## Chapter 1

Non-Newtonian fluids are of cardinal importance in all our lives, happening in such diverse fortunes as nutrient ( mayonnaise, cheese, cocoa, etc. ) , biofluids ( blood, mucin, synovial fluid, saliva, etc. ) , personal attention merchandises ( shampoo, toothpaste etc. ) , electronic and optical stuffs ( liquid crystals ) and polymers that are largely relevant to industries.

Extensional distortions of these fluids are by and large present and frequently play a important function in finding the flow kineticss in many of import industrial, technological and biologically relevant procedures including porous media flows ( enhanced oil recovery and filtration ) , particle suspension/sedimentation, ink-jet printing, fibre-spinning, blood and mucin flows, among many others. Since these sorts of procedures frequently inherently involve the flow of complex polymeric fluids, quantifying the extensional viscousness and understanding the elastic response of polymer molecules to such extensional flow Fieldss and how this modifies the local flow construction are inquiries of cardinal importance. However, for dilute solutions with low viscousness ( 1mPa·s to 1Pa·s ) , these remain some of the most ambitious undertakings of rheometer development. Odell and Carrington [ 1 ] have made a good advancement in measuring of the extensional viscousness in low-viscosity fluids with cross-slot geometries organizing dead point.

## 1.1.2. Entry Flow of Non-Newtonian Fluids

Entry flow through an disconnected contraction is a typical geometry for analyzing benchmark flow job that can capture the cardinal phenomena of viscoelastic fluid flow. Study of this flow job can be traced back to the late 1800s, when early workers like Hagenback, Boussinesq, and Couette were interested in mensurating the force per unit area bead across round entry flows, motivated by a demand to develop a capillary rheometer for accurate viscousness measuring of Newtonian fluids [ 2 ] . By correlating the mensural force per unit area bead across a contraction, this method besides has been used to research the extensional belongingss of complex fluids in macroscale geometries [ 3-5 ] .

This typical geometry has besides been widely used to analyze the non-linear flow phenomena associated with unstable snap in meeting flows [ 6-9 ] . In the contraction geometry, the fluid is forced to speed up spatially as it enters the smaller downstream channel. The flow is dominated by elongational flow at the center line and shear flow at the walls. The complex fluid flow near to the contraction part is extremely extension dominated and exhibits elastic flow phenomena. Entry flow is hence a good benchmark trial for analysing the consequence of elongational viscousness on polymer solution flows, and besides serves as a benchmark flow job for numerical simulations [ 10-11 ] .

## 1.1.3 The Advantages of Using Micro-Fabricated Devicess

The importance of the geometric graduated table in micro-hydrodynamics has been of peculiar involvement over the past decennary. Compared to macro-scale systems, microfluidics can cut down the infinite, labor, clip and reagent ingestion greatly, which facilitates system integrating particularly for biological and chemical procedures.

It has been confirmed that the traditional government equations including the continuum equation and the Navier-Stokes equations are basically valid for microscale Newtonian fluid flow without the non-conservative forces ( e.g. Electrokinetics ) [ 13-15 ] . In fact, most fluids processed in microfluidic devices are likely to exhibit a complex micro-structure and non-linear phenomena by the extra elastic constituent [ 16-17 ] . With visual image methods their experimental consequences show that microscale geometries can ensue in unusually different flow phenomena from the macroscale, which is relevant to ultimate applications such as the lab-on-a-chip [ 18 ] , the high velocity electro-spraying [ 19 ] , electro-spinning and inkjet printing [ 20 ] typically using aqueous fluids incorporating low concentrations of high molecular weight polymers [ 21 ] . Many of these applications affecting viscoelastic fluids have a complex micro-structure [ 22 ] .

The typical radius of rotation of a polymer concatenation or a characteristic radius of a suspended atom, ranges from 1nm to 10µm. These micro-structures may take to flux inhomogeneities. Furthermore, as the characteristic length-scale of the flow geometry attacks that of the unstable micro-structure, physical parturiency can change the dynamical development of the microstructure [ 23-24 ] and must be taken into history when sing the majority response. Therefore, it is desirable to understand the majority flow of complex liquids on little graduated tables to optimise the design and execution of microfluidic systems.

