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Drops and jets of complex fluids
Javadi, A.
Publication date
2013
Link to publication
Citation for published version (APA):
Javadi, A. (2013). Drops and jets of complex fluids.
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1.
General Introduction
Drops and jets are perhaps the most common shapes a liquid can take when it leaves its reservoir or pipe line. Generally, drops are round-shaped portions of liquid floating in another fluid, like air, or adhering to a solid surface and jets are columnar stream of liquid. Small drops (of size . 1 mm), usually take the shape of a sphere in air (or another fluid), and the shape of a spherical cap on a solid surface, while jets usually take the shape of a straight or bent cylinder. Despite their simple shapes, the dynamics of drops and jets in different situations are quite complex. These complexities have been the focus of a good portion of the literature in soft matter science, especially in recent times. As an example, wetting and spreading of drops is of major interest in both science and technology. Many practical processes require the wetting or spreading of a drop on a solid. The liquid drop maybe a paint, a lubricant, an ink or a dye (1). The solid may either show a simple surface or be finely divided (suspensions, porous media, fibers). Buckling, coiling and folding of viscous jets when they impact a solid surface are other examples of complex situations. In this thesis, we will explore some of these complexities of drops and jets of both Newtonian (water and silicon oil) and non-Newtonian fluids (ferrofluid, polymer solutions and foam).
1.1
Drops
This thesis (like most of the literature) will consider small drops of size ∼ 1 mm and thin jets of diameter . 1 cm. In these cases, the surface to volume ratio is relatively large. Therefore, the surface properties of the fluid, and in particular the surface tension, would be an important parameter of the problem. Surface tension is in fact responsible for the shape of liquid droplets and the breakup of liquid jets into drops.
Liquid
Figure 1.1: The forces acting on a liquid molecule inside the bulk and on the surface.
1.1.1
Surface tension
Surface tension (γ) is measured as the energy required to increase the surface area of a liquid by a unit of area. The surface tension of a liquid results from an imbalance of intermolecular attractive forces, such as the Van der Waals forces, between molecules. A molecule in the bulk liquid experiences cohesive forces with other molecules in all directions, while a molecule at the surface of a liquid experiences only net inward cohesive forces (Fig. 1.1).
The unbalanced attraction of molecules at the surface of a liquid tends to pull the molecules back into the bulk of the liquid, leaving the minimum number of molecules on the surface. Energy is required to increase the surface area of a liquid because a larger surface area contains more molecules in the unbalanced situation. This is the reason why a small drop has a sphere shape (in equilibrium conditions). A sphere has the minimum free surface (compared to other geometrical shapes) for a given volume of liquid. It is also the reason why a cylindrical jet of a liquid tends to break into spherical drops: surface minimization.
Gravity can play a role in the shape of a static drop. A bond number is defined as the ratio of gravity to surface tension forces. For a drop of size R it can be written as
Bo = ρgR
2
γ , (1.1)
where ρ is density, g the gravity acceleration and γ the surface tension. Since Bo ∝ R2, for small sizes the gravity is negligible and surface tension determines
the shape of the drop. For instance, a water drop of size R = 1 mm, has a Bond Number Bo ∼ 0.1.
1.1. Drops
Figure 1.2: A small droplet in equilibrium over a horizontal surface. (a) and (b) correspond to partial wetting, the trend towards wetting being stronger in (a) than in (b). (c) corresponds to complete wetting (θe= 0) (1).
1.1.2
Wetting
A liquid (L) when deposited on a flat, impermeable, solid surface (S), may show two types of equilibrium behaviour: partial wetting (Fig. 1.2(a) and (b)) or total wetting (Fig. 1.2(c)). The choice is dictated by interfacial energies γSL, γSV and
γLV, where V stands for the vapor phase.
When the combination:
S = γSV − (γSV + γLV), (1.2)
is positive, the energy of the solid/vapor interface is lowered by interaction of a flat liquid film: this corresponds to complete wetting. But when S is negative, a liquid drop does not spread on the solid: it terminates in the form of a wedge, with a well-defined contact angle θe (Fig. 1.2). We call this partial wetting.
Balancing the tensions γ (projected along the solid surface, which defines the al-lowed direction of motion) Young found the relation (2)
γSV − γSL= γLVcosθe. (1.3)
Equation (2.12) is best derived by considering a reversible change in contact line position, using global energetic arguments (1). Thus the nature of the contact line
U
Figure 1.3: A wedge of liquid moving with velocity U.
region, over which intermolecular forces are acting, is not considered. Accordingly, θe is understood to be measured macroscopically, on a scale above that of
long-ranged intermolecular forces.
