# Extensional Deformations Of These Fluids Biology Essay

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## Chapter 1

Non-Newtonian fluids are of fundamental importance in all our lives, occurring in such diverse circumstances as food (mayonnaise, cheese, chocolate, etc.), biofluids (blood, mucin, synovial fluid, spittle, etc.), personal care products (shampoo, toothpaste etc.), electronic and optical materials (liquid crystals) and polymers that are mostly relevant to industries.

Extensional deformations of these fluids are generally present and often play a crucial role in determining the flow dynamics in many important industrial, technological and biologically relevant processes 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 kinds of processes often inherently involve the flow of complex polymeric fluids, quantifying the extensional viscosity and understanding the elastic response of polymer molecules to such extensional flow fields and how this modifies the local flow structure are questions of fundamental importance. However, for dilute solutions with low viscosity (1mPa·s to 1Pa·s), these remain some of the most challenging tasks of rheometer development. Odell and Carrington [1] have made a good progress in measurement of the extensional viscosity in low-viscosity fluids with cross-slot geometries forming stagnant point.

## 1.1.2. Entry Flow of Non-Newtonian Fluids

Entry flow through an abrupt contraction is a typical geometry for studying benchmark flow problem that can capture the key phenomena of viscoelastic fluid flow. Study of this flow problem can be traced back to the late 1800s, when early workers like Hagenback, Boussinesq, and Couette were interested in measuring the pressure drop across circular entry flows, motivated by a need to develop a capillary rheometer for accurate viscosity measurement of Newtonian fluids [2]. By correlating the measured pressure drop across a contraction, this method also has been used to explore the extensional properties of complex fluids in macroscale geometries [3-5].

This typical geometry has also been widely used to study the non-linear flow phenomena associated with fluid elasticity in converging flows [6-9]. In the contraction geometry, the fluid is forced to accelerate spatially as it enters the smaller downstream channel. The flow is dominated by elongational flow at the centerline and shear flow at the walls. The complex fluid flow near to the contraction region is highly extension dominated and exhibits elastic flow phenomena. Entry flow is therefore a good benchmark test for analyzing the effect of elongational viscosity on polymer solution flows, and also serves as a benchmark flow problem for numerical simulations [10-11].

## 1.1.3 The Advantages of Using Micro-Fabricated Devices

The importance of the geometric scale in micro-hydrodynamics has been of particular interest over the past decade. Compared to macro-scale systems, microfluidics can reduce the space, labour, time and reagent consumption greatly, which facilitates system integration especially for biological and chemical processes.

It has been confirmed that the traditional governing equations including the continuum equation and the Navier-Stokes equations are essentially 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 additional elastic component [16-17]. With visualization methods their experimental results show that microscale geometries can result in remarkably different flow phenomena from the macroscale, which is relevant to ultimate applications such as the lab-on-a-chip [18], the high speed electro-spraying [19], electro-spinning and inkjet printing [20] typically utilizing aqueous fluids containing low concentrations of high molecular weight polymers [21]. Many of these applications involving viscoelastic fluids have a complex micro-structure [22].

The typical radius of gyration of a polymer chain or a characteristic radius of a suspended particle, ranges from 1nm to 10µm. These micro-structures may lead to flow inhomogeneities. Moreover, as the characteristic length-scale of the flow geometry approaches that of the fluid micro-structure, physical confinement can alter the dynamical evolution of the microstructure [23-24] and must be taken into account when considering the bulk response. Therefore, it is desirable to understand the bulk flow of complex liquids on small scales to optimize the design and implementation of microfluidic systems.

Due to the small dimensions of microscale devices, the magnitude of viscoelastic effects in dilute polymer solutions can be enhanced. The decrease in scale can significantly increase the shear rate (), which is defined as, where U is a characteristic velocity difference acting over a characteristic distance L. In entry flow experiments the reduction in the scale of the geometry also provides access to increased Weissenberg number (Wi) regimes while reducing the order of the Reynolds number (Re), leading to high elasticity numbers (El=Wi/Re). From the summary of earlier entry flow studies performed in macroscale geometries, for shear-thinning fluids, the fluids worked in the ranges of Wi < 10 and Re < 1000.

Rodd et al. [25] studied the flow of shear-thinning fluids through micro-fabricated planar abrupt contraction-expansion, achieving the high Weissenberg number (0 ≤ Wi ≤ 548) regime, while maintaining moderate Reynolds numbers (0.44 ≤ Re ≤ 64), obtaining elasticity numbers (El=Wi/Re) up to 89. Gulati et al. [26] conducted an experimental study of the flow of large molecular fluids, such as DNA solutions, through a micro-contraction at very high Weissenberg numbers (0.8 < Wi < 629) while keeping the Reynolds number very low (6.0-10-7 < Re < 9.8-10-2). Extremely high elasticity numbers were accessed, 1.4-103 < El < 1.4-106. The micro-fabricated geometries greatly expand the ranges of Wi and El, in which both creeping flow and the interplay region between inertia and elasticity are approached. In fact, studies of the flow of polymer solutions through micro-contractions are still very few. In this thesis, the schematic sketch of the micro-fabricated flow cell is displayed in Fig. 1.1 (a). Fig. 1.1 (b) shows the SEM (Scanning Electron Microscope) image of one of the geometries used in this thesis.

