Drops and jets of complex fluids - 3: Dynamics of ferrofluid drops

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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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3.

Dynamics of Ferrofluid drops

3.1

Introduction

In the previous chapter, we focused on the Laplace pressure inside a liquid drop deposited on a solid surface. We showed how the wetting properties of the solid surface and the pinning of contact line can be used to manipulate the Laplace pressure inside drops and hence the flow rate of our capillary micropump. In doing this, we did not take into account the effect of contact line dynamics of the drops, because the contact line speed was very small.In this chapter, we focus on the contact line dynamics of a drop.

Understanding the dynamics of contact lines has become of a major interest as it plays an important role in problems related with industrial applications (31). A large research effort has gone into the control of the interaction between solid substrate and liquids in order to help or prevent the spreading of liquids. Coating and drying solid surfaces are important problems as well. Often, these phenomena are controlled by what happens at the frontier between the wetted and dry part of the substrate: i.e. the contact line (C.L).

One of the key questions is how a contact line changes in response to the dynamics of the spreading/withdrawal of the liquid. Several features emerge: the wriggling dynamics of a C.L. on heterogeneous substrate, or the instability of a receding con-tact line leading to the formation of a film, known as the Landau-Levitch transition (37).

The study reported in the present chapter deals with the interaction between the shape and the dynamics of the C.L. But, instead of imposing the motion between liquid and solid, as is usually done, we impose the shape of the contact line and ob-serve the consequences on its dynamics. To achieve this goal, we have built objects we have called “shaped drops”. These are obtained by pouring some ferrofluid on a small neodynium magnet. The ferrofluid wraps around the magnet, and forms a film between the bottom of the magnet and the supporting substrate. Thus,

the magnet levitates in the core of a drop that keeps the geometry imposed by the magnet. Therefore, contact lines of unusual shapes can be made by changing the geometry of the magnet as desired. As a result, one can study the motion of contact lines of arbitrary shapes (Fig. 3.8).

3.2

Ferrofluids

The study of various fields and interactions with fluids may be divided into three main categories (38):

1. electrohydrodynamics (EHD), the branch of fluid mechanics concerned with electric force effects;

2. magnetohydrodynamics (MHD), the study of interaction between magnetic fields and fluid conductors of electricity;

3. ferrohydrodynamics (FHD), the study of fluid motion influenced by strong forces of magnetic polarization.

It is important here to emphasize the difference between ferrohydrodynamics and the relatively better known discipline of megnetohydrodynamics. In MHD the body force acting on the fluid is the Lorentz force that arises when electric current flows at an angle to the direction of an imposed magnetic field. However, in FHD there need not be an electric current flowing in the fluid and usually there is none. The body force in FHD is due to polarization force, which in turn requires material magnetization in the presence of magnetic field gradients or discontinuities. Ferro-hydrodynamics began to be developed in the early to mid-1960s, motivated initially by the objective of converting heat to work by no mechanical parts (39).

Several types of magnetic fluids can be used for FHD; the principal type is col-loidal ferrofluid. A colloid is a suspension of finely divided particles in a continuous medium, including suspensions that settle out slowly. However, a true ferrofluid does not settle out, even though a slight concentration gradient can become es-tablished after long exposure to a force field (gravitational or magnetic). Such ferrofluids are composed of small particles (3-15 nm) particles of solid, magnetic, single-domain particles coated with a molecular layer of dispersant and suspended in a liquid carrier (see Fig. 3.1). Thermal agitation keeps the particles suspended because of Brownian motion, and the coating prevent the particles from sticking to each other.

The colloidal ferrofluid must be synthesized, since it is not found in nature. A typical ferrofluid contains 1023 _{particles per cubic meter and is opaque to visible} light.