1.1.3
Dynamic of partial wetting
Equation (2.12) holds at equilibrium. What happens if we move out of equilibrium, for instance by forcing a droplet on a surface as in chapter 3 of this thesis ? Let us discuss this for the case of partial wetting.
If the contact line of Fig. 1.3 moves at a velocity U , we expect a dissipation per unit length
T ˙S = F U, (1.4)
where F is the non-compensated Young force:
F = γSV − γSL− γLVcosθd= γLV(cosθe− cosθd), (1.5)
θd being the dynamic contact angle. If we can find the dissipation mechanism, we
end up with a relation between the driving force and the velocity.
The dissipation may have different origins: either molecular processes very near the contact line, or viscous processes in the whole moving fluid. The first may be sensitive to the chemical details of the molecules making liquid and the solid. The second is more universal. There is one limit, where viscous flows must be dominant: namely when the dynamic contact angle is small (θd 1). We can understand
this by the following argument.
Inside the moving wedge of Fig. 1.3, the velocities u range from u ∼ U at the free surface and u ∼ 0 at the lower surface. Therefore the viscous dissipation is of order
T ˙S = Z dxη U y 2 y, (1.6)
where y = θdx is the local thickness. Equation (1.6) gives a logarithmic integral
l = ln(xmax/xmin). Inserting the correct coefficients:
T ˙S = 3lηU
2
θd
1.2. Jets
U
Figure 1.4: Dynamic contact angle θd versus velocity U.
The logarithmic factor l is of order 12; it has worried the experts in fluid mechanics for many years. However, it is not the dominant feature of equation (1.7). The really important feature is the presence of θd in the denominator. At small wedge
angles, the viscous dissipation becomes very large, and dominates over all molecular processes.
Combining equations (1.4) and (1.7), we end up with a basic dynamic formula for partial wetting (3):
F ≡ γ(cosθe− cosθd) =
3lη θd
U, (1.8)
valid for θd 1 (γ ≡ γLV). A vast number of experiments can be understood in
these terms (4).
Fig. 1.4 shows the relation between U and θd in partial wetting. Of course, U
vanishes at the equilibrium angle U (θe) = 0, but it also vanishes at small θd, where
the dissipation is large.
1.2
Jets
Jets are present in our everyday environment in kitchens (Fig. 1.5(a)), showers, pharmaceutical sprays and cosmetics. The study of jets is also motivated by many
practical applications such as improving and optimizing liquid jet propulsion, diesel engine technology, agricultural sewage and irrigation (Fig. 1.5(b)), manufacturing (Fig. 1.5(c)), powder technology, ink-jet printing (Fig. 1.5 (d)), medical diagnos-tics and DNA sampling. On the other hand, jet dynamics probes a wide range of physical properties, such as liquid surface tension, viscosity or non-Newtonian rheology and density contrast with its environment.
Here, our focus will be on the dynamics of falling jets. When a vertical jet em-anates from a nozzle and falls under gravity, it exhibits a number of complex and interesting phenomena downstream. Depending on the boundary conditions and parameters of the problem, such as viscosity, flow rate and falling height, different phenomena are observed:
Break-up: If the height of the fall is large enough, the inevitable destiny of the liquid jet is breakup. The jet breaks into larger main drops and some smaller satellite drops downstream (Fig. 1.6). The main driving force here is the surface tension. Inertia and viscosity oppose the breakup; jets with larger viscosity and flow rate have larger breakup lengths. Supposedly, gravity helps the breakup by thinning the profile of the jet. But since it irons out the perturbations which lead to the breakup of the jet at the same time, it can actually make the breakup length larger (see Chapter 4).
Hydraulic jump: When a falling cylindrical jet impacts a solid surface before its breakup, different behaviours are observed. Typically, hydraulic jump occurs for high flow rates and low viscosities (Fig. 1.5(a)): right after the impact the jet spreads symmetrically in a thin layer, then there is an abrupt increase in the fluid depth at a well-defined Radius from the impact point Rj. This abrupt rise of the
fluid surface, flowing from a shallower and higher velocity to a deeper and lower velocity zone is called hydraulic jump.
