Droplet production in 2D geometries or 2D geometry with discontinuity 

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Increased permeability and reduced hydrodynamic dispersion

The presence of slip eect allows generation of enhanced permeability in porous media. In a pressure drop ow, when there is slip, the velocity of uid is increased by a factor 1+6b=h compared with no slip, with b is the slip length, and h the channel characteristic length [12]. Petroleum discovery can take advantage of the slip phenomenon and inject uids which slip on rock surfaces into the underground, so that with a constant pressure application, uids can go further and faster. The slip eciency can be estimated as b=h, the eect of increased permeability becomes signicant in uid path with sub-micrometric dimension.
Besides, presence of slip reduces dispersion of components (molecules, particles) in the uid. High dispersion close to interface is due to non-slip boundary condition, where the relativedierence of velocity V = (vmax 􀀀 vmin)=vmax is maximized. However, this is reduced to V = (1 + 4b=h)􀀀1 for slipping channel [12].

Enhanced interfacial transport

Flow can be driven in a channel not only by pressure gradient, some alternative methods can generate ow at the interface of uid and solid, this eect realises the interfacial transportation. With the presence of hydrodynamic slip, transportation can sometimes be largely facilitated. Taking the example of electro-osmosis [69], in which case the ow is generated by an electric eld. Due to charged solid surface in contact with uid, particles and molecules with charges in the uid are reorganized geometrically, and form a electrical double layer near the surface. This layer is characterized by a length scale called Debye length D, outside this layer the environment is electrically neutral, while inside there is an electrical potential. The uid motion induced by stream-wise electrical eld is restricted into the double layer. The velocity of a charged particle is determined by the balance between electrical force fe = q E = V0 D E, and the viscous drag in the uid f = veo=(D), being the dielectric permittivity of the uid, V0 the potential of solid surface, the viscosity. With the presence of hydrodynamic slip, the balance transforms to: veo D + b 􀀀 V0 DE.

Slippage of polymers

The study of polymer ow along solid surface has signicance in industrial production. For instance in the polymer extrusion procedure, ow instabilities have been observed at given shear stress range. Kalika and Denn [71] studied capillary extrusion of polyethylenes, and reported oscillation on throughput of the extruder at certain range of pressure. The oscillation is attributed to the imperfections of the surface, which induces the stick-slip phenomenon of the polymer ow. Vinogradov et al. [137] studied ow of polybutadiene melts and reported that above a critical shear stress the throughput of extruder jumps to a higher value, this phenomenon of spurt is a macroscopic interpretation of polymer slippage.
The theoretical expectations and experiments on the polymer-surface interaction and its in uence on polymer slippage are reviewed by Leger et al. [79].

Formation of polymer layer on solid surface

The discussion of polymer slippage is based on the formation of a polymer layer on the surface. The layer can be formed by chemical methods [79], by grafting the end of polymer chains with high density onto chemically treated surfaces. In the case of an attractive surface, no chemical method is needed to produce irreversible adhesion. Taking the example of PDMS melt on silica surface, polymer chains form hydrogen bonds between silanol sites of silica surface and the oxygen atoms of the backbones. The adsorption is irreversible, and allows the thickness of formed layer h N1=27=8 0 , where N the index of polymerization, and 0 the polymer volume fraction. To the contrary, the reversible adsorption is based on thermodynamic equilibrium near the surface, where polymer chains exchange permanently between the solid surface and the bulk [38].
Polymer molecules adsorbed on solid surface increase ow resistance, thus reduce ow rate in the capillary. Reported works utilized this fact to measure the eective hydrodynamic thickness eH of adsorbed layer, according to equation: eH=R = 1􀀀(Qa=Q)1=4, where R the radius of the capillary, Qa and Q the ow rate after and before the adsorption of polymer respectively [30]. De Gennes [38] constructed equilibrium concentration prole of polymers near the wall, and characterised the wall by the \free energy of sticking », which is negative for adsorption and positive for desorption. He concluded that the eective hydrodynamic thickness depends on the largest loop of the adsorbed chain, which scales like the gyration raduis of a free polymer chain in solvent.
Previous works suggested that the presence of polymer chain on the solid induces surface roughness and suppresses slip [106, 153]. This fact expressed on the velocity prole of polymer solution close to the wall as a negative slip length [14]. Since then, no detailed work has been done in order to determine the exact negative slip length of polymer solution on the wall with satisfactory precision.