3.2.1

Stability requirements

Dimensional reasoning may be used to arrive at criteria for physicochemical sta-bility. To begin it is useful to write expressions for various energy terms. These energies per particle are (38):

3.2. Ferrofluids

Figure 3.1: Schematic representation of coated, subdomain, magnetic particles in a colloidal ferrofluid. Collisions of suitably coated particles are elastic (38).

thermal energy = kT magnetic energy = µ0M HV gravitational energy = ∆ρV gL

where k is Boltzman’s constant and equals 1.38×10−23 N · M · K−1, T is abso-lute temperature in Kelvin, µ0 is the permeability of free space an has the value 4π × 10−7 H · m, volume V = πd3_{/6 for a spherical particle of diameter d, and} L is the elevation in the gravitational field. Ratios of one term to another yield dimensionless quantities that inform about the stability of the ferrofluids.

Stability in a magnetic field gradient

Consider the stability against settling of particles in a field gradient due to an ex-ternal magnetic source. Particles are attracted to the higher intensity regions of a magnetic field, while thermal motion counteracts the the field force and provides statistical motions that allow the particles to sample all portions of the fluid vol-ume. The magnetic energy µ0M HV represents the reversible work in removing a magnetized particle from a point in the fluid, where the field is H, to a point in the fluid that is outside the field:

W = − Z

(µ0M dH

Provided some part of ferrofluid volume is located in a field-free region, then sta-bility against segregation is favored by a high ratio of the thermal energy to the magnetic energy: thermal energy magnetic energy = kT µ0M HV ≥ 1. (3.2)

Rearranging and substituting for the volume of a sphere gives an expression for the maximum particle size:

d ≤ (6kT /πµ0M H)1/3. (3.3)

Consider the conditions existing in a beaker of magnetic fluid containing mag-netite (F e3O4) particles subject to magnetic gradient field of a typical hand-held permanent magnet:

H = 8 × 104A · m−1, (3.4)

M = 4.46 × 105_{A · m}−1_{, (3.5)}

T = 298 K, (3.6)

the particle size then follows as d ≤ 8.1 × 10−9 _{m or 8.1 nm. The actual particle} size of stable colloids range up to about 10 nm.

Stability against settling in a gravitational field

The relative influence of gravity to magnetism is described by the ratio gravitatinal energy

magnetic energy =

∆ρgL µ0M H

. (3.7)

Again for a beaker of fluid, typical values of parameters are L = 0.05 m and ∆ρ = ρsolid− ρf luid = 4300 kg m−3; with g = 9.8 ms−2, the ratio is 0.047. Thus gravity is less of a threat to the segregation of these magnetic fluid than is magnetic field.

3.2.2

Some applications of ferrofluids

Applications of ferrofluids span a very wide range. Commercial use presently in-cludes novel zero leakage rotary shaft seals used in computer disk drives (Bailey 1983) (40), vacuum feed-throughs for semiconductor manufacturing and related uses (Moskowitz 1975) (41), and more. Also in use are liquid cooled loud-speakers that employ mere drops of ferrofluid to conduct heat away from the speaker coils (Hathaway 1979) (42). This innovation increases the amplifier power the coils can accommodate and hence the sound level the speaker produces. A magnetic field

3.2. Ferrofluids

Figure 3.2: Shape of the drops at rest, when ferrofluid is added to a magnet with height l=2 mm and diameter d=10 mm, on a horizontal glass surface. a) The magnet. b) Added fluid volume = 400 µl, c)1000 µl, d) 2800 µl.

5 10 15 20 25 30 35 40 5 10 15 20 25 30 X(mm) Y(mm)

Figure 3.3: Experimentally obtained surface shapes for the magnet l = 2 mm, d = 10 mm on a solid plastic surface with volumes of fluid: 0, 600, 1200, 1800, 2400, 3000, 3600, 4400 µl.

can also pilot the path of a drop of ferrofluid in the body, bringing drugs to a tar-get site (Morimoto, Akimoto and Yotsumoto 1982) (43) , and ferrofluid serves as a tracer of blood flow in noninvasive circulatory measurements (Newbower 1972) (44).

An especially promising application under study is the use of magnetic fluid ink for high speed, inexpensive, silent printers (Maruno, Yubakami and Soga 1983) (45) . In one type of these printers, as many as 104_{drops per second issue from a} tiny orifice and are guided magnetically to form printed characters on a substrate (Kuhn and Myers 1979) (46).

