ~ 15

Author(s):

Alice B. Thompson ◽

Anne Juel ◽

Andrew L. Hazel

Keyword(s):

Pressure Jump ◽

Channel Depth ◽

Bifurcation Structure ◽

Transverse Curvature ◽

Averaged Equations ◽

Curvature Term ◽

Air Liquid Interface ◽

Spatially Varying ◽

Averaged Model ◽

Mobility Coefficient

AbstractWe consider the propagation of an air finger into a wide fluid-filled channel with a spatially varying depth profile. Our aim is to understand the origin of the multiple coexisting families of both steady and oscillatory propagating fingers previously observed in experiments in axially uniform channels each containing a centred step-like occlusion. We find that a depth-averaged model can reproduce all the finger propagation modes observed experimentally. In addition, the model reveals new modes for symmetric finger propagation. The inclusion of a spatially variable channel depth in the depth-averaged equations leads to: (i) a variable mobility coefficient within the fluid domain due to variations in viscous resistance of the channel; and (ii) a variable transverse curvature term in the dynamic boundary condition that modifies the pressure jump over the air–liquid interface. We use our model to examine the roles of these two distinct effects and find that both contribute to the steady bifurcation structure, while the transverse curvature term is responsible for the distinctive oscillatory propagation modes.

Download Full-text

A Depth-Averaged Model for Electrokinetic Flows in a Thin Microchannel Geometry

Fluids Engineering ◽

10.1115/imece2004-61017 ◽

2004 ◽

Cited By ~ 1

Author(s):

Hao Lin ◽

Brian D. Storey ◽

Juan G. Santiago

Keyword(s):

Three Dimensional ◽

Momentum Equation ◽

Plane Motion ◽

Width Ratio ◽

Three Dimensional Modeling ◽

Depth Direction ◽

Averaged Equations ◽

Electrokinetic Flows ◽

Averaged Model ◽

Shallow Channel

We have developed a generalized electrokinetic model suitable for the study of microchannel flows with conductivity gradients and shallow channel depths. An asymptotic analysis was performed with channel depth-to-width ratio as the smallness parameter, and three dimensional transport equations are reduced to a set to depth-averaged equations governing flow dynamics in the streamwise-spanwise plane of a shallow channel. The momentum equation uses a Darcy-Brinkman-Forchheimer type formulation, and the convective-diffusive transport of the conductivity field in the depth direction manifests itself as a dispersion effect on in-plane motion. Accuracy of the model was assessed by comparing the numerical results with direct numerical simulations. These depth-averaged equations provide the accuracy of three-dimensional modeling with a convenient quasi-two-dimensional equation set applicable to a fairly wide class of microfluidic devices.

Download Full-text

Multiscale modelling of hydraulic conductivity in vuggy porous media

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2013.0383 ◽

2014 ◽

Vol 470 (2162) ◽

pp. 20130383 ◽

Cited By ~ 13

Author(s):

K. R. Daly ◽

T. Roose

Keyword(s):

Porous Media ◽

Soil Aggregates ◽

Three Dimensional ◽

Computation Time ◽

X Ray ◽

Averaged Equations ◽

Full Structure ◽

X Ray Computed ◽

Flow Pathways ◽

Averaged Model

Flow in both saturated and non-saturated vuggy porous media, i.e. soil, is inherently multiscale. The complex microporous structure of the soil aggregates and the wider vugs provides a multitude of flow pathways and has received significant attention from the X-ray computed tomography (CT) community with a constant drive to image at higher resolution. Using multiscale homogenization, we derive averaged equations to study the effects of the microscale structure on the macroscopic flow. The averaged model captures the underlying geometry through a series of cell problems and is verified through direct comparison to numerical simulations of the full structure. These methods offer significant reductions in computation time and allow us to perform three-dimensional calculations with complex geometries on a desktop PC. The results show that the surface roughness of the aggregate has a significantly greater effect on the flow than the microstructure within the aggregate. Hence, this is the region in which the resolution of X-ray CT for image-based modelling has the greatest impact.

Download Full-text

Sensitivity of Saffman–Taylor fingers to channel-depth perturbations

Journal of Fluid Mechanics ◽

10.1017/jfm.2016.131 ◽

2016 ◽

Vol 794 ◽

pp. 343-368 ◽

Cited By ~ 13

Author(s):

Andrés Franco-Gómez ◽

Alice B. Thompson ◽

Andrew L. Hazel ◽

Anne Juel

Keyword(s):

Aspect Ratio ◽

Experimental System ◽

Quantitative Agreement ◽

Relative Depth ◽

Channel Depth ◽

Aspect Ratios ◽

A Value ◽

Unstable Solutions ◽

Extreme Sensitivity ◽

Averaged Model

We examine the sensitivity of Saffman–Taylor fingers to controlled variations in channel depth by investigating the effects of centred, rectangular occlusions in Hele-Shaw channels. For large occlusions, the geometry is known to support symmetric, asymmetric and oscillatory propagation states when air displaces a more viscous fluid from within the channel. A previously developed depth-averaged model is found to be in quantitative agreement with laboratory experiments once the aspect ratio (width/height) of the tube’s cross-section reaches a value of 40. We find that the multiplicity of solutions at finite occlusion heights arises through interactions of the single stable and multiple unstable solutions already present in the absence of the occlusion: the classic Saffman–Taylor viscous fingering problem. The sequence of interactions that occurs with increasing occlusion height is the same for all aspect ratios investigated, but the occlusion height required for each interaction decreases with increasing aspect ratio. Thus, the system becomes more sensitive as the aspect ratio increases in the sense that multiple solutions are provoked for smaller relative depth changes. We estimate that the required depth changes become of the same order as the typical roughnesses of the experimental system ($1~{\rm\mu}\text{m}$) for aspect ratios beyond 155, which we conjecture underlies the extreme sensitivity of experiments conducted in such Hele-Shaw channels.

Download Full-text

A Finite Difference Algorithm Applied to the Averaged Equations of the Heat Conduction Issue in Biperiodic Composites—Robin Boundary Conditions

Materials ◽

10.3390/ma14216329 ◽

2021 ◽

Vol 14 (21) ◽

pp. 6329

Author(s):

Ewelina Kubacka ◽

Piotr Ostrowski

Keyword(s):

Heat Conduction ◽

Finite Difference ◽

Boundary Conditions ◽

Robin Boundary Conditions ◽

Temperature Field Distribution ◽

Averaged Equations ◽

Tolerance Modelling ◽

Model Equations ◽

Robin Boundary ◽

Averaged Model

This note deals with the heat conduction issue in biperiodic composites made of two different materials. To consider such a nonuniform structure, the equations describing the behavior of the composite under thermal (Robin) boundary conditions were averaged by using tolerance modelling. In this note, the process of creating an algorithm that uses the finite difference method to deal with averaged model equations is shown. This algorithm can be used to solve these equations and find out the temperature field distribution of a biperiodic composite.

Download Full-text