Theoretical and experimental study of polymer slippage on solid surface

The widely accepted theory about polymer slippage on an interface is that developed by de Gennes and Brochard [20], they assumed a situation that shear induced polymer melt owing above a solid surface with grafted polymer chains. The density of grafting is small so as to avoid interaction between the grafted chains. Without applying a shear stress to the owing polymer, the conformation of the tethered chains is dominated by entropy, and exist in shape of mushrooms. Once been applied a shear stress , the state of entanglement and disentanglement between ow polymer and grafted chains induces dierent slip length and surface velocity. Three regimes have been distinguished. At shear stress smaller thana critical shear stress , the owing system is under entangled regime. The structure near the surface can be approximated as a cylinder with length L and diameter D, which consists of the elongated tethered chain with a number of owing chains entangled with it. This situation can be described by equating the elastic force Fel = 3kT R2 0L with viscous forceFvis = V (R2 0 + L2)1=2, with R0 the entropy dominated coil size of grafted chain, and V the velocity on the surface (i.e. the slip velocity). The elongation L increases with surface velocity, until the state of entanglement is not stable any more, this critical surface velocity is V = kT Za2 , where Z the index of polymerization of the tethered chain. At > , a much weaker friction is exerted between owing chains and tethered ones. The friction can be approximated as a sum of contribution of single monomers according to Rouse model. By equating this friction with elastic force, the smaller elongation LR is obtained with LR = L V V1 , where V1 = V 1N1=2 e , with 1 the viscosity of liquid of monomers, Ne the number of entanglement of tethered chains. In this marginal regime, owing chainsdisentangle and re-entangle with tethered chains, resulting in a jump of surface velocity at critical shear stress. In the regime of strong shear ow, the shear stress grows even higher, the owing chains disentangle with tethered chains, the slip velocity continues increasing with shear stress, but the slip length stays independent from the slip velocity. The force balance which describes the three regimes is summarized by [20]: kmV + kTL(V ) R2 0 = = V b(V ).

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Motivation of study polymer slippage on solid surface

Considering the theoretical and experimental studies of polymer slippage on solid surface previously mentioned, the slip length are claimed to be positive in the order of 1 m at low shear rate. Besides, adsorbed polymer chain on the solid is claimed to induce surface roughness and inhibit slip. The slip length is tightly related with polymer chain interaction with the solid surface (ie. repulsion, adsorption, etc.). TIRF method provides the opportunity to study velocity prole of polymer solution in the area close to solid surface within 500 nm, so that to illustrate polymer behaviour in this zone. One of the aims of this present thesis is to show for the rst time 3-D velocity eld of polymer solution, and correlate it with the interaction mechanism between polymer chain and solid surface.

Experimental methods of investigation of slippage Summary

In this chapter the experimental methods aimed at slip length determination are reviewed. The indirect methods are based on measurements of quantities which are aected by the presence of slippage. Some of this methods require pre-estimated values of uid property, which allows the induction of slip length. The most ecient method is based on Surface Force Apparatus [33], which does not need information about uid property, and provide a resolution of measurement 2 nm. A common backdrop of the indirect methods consists of the impossibility to study ow prole deeper into the uid, so that these methods are restricted to simple homogeneous uid. The study of complex uids which exhibits specic ow prole requires direct methods which allow in-situ measurements. The direct methods rely on tracking the motion of passive tracers in the uid. Since the tracers have certain size distribution and possess a number of charges on the surface, their diusion, hydrodynamic interaction with the uid and with the solid surface will introduce biases on the measured uid velocity. Corrections need to be elaborated. The resolution on the direction normal to the solid surface is a principal challenge to overcome in order to obtain highly resoluted slip length. These methods will be discussed in detail below.