Figure 3.4: Ferrofluid added to the magnet l = 2 mm, d = 10 mm suspended in the air with its symmetry axis parallel to the gravity direction with different volumes of the fluid a) v = 400 µl b) 1200 µl c) 2600 µl d) 4000 µl.

3.3

Drop Characteristics

In our experiments, we have used a commercial ferrofluid (EFH1 by Ferrotec) on a flat NdFeB magnet with a thickness of 1 mm and typical size of 1 cm (Fig. 3.2 and 3.3). The overall shape of the liquid surface of the drop is strongly dominated by the magnetic forces. We have checked that gravity has only a small influence by obtaining a similar drop shape on a magnet suspended by strings (Fig. 3.4 and 3.5). The capillary forces appear to play a role only in the vicinity of a substrate to which the liquid surface connects under a small contact angle. The interface is there deformed over a typical distance of 0.1 mm, that we can call a magneto-capillary distance lmc= γ/µ0M H, where γ is the surface tension, H the magnetic field and M is the magnetic polarisation of the ferrofluid. The measured lmc is consistent with γ = 0.025 N.m−1, H = 2 × 103A.m−1and M = 105A.m−1. We measured the elevation of the magnet inside the drop by attaching a flat index under the magnet poking out of the ‘shaped drop’ surface (Fig. 3.7).

It would be an interesting challenge to find the equilibrium shape of the ferrofluid around magnets of different geometries. In principal this can be done by minimizing the magnetic bulk energy along with the free surface energy of the ferrofluid. We studied these shapes, for different magnet geometries both when the magnet was on a solid surface and when it was suspended in air.

Throughout this section and the next, l will stand for the height of the cylindrical magnet, d for its diameter and v for the volume of the ferrofluid added to the magnet.

We checked that the viscosity of our ferrofluid is not affected by the magnet. Ac-cording to Shliomis (47), the change in effective viscosity due to the magnetic field |H| = H and the magnetic moment of the particles M , if the Langevin parameter ξ = M H/kT  1, is

∆η ∼=3 2ηφξ

3.3. Drop Characteristics

5 10 15 20 25 −2 0 2 4 6 8 10 12 14 16 18 20 X(mm) Y(mm)

Figure 3.5: Disk magnet of l = 2 mm, d = 10 mm suspended in the air in horizontal position (with its symmetry axis parallel to the gravity direction), the volumes are : 200, 600, 1200, 1800, 2400, 3000, 3600, 4000 µl, respectively. 0 200 400 600 800 1000 1200 1400 1600 1800 0.0 0.5 1.0 1.5 2.0 2.5 C m = A + B*v A -0.31036 ±0.14343 B 0.00146 ±0.00014 Experiments Linear fit C m ( m m ) v ( L

Figure 3.6: Vertical component of the center of mass of the ferrofluid on the magnet l = 2 mm, d = 10 mm as a function of the fluid volume.

Figure 3.7: Measurement of elevation height of the magnet, the wire is attached to the bottom of the magnet l=1, d=10 mm, and it is sticking out.

where φ is the particle volume concentration and δ is the angle between the vectors H and Ω = ∇ × v/2. Since in our experiments φ ∼ 10−1, ξ ∼ 10−1 and sin2δ ∼ 1 at most, hence ∆η ∼ 10−3_{η. This makes the influence of magnetization on viscosity} negligible in our experiments. Our test fluid is a Newtonian fluid with η = 7.8 mPa.s.

3.4

Dynamics:

Sliding Shaped drops on an

in-clined plane

Since the magnet is not touching the surface, the overall drop has a very small static friction coefficient, so that it will slide down the plate even with the slightest inclinations. In our experiment, we let the drop slide down a flat plate with an inclination with respect to the horizontal θ ranging between 2.2◦ and 18◦. The

motion of the drop is observed on a typical distance of 70 cm. After a short

distance, of the order of 10 cm, the drop has a constant velocity. The most striking observation is the direction taken by the drops.