Due to the little dimensions of microscale devices, the magnitude of viscoelastic effects in dilute polymer solutions can be enhanced. The lessening in graduated table can significantly increase the shear rate ( ) , which is defined as, where U is a characteristic speed difference moving over a characteristic distance L. In entry flow experiments the decrease in the graduated table of the geometry besides provides entree to increased Weissenberg figure ( Wi ) governments while cut downing the order of the Reynolds figure ( Re ) , taking to high snap Numberss ( El=Wi/Re ) . From the sum-up of earlier entry flow surveies performed in macroscale geometries, for shear-thinning fluids, the fluids worked in the scopes of Wi & A ; lt ; 10 and Re & A ; lt ; 1000.

Rodd et Al. [ 25 ] studied the flow of shear-thinning fluids through micro-fabricated planar disconnected contraction-expansion, accomplishing the high Weissenberg figure ( 0 ? Wi ? 548 ) government, while keeping moderate Reynolds Numberss ( 0.44 ? Re ? 64 ) , obtaining snap Numberss ( El=Wi/Re ) up to 89. Gulati et Al. [ 26 ] conducted an experimental survey of the flow of big molecular fluids, such as DNA solutions, through a micro-contraction at really high Weissenberg Numberss ( 0.8 & A ; lt ; Wi & A ; lt ; 629 ) while maintaining the Reynolds figure really low ( 6.0-10-7 & A ; lt ; Re & A ; lt ; 9.8-10-2 ) . Highly high snap Numberss were accessed, 1.4-103 & A ; lt ; El & A ; lt ; 1.4-106. The micro-fabricated geometries greatly expand the scopes of Wi and El, in which both crawling flow and the interplay part between inactiveness and snap are approached. In fact, surveies of the flow of polymer solutions through micro-contractions are still really few. In this thesis, the conventional study of the micro-fabricated flow cell is displayed in Fig. 1.1 ( a ) . Fig. 1.1 ( B ) shows the SEM ( Scaning Electron Microscope ) image of one of the geometries used in this thesis.

Fig.1.1. ( a ) Conventional diagram of the two-dimensional micro-fabricated contraction – enlargement ; wc is the contraction breadth, wu the upstream breadth, Lc the downstream length and H is the unvarying deepness of the channel ; ( B ) SEM image of the planar contraction with contraction ratio ( ? = 8:1 ) .

Furthermore, the length of a supermolecule in microscale extensional flow can, under particular fortunes ( such as at a stagnancy point ) , approach the to the full extended contour length and may near the characteristic dimensions of the microscale flow channels. The effects of the flow on the conformation of a individual Deoxyribonucleic acid molecule have been by experimentation shown in assorted flow types, e.g. shear flow [ 27-30 ] , elongational flow [ 31-33 ] and assorted flow [ 34 ] . Deoxyribonucleic acid molecules stretch out in speed uping flow parts and kick in slowing flow or dead parts [ 35 ] . All these geographic expeditions provide chances to understand the cardinal natural philosophies of viscoelastic fluid flows.

The measuring of shear viscousness at high shear rates and of extensional viscousness are still disputing for commercial rheometers. Microfabricated devices have been used to mensurate the rheological response of complex fluids by accomplishing shear rates up to 106s-1 [ 36-38 ] . Extensional constituents were besides studied by utilizing a assorted flow geometries in microscale including contraction – enlargement and sink flows, bifurcations ( ‘T ‘ or ‘Y ‘ junctions ) , flow around obstructions [ 39 ] and stagnancy points in cross-slot [ 40-41 ] , in which the inertial effects were greatly reduced. All these plants indicate that microfluidics could supply an first-class platform for the development of an extensional rheometer for low viscousness dilute polymer solutions. At present commercial extensional rheometers are non able to run in this government.

In this thesis, both the extensional flow behavior in the cross-slot and planar contraction geometries in microscale were studied.

## 1.2 Dimensionless Parameters

Three of import dimensionless measures are used in this thesis to qualify polymer solution flow governments through contraction-expansion geometries, as advocated by Boger [ 42 ] . These are the Weissenberg figure ( Wi ) , the Reynolds figure ( Re ) , and the snap figure ( El ) .

Harmonizing to the flow cell dimensions given in Fig.1.1, Eqs. ( 1.2 ) – ( 1.5 ) provide the definitions of these non-dimensional Numberss. For contraction entry flow, the Weissenberg figure is used to measure the elastic effects, and defined by the ratio of the relaxation clip of the fluid and local flow clip graduated tables due to a local shear rate. In this thesis, it is defined in footings of polymer solution characteristic relaxation clip and the mean shear rate ( Eq.1.1 ) in the contraction pharynx:

( 1.1 )

( 1.2 )

Where, is the mean speed, wc the contraction breadth, h the deepness of the channel and Q is the volumetric flow rate.

The Deborah figure ( De ) is defined as the ratio of a relaxation clip to the characteristic clip graduated table ( ) of an experiment ( or a computing machine simulation ) examining the response of the stuff [ 43 ] :

( 1.3 )

Where, is the relaxation clip graduated table and tp refers to the clip graduated table of observation. The value of Wi is equal to that of De from the definitions, nevertheless, they have different physical readings. The Weissenberg figure indicates the grade of anisotropy or orientation generated by the distortion, and is appropriate to depict flow with a changeless stretch history, normally restricted to steady flows and used in some particular instances, such as simple shear. In contrast, the Deborah figure should be used to depict flows with a non-constant stretch history, and physically represents the rate at which elastic energy is stored or released. Hence, the Weissenberg figure is used in the entry flow surveies of this thesis.

The Reynolds figure is used to measure the inertial consequence and defined in footings of the mean speed in the contraction pharynx:

( 1.4 )

In which, the hydraulic diameter, Dh, is given by, and are the unstable denseness and the zero-shear viscousness, severally. For viscoelastic flow, the Reynolds figure represents the comparative importance of inertial to syrupy forces because it non-dimensionalizes the impulse balance equation.

In order to measure the comparative importance of elastic emphasiss to inertia effects, the snap figure is defined as in Eq. ( 1.5 ) , which represents the ratio of Weissenberg figure to Reynolds figure:

( 1.5 )

It is independent of the fluid kinematics since both Wi and Re vary linearly with characteristic speed, but depends on the belongingss of the fluid and the characteristic length graduated tables of the device. This figure was used in [ 25 ] to stand for the flight of a set experiment with a given viscoelastic fluid through the Wi-Re operating infinite.

Besides the dimensionless Numberss used to depict the flow kineticss, the whirl growing behaviour observed in viscoelastic entry flows is besides characterised preponderantly in footings of a dimensionless whirl length. Boger [ 44 ] presented the basic elements for laminar flow through an disconnected round contraction as shown in Fig. 1.2, in which Lv is the length of the whirl from the internal corner to the point of withdrawal. Harmonizing to this constellation, for a two-dimensional contraction-expansion ( illustrated in Fig. 1.1 ) used in this thesis, the dimensionless whirl is quantified as the axial distance upstream from the contraction plane at which the primary flow foremost detaches from the wall:

( 1.6 )

Obviously, for all definitions of non-dimensional parametric quantities, we use the dimension of contraction pharynx wc alternatively of the upstream Wu. In fact, to correlate elastic effects in two-dimensional entry flow, the Weissenberg Numberss are largely expected to be evaluated in both the upstream and contraction pharynx. In [ 25 ] , they mentioned that both Wiu and Wic can be used as an upper and lower bounds of the true magnitude of viscoelastic effects in the entry part. They finally adopted the downstream Weissenberg figure, Wic, to characterize the entry flow, where Wiu can be deduced from the contraction ratio ( ? ) , Wiu = Wic/i?? . Therefore, when comparing the consequences of different literature, we should pay attending to the definition of Wi.

Fig.1.2. Basic elements of an entry flow for from a big tubing through an disconnected into a smaller tubing. Taken from [ 44 ] .