Buckling: At the other extreme, at low flow rates and very high viscosities, the thin jet buckles as it impinges the plate; an interesting solid-like behaviour. As the flow continues a rotating helical coiling or folding of the thin filament is observed. Here, the jet exhibits very rich dynamics and different regimes of motion.
1.3
Contents
In chapter 2 of this thesis we show how a simple feature of water droplets on a surface, i.e. Laplace pressure, can be exploited to build a micropump. We also use electrowetting on water droplets to make a bi-directional micropump. The dynamics and velocity of the contact line of the drops does not play an important role in this chapter. Instead, we show that wetting properties are the key to controlling the flow. In the next chapter, sliding drops on an inclined plane are
1.3. Contents
(a)
(b)
(c)
(d)
Figure 1.5: a) A jet of tap water falling into a sink. The jet is too thick and its falling time too short for breakup to occur, yet it has become rough. The continuous jet hits the sink floor, where it expands radially in the form of a thin sheet boarded by a hydraulic jump. b) Sprays produced by jets are widely used in agricultural irrigation. c) Higher speed water jets are also used to cut tissues, meat, and even metal plates. d) Drops emerging from a bank of ink-jet nozzles. The drop heads are 50 µm across and the tails are less than 10 µm wide (5).
Figure 1.6: Plate I of Rayleigh’s ‘some applications of photography’ (6) showing: a) the destabilization of a jet of air into water and b) of a water jet in air. Rayleigh notes that the air jet destabilizes faster than the water jet. c), d) The breakup of a falling jet. Larger main and smaller satellite drops can be seen in these pictures.
1.3. Contents
studied to investigate the dynamics of wetting. Ferrofluid and magnets are used to make drops which do not follow the steepest decent.
The subsequent chapters make a study of 3 scenarios which can occur for a falling jet; the breakup of viscous threads of silicone oil, the spread of a non-Newtonian liquid jet (dilute polymer solutions) over a horizontal plate and the coiling of yield stress fluids (foam and gel). A more detailed abstract of what is done in each chapter is as follows:
Chapter 2: We investigate capillary pumping in microchannels both experimen-tally and numerically. Putting two droplets of different sizes at the in/outlet of a microchannel, will generally produce a flow from the smaller droplet to the larger one due to the Laplace pressure difference. We show that an unusual flow from a larger droplet into a smaller one is possible by manipulating the wetting properties, notably the contact line pinning. In addition, we propose a way to actively control the flow by electrowetting.
Chapter 3: ‘Shaped drops’ are made by adding ferrofluid to small magnets (∼ 1 cm). The important feature of these drops is that the magnet preserves the shape of the drop and the contact line. Therefore, we can impose the shape of the contact line and observe the consequences on the dynamics of sliding drops on an inclined plane. In this chapter, we observe how a liquid object, submitted to its weight and viscous forces, can adopt a different direction than the steepest descent.
Chapter 4: Thin jets of viscous fluid like honey falling from capillary nozzles can attain lengths exceeding 10 m before breaking up into droplets via the Rayleigh-Plateau (surface tension) instability. Using a combination of laboratory exper-iments and WKB analysis of the growth of shape perturbations on a jet being stretched by gravity, we determine how the jet’s intact length lb depends on the
flow rate Q, the viscosity η, and the surface tension coefficient γ. In the asymp-totic limit of a high-viscosity jet, lb∼ (gQ2η4/γ4)1/3, where g is the gravitational
acceleration. The agreement between theory and experiment is good except for very long jets.
Chapter 5: This chapter contains a brief review of hydraulic jump for viscous and inviscid flow. Expressions for the radius of the jump Rj(from the impact point
of the jet) are derived. Correction to the viscous expressions due to the surface tension is also presented. The same approach is pursued for the case of non-Newtonian power law fluids. Some data for the hydraulic jump of dilute polymer solutions are also shown, however, these experiments are still incomplete.
Chapter 6: We present an experimental investigation of the coiling of a filament of a yield stress fluid falling on a solid surface. We use two kinds of yield stress fluids: shaving foam and hair gel, and show that the coiling of the foam is similar
to the coiling of an elastic rope. Two regimes of coiling (elastic and gravitational) are observed for the foam. Hair gel coiling, on the other hand, is more like the coiling of a liquid system; here we observe viscous and gravitational regimes. No inertial regime is observed for either system because of instabilities occurring at high flow rates or the break up of the filament in large heights.