Fig.1.1. (a) Schematic diagram of the planar micro-fabricated contraction - expansion; wc is the contraction width, wu the upstream width, Lc the downstream length and h is the uniform depth of the channel; (b) SEM image of the planar contraction with contraction ratio (β = 8:1).

Furthermore, the length of a macromolecule in microscale extensional flow can, under special circumstances (such as at a stagnation point), approach the fully extended contour length and may approach the characteristic dimensions of the microscale flow channels. The effects of the flow on the conformation of a single DNA molecule have been experimentally shown in various flow types, e.g. shear flow [27-30], elongational flow [31-33] and mixed flow [34]. DNA molecules stretch out in accelerating flow regions and recoil in decelerating flow or stagnant regions [35]. All these explorations provide opportunities to understand the fundamental physics of viscoelastic fluid flows.

The measurement of shear viscosity at high shear rates and of extensional viscosity are still challenging for commercial rheometers. Microfabricated devices have been used to measure the rheological response of complex fluids by achieving shear rates up to 106s-1 [36-38]. Extensional components were also studied by using a various flow geometries in microscale including contraction - expansion and sink flows, bifurcations ('T' or 'Y' junctions), flow around obstacles [39] and stagnation points in cross-slot [40-41], in which the inertial effects were greatly reduced. All these works indicate that microfluidics could provide an excellent platform for the development of an extensional rheometer for low viscosity dilute polymer solutions. At present commercial extensional rheometers are not able to operate in this regime.

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

## 1.2 Dimensionless Parameters

Three important dimensionless quantities are used in this thesis to characterize polymer solution flow regimes through contraction-expansion geometries, as advocated by Boger [42]. These are the Weissenberg number (Wi), the Reynolds number (Re), and the elasticity number (El).

According to the flow cell dimensions given in Fig.1.1, Eqs. (1.2)-(1.5) provide the definitions of these non-dimensional numbers. For contraction entry flow, the Weissenberg number is used to evaluate the elastic effects, and defined by the ratio of the relaxation time of the fluid and local flow time scales due to a local shear rate. In this thesis, it is defined in terms of polymer solution characteristic relaxation time and the average shear rate (Eq.1.1) in the contraction throat:

(1.1)

(1.2)

Where, is the average velocity, wc the contraction width, h the depth of the channel and Q is the volumetric flow rate.

The Deborah number (De) is defined as the ratio of a relaxation time to the characteristic time scale () of an experiment (or a computer simulation) probing the response of the material [43]:

(1.3)

Where, is the relaxation time scale and tp refers to the time scale of observation. The value of Wi is equal to that of De from the definitions, however, they have different physical interpretations. The Weissenberg number indicates the degree of anisotropy or orientation generated by the deformation, and is appropriate to describe flow with a constant stretch history, usually restricted to steady flows and used in some special cases, such as simple shear. In contrast, the Deborah number should be used to describe flows with a non-constant stretch history, and physically represents the rate at which elastic energy is stored or released. Hence, the Weissenberg number is used in the entry flow studies of this thesis.

The Reynolds number is used to evaluate the inertial effect and defined in terms of the average velocity in the contraction throat:

(1.4)

In which, the hydraulic diameter, Dh, is given by, and are the fluid density and the zero-shear viscosity, respectively. For viscoelastic flow, the Reynolds number represents the relative importance of inertial to viscous forces because it non-dimensionalizes the momentum balance equation.

In order to evaluate the relative importance of elastic stresses to inertia effects, the elasticity number is defined as in Eq. (1.5), which represents the ratio of Weissenberg number to Reynolds number:

(1.5)

It is independent of the fluid kinematics since both Wi and Re vary linearly with characteristic velocity, but depends on the properties of the fluid and the characteristic length scales of the device. This number was used in [25] to represent the trajectory of a set experiment with a given viscoelastic fluid through the Wi-Re operating space.

Besides the dimensionless numbers used to describe the flow dynamics, the vortex growth behaviour observed in viscoelastic entry flows is also characterised predominantly in terms of a dimensionless vortex length. Boger [44] presented the basic elements for laminar flow through an abrupt circular contraction as shown in Fig. 1.2, in which Lv is the length of the vortex from the internal corner to the point of detachment. According to this configuration, for a planar contraction-expansion (illustrated in Fig. 1.1) used in this thesis, the dimensionless vortex is quantified as the axial distance upstream from the contraction plane at which the primary flow first detaches from the wall:

(1.6)

Obviously, for all definitions of non-dimensional parameters, we use the dimension of contraction throat wc instead of the upstream wu. In fact, to correlate elastic effects in planar entry flow, the Weissenberg numbers are mostly expected to be evaluated in both the upstream and contraction throat. 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 region. They eventually adopted the downstream Weissenberg number, Wic, to characterise the entry flow, where Wiu can be deduced from the contraction ratio (β), Wiu = Wic/ï¢. Therefore, when comparing the results of different literature, we should pay attention to the definition of Wi.

Fig.1.2. Basic elements of an entry flow for from a large tube through an abrupt into a smaller tube. Taken from [44].