Method based on surface force measurement

The stream-wise hydrodynamic force exerted on a sphere which is approaching perpendicularly to the solid surface with velocity V is expressed by the formula: Fh = 6rV r h with uid viscosity, r sphere radius, h the separation distance. This formula is based on no-slip boundary condition on the surface, however this force may be modulated with presence of slippage [6, 14, 33]. The author applied Vinogradova’s theory [138], which incorporated slip boundary condition into the drainage force formula: Fh = 6r2Vh f, with f a correction factor for slip on both surfaces f = h 3b [(1 + h   ) ln(1 + 6b h ) 􀀀 1].

Table of contents :

1 General introduction 
1.1 Device Fabrication
1.2 Application
1.2.1 An alternative to study uid physics on micro and nano scale
1.2.2 Droplet microuidics
1.2.3 Capacity of modeling networks in cores for petroleum application
1.3 The objectives and organisation of thesis
I Experimental study of slippage of Newtonian uids and polymer solution 
1 Presentation of the state of the art 
1.1 Slippage of Newtonian uid
1.1.1 Theoretical expectations
1.1.2 What physical factors in uence slippage
1.1.3 Applications of slippage
1.2 Slippage of polymers
1.2.1 Formation of polymer layer on solid surface
1.2.2 Theoretical and experimental study of polymer slippage on solid surface
1.2.3 Motivation of study polymer slippage on solid surface
1.3 Experimental methods of investigation of slippage
1.3.1 Indirect methods
1.3.2 Direct methods
1.3.3 Motivation of high resolution slip length measurement based on TIRF method
2 Experimental study of slippage 
2.1 Experimental setup
2.1.1 Illumination setup
2.1.2 Acquisition setup
2.1.3 Pressure applying setup
2.1.4 Studied solutions and seeding particles
2.1.5 Channel geometry
2.1.6 Preparation of surfaces
2.2 Measurement technique
2.2.1 Incident angle measurement
2.2.2 Acquisition of data
2.2.3 Laser waist and intensity calibration
2.3 Detection method
2.4 Analysis procedure
2.4.1 Numerical data treatment
2.4.2 Determination of position of the wall
2.4.3 In uence of nite particle size in evanescent eld on the altitude determination
2.4.4 In uence of bleaching on owing particles
2.4.5 In uence of focusing on particle altitude determination
2.4.6 Electrostatic forces
2.4.7 Rheological determination of solution viscosity
2.5 Results
2.5.1 Raw results before correction
2.5.2 Langevin simulation
2.5.3 Results after correction
2.5.4 Results with Polyethylene solution on hydrophilic surface
2.6 Conclusion
II Physics of drop formation at a step 
1 Introduction 
1.1 Droplet production in 2D geometries or 2D geometry with discontinuity
1.1.1 From dripping to jetting
1.1.2 Pinching mechanism and drop size
1.2 Production of droplets by step emulsication
1.3 Theories developed for drop formation at a step
1.3.1 Previous theories on step emulsication
1.3.2 Low-Ca approximative theory (by A. Leshansky)
1.3.3 Motivation of the project
2 Experimental study of step emulsication 
2.1 Experimental setup
2.2 Transition from step emulsication to large drops
2.3 Step emulsication-dynamics of pinching
2.4 Droplet size
2.5 Conclusion
III Flow and nano particles transport in sub-micrometric models of reservoirs 
1 Project background 
2 AEC nano particles in patterned micro models 
2.1 Preliminary results
2.2 Transport of nano particles in patterned micro models
2.3 Conclusion
A Clogging of polystyrene particles in micro uidic channels 
A.1 Introduction
A.2 Experiments on clogging
A.2.1 Extrinsic clogging
A.2.2 Intrinsic clogging
A.3 Conclusion


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