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## Flows in complex geometries

In the fluid flows in networks, complex dynamics can always result from the nonlinear relation between the local pressure and flow rate, for example when the flow propagates in compliant conduits (Hazel & Heil, 2003) or in the presence of immiscible interfaces (Howell et al., 2000). In such situations, the local flow-rate fluctuations lead to instantaneous, global reequilibration of the pressure, thus producing long-range couplings in the flow. This can play a dominant role for flows in porous or biological media, where the evolution of multiphase flows occurs through a competition between viscous and capillary effects, both of which are influenced by the local details of the geometry. Generalizing these studies can yield understandings of many mysterious phenomena in nature: for example, the water transport in plants (West et al., 1999; McCulloh et al., 2003), the physiological organization of animals’ vascular systems (Murray, 1926) and the exploration of optimal transport networks (Katifori et al., 2010; Corson, 2010). Furthermore, the large scale rearrangement in complex geometries, involving the modeling of fluid transport, social interactions and disease propagation in networks, has also been a challenging issue of long standing interest with many related applications (Strogatz, 2001; Newman, 2003).

### Preliminary understanding of the physics

Many efforts have been made to study the contact of two kinds of liquids and a solid surface in the previous decades (Dussan, 1979). However, the difficulties arise due to the interactions between the immiscible interfaces and the complex geometry, which can significantly affect the statics and the dynamics of the liquid flow. Some fundamental results are found in previous studies about the moving contact line on horizontal (Tanner, 1979) or rough surfaces (Jansons, 1985) and in capillaries (Hoffman, 1975). When considering the flow in a confined geometry, the contact dynamics as well as the geometry constraint become more important (Ajaev & Homsy, 2006). Bretherton (1961) made great contribution to the investigation about the motion of a long bubble in tubes. The nonlinear relation between the driving pressure P and bubble’s capillary number Ca was found to be P ∝ Ca2/3 when the capillary number is very small (Ca < 5 × 10−3). Predictions of the bubble profile in the capillary tube could be made by using this model. Later on, Wong et al. (1995) studied the bubble motions in different polygonal capillaries and the pressure drop across the bubble was found to scale as Ca2/3. A similar problem was investigated by Hodges et al. (2004) who considered a viscous drop in a cylindrical tube and the liquid viscosity was found to have an effect on the film thickness around the drop. Recently, some characteristics of the drop motion in milli- or micro-sized tubes have been found (Stone et al., 2004; Fujioka & Grotberg, 2004; Fuerstman et al., 2007). The pressure jump in the tube due to the presence of drops was studied numerically (Fujioka & Grotberg, 2004) and experimentally (Aussillous & Quéré, 2000; Pehlivan et al., 2006). A nonlinear relationship between pressure difference and flow rate of the drop was introduced by surface tension, through the addition of Laplace pressure (Bico & Quéré, 2001; Ody et al., 2007). For a single plug of wetting liquid in a straight rectangular microchannel, the relation between the driving pressure and the plug’s capillary number was obtained by taking into account the changes of the interface curvatures in both Plane A and B, as shown in figure 1.9. While the plug was advancing, the front interface, illustrated by solid curves deformed and exhibited a non-zero contact angle with the channel wall although the liquid was wetting. Meanwhile, a thin film of liquid was deposited on the walls of the channel behind the plug, which changed the curvature of the rear interface illustrated by dashed curves. Based on this approach, the following formula was found to be valid for a long plug of wetting liquid moving at low capillary numbers in a rectangular microchannel whose width was much larger than the height: Pdr = FCa + GCa2/3, in which Pdr was the driving pressure and F,G were functions of the channel geometry, the physical and geometrical properties of the plug.

#### Nonlinear relation between pressure and flow rate

The presence of one liquid plug in a straight channel with rectangular cross-section introduces a resistance to flow through its viscosity and surface tension. The relation between the viscocapillary pressure Pvc and the plug’s capillary number Ca= ηU/γ can be written as (Ody et al., 2007): Pvc = F(w)LCa + G(w)Ca2/3, (2.1) in which η and γ are the viscosity and surface tension of the liquid, U is the plug velocity, L is the plug length, F(w) and G(w) are expressed as follows: where h and w are the height and the width of the channel. β is a nondimensional coefficient obtained from Bretherton’s and Tanner’s laws (Hoffman, 1975; Bico & Quéré, 2001). For the same liquid, F(w) and G(w) are functions of the channel geometry. Equation (2.1) is valid for a long plug that moves in a rectangular channel with large aspect ratio w/h. The linear part represents viscous effects, which change with the channel geometry and plug length. Meanwhile, the capillary effects introduced by the air-liquid interfaces are expressed by the nonlinear part, which also changes with the channel geometry.

**Flow rate evolution under constant pressure**

When a single plug is pushed at a constant pressure, the velocity of its daughters in generation i, Ui, varies with the generation number, which results from the variation of F(w) and G(w) in (2.1) due to channel sizes and plug lengths. For every single daughter plug in a given generation i, (2.1) can be rewritten as: Pdr = F(wi)LiCai + G(wi)Cai 2/3. (2.4)

Here Pdr is the constant driving pressure, wi is the channel width in the ith generation and Li is the length of the daughter plug with the assumption of equal division at every bifurcation. Cai is thus calculated based on the velocity of a daughter plug Ui, which is shown in figure 2.6. All the parameters related to the calculation are given in table 2.1. The velocities in the first

**Table of contents :**

**1 Introduction **

1.1 Motivation of the study

1.2 Two-phase flows in networks

1.2.1 Liquid plug in the pulmonary airway

1.2.2 Immiscible displacement in porous media

1.2.3 Microfluidic approach

1.2.4 Flows in complex geometries

1.3 Preliminary understanding of the physics

1.4 Outline of the thesis

**2 Models of the plug motion **

2.1 Description of the network

2.2 Plug in the straight channel

2.2.1 Nonlinear relation between pressure and flow rate

2.2.2 Flow rate evolution under constant pressure

2.2.3 Pressure evolution under constant flow rate

2.2.4 Applications

2.3 Passage through a bifurcation

2.3.1 Position-dependent capillary pressure difference

2.3.2 Variation of threshold pressures in the network

**3 Experimental setups **

3.1 Network concerned

3.1.1 Microfabrication

3.1.2 Network geometry

3.2 Setups

3.3 Preliminary studies

3.3.1 Measurements

3.3.2 Interface tracking

**4 One plug injected into the network **

4.1 Experimental observations

4.2 Flow evolutions under different driving conditions

4.2.1 Constant pressure driving

4.2.2 Constant flow rate driving

4.2.3 Flow patterns in the narrowing network

4.3 Flow organization in different networks

4.3.1 Network with narrowing channels

4.3.2 Network with widening channels

4.3.3 Global organization of the flow

4.4 Summary

**5 A train of plugs in the network **

5.1 Empirical resistance to flow associated with one plug

5.2 Two plugs injected into the network

5.2.1 Flow rate dependence on the initial distance between plugs

5.2.2 Applicability of the empirical relation

5.3 A train of plugs injected

5.4 Summary and discussion

**6 Airway reopening through cascades **

6.1 Introduction

6.2 Experimental setups

6.3 Cascades of plug ruptures

6.3.1 Cascades in a straight channel

6.3.2 Observations in a network

6.4 Summary

**7 Conclusions **

**Bibliography**