As a drop made with a disc-shaped magnet slides down the plate along the steepest descent, drops that do not have a plane of symmetry will take other directions. As an example, a drop built from a Tetris-like assembly of four square magnets in a shape of a ‘S’ deviates to the left of the the steepest descent under a deviation angle ϕ. The symmetric shape ‘Z’ deviates to the right, taking a path symmetric to the one of the ‘S’ drop with respect to the steepest decent (Fig. 3.8). A more surprising behaviour is observed with drops made from a half-disc magnet (obtained by breaking a disc magnet in half). When put in motion, its planar symmetry is broken, each drop deviates from the steepest slope with a 50-50 distribution to the left and the right side.

Due to wetting, the magnet leaves a thin film of fluid behind as it slides down the surface. We have weighted the mass of fluid left behind, on a distance of 50 cm, a drop loses around 3% of its total mass which can be assumed constant.

3.4. Dynamics: Sliding Shaped drops on an inclined plane

d

c

b

a

Figure 3.8: Top: picture of the magnet used to make shaped drops. Bottom: Pictures of the moving shaped drops, the vertical side of the picture is parallel to the steepest descent. a) A drop made from a disc slide down the steepest decent. A drop made from four square magnets (sticking together by dipolar interaction) in a shape of a ‘S’ Tetris piece (b) deviates to left, while a ‘Z’ piece (c) deviates in the symmetric direction. Made from a half-disc magnet (d) deviates from the steepest descent with a constant angle, but with a random side.

Figure 3.9: The schematic of the setup for the ferrofluid drop moving on the tilted plate

Figure 3.10: The movie of the fluid ball (magnet geometry: l =2, d =8 mm), sliding down the tilted plate. With tilt angle: 0.149 rad. Rate of the movie capture is 125 frames/s. The camera is rotated to make the horizontal axis of movie frames parallel to the shaped drop’s velocity.

The motion was captured with a CCD at 125 frames/s, placed vertical to the path of the ball, to measure its speed. The setup and a sequence of our movies with the drop moving on the tilted plate are shown in Fig . 3.9 and 3.10, respectively. To measure the location of the drop in each frame we used a technique to achieve

sub-pixel accuracy (Fig. 3.11). This method helps for getting smoother time

derivatives of position. We used the function ‘find-edge’ in ‘Labview’, which ba-sically assigns higher values to the edges in the image, by taking a derivative of the image matrix values (Fig. 3.11c). Apparently, The edges of the drop has a thickness (of a few pixels), and there is a maximum value somewhere in the mid-dle, the coordinate of this maximum is the exact location of the edge of the drop (Fig. 3.11d). By fitting a gaussian curve to the points nearby each peak, we find the position of the edge (and hence the drop) with sub-pixel accuracy (Fig. 3.11). When performed at different slopes θ the velocity is found to be proportional to sin θ (Fig. 3.12). For a given drop (same magnet and amount of ferrofluid), the deviation angle ϕ is independent of the slope.

3.4.1

Discussion on the acting forces

To understand this surprising behaviour, we have to list the forces acting on the drop. As it reaches a constant velocity, the net force acting on the shaped drop vanishes. The component of its weight parallel to the plate is mg sin θ. One may

3.4. Dynamics: Sliding Shaped drops on an inclined plane

Figure 3.11: a) The l =2, d = 8 mm on the tilted plate while moving down. Ferrofluid mass is 1.005 g. b) The intensity values on the line defined in (a). c) The edges of the image (a). d) The intensity values on the line defined in (c), this diagram can be considered as the derivative of the diagram (b). The precise position of the borders of the droplet is the location of the peaks in this diagram. Sub-pixel accuracy is achieved by fitting a gaussian curve to the points nearby each maximum peak, and taking the curve’s peak x-coordinate.

0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 sin V ( m / s ) l=1 mm, d=10 mm, m=1.216 g l=2 mm, d=8 mm, m=1.00 g l=1 mm, d= 6 mm, m=0.475 g V ( m / s ) l=1 mm, d= 10 mm, m=0.51

Figure 3.12: Steady velocity of the drop as a function of Sine of the tilt angle of the plate. Top: different disk magnets. The total mass of the shaped drop is m. Bottom: a half disk magnet.

3.4. Dynamics: Sliding Shaped drops on an inclined plane

assume that the resisting force originates from the viscous shearing in the fluid between the bottom of the magnet and the plate. As a typical case, let’s consider a drop made with a disc magnet with a diameter d = 10 mm and a total mass m = 1.216 g. The magnet is then elevated by h = 0.183 mm. On a plate tilted at an angle of 5.6◦, the drop adopts a velocity V = 28 cm/s. The resulting force should be

Fshear= ηπd2_V

4h = 9.4 × 10

−5_{N, (3.9)}

which is much smaller than mg sin θ = 1.19 × 10−3N. Thus the resistance is

governed by the flow at its perimeter (this last term will be preferred to “contact line” as the drop, leaving a film, does not have a receding C.L.). We propose a simple model for this force, Fp. Suppose that the force acting on an element of perimeter has the form proposed by Huh and Scriven (48) or Cox (49), the force in this model is proportional to the velocity (as described in Chapter 1). Here, the relevant velocity is the component normal to the contact line. This choice is justified by the fact that in the vicinity of the contact the internal motion of the fluid is perpendicular to the contact line (50). Each element of length ds of the perimeter of the drop is submitted to an elementary force

dF = −Ψa(V · n) nds or dF = Ψr(V · n) nds, (3.10)

where V is the velocity vector of the center of mass of the drop, n is the local unit vector normal to the perimeter, in the plate’s plane, pointing outside the drop. The coefficient Ψa is used for the advancing parts of the perimeter, and Ψr for the receding parts of the drop, where a film is deposited. Ψa and Ψr have dimensions of viscosity and are in principle dependent to the geometry and hydrodynamic properties of the fluid surface in vicinity of the solid substrate. As can be seen in Fig. 3.13, the shape of the surface near the substrate is different for the advancing and receding parts, and hence Ψaand Ψrare not identical in general. The total friction force is obtained by integrating over the front, advancing, part of the perimeter (index a) and its back, receding portion (index r).

F = −Ψa Z a (V · n) nds + Ψr Z r (V · n) nds. (3.11)

3.4.2

Extracting the perimeter

To be able to perform this integration, we need to know the shape of the perimeter. The visualization of the perimeter is not a simple task for the following reasons: (i ): parts of the drop overhangs above the surface, the perimeter cannot be seen from above, Fig. 3.14a, (ii ) the ferrofluid is an opaque, black liquid, there is no clear contrast between the free surface and the contact area, (iii ) the smooth liquid surface does not scatter light but reflects it. The experiments has thus been performed on a glass plate. The drop, observed through the plate, slide between

two lines of square LEDs that provide an extended and grazing light source (Fig. 3.14b). Their reflection on the liquid free surface is observed as bright spots whose position on the drop change as it moves. To build the picture shown in Fig. 3.15, we have superposed successive images shifted so that the drop appears at the same position. In the reconstructed image, the part that is not showing that reflection of the LEDs is the contact area, Fig. 3.15a.

3.4.3

Analyzing the data

We focus our study on two types of shaped drops: the ones from disc and half-disc magnets. The observed perimeter of the half-disc-like drop is circular within the imaging precision. In our model, the resisting force is simply

F =π

2Rp(Ψa+ Ψr)V, (3.12)

where Rpis the radius of the contact area. To be able to have the values of Ψaand Ψrindividually, we study the motion of the half-disk like drop taken under the same conditions (same slope, same elevation h, same magnet model). After the extraction of the drop perimeter, one can perform the two integrations, R

a(V · n) nds and R

r(V · n) nds. Ψa and Ψr are obtained by equating F, as expressed in eq. (3.11) with the component of the weight along the plate.

From Fig. 3.16a and b it is evident that the values of Ψa and Ψr are independent of the tilt angle of the plate and hence the velocity of the drop, as expected. Fig 3.16c and d show that Ψa and Ψr are changing very smoothly with the elevation

V

a

a

b

c

Figure 3.13: a. Side view of a shaped drop made from a disc magnet with diameter of 10 mm and 400 µL of ferrofluid. The white dashed line represented the approx-imate position of the magnet inside the drop. The drop is currently moving at a velocity V = 24 cm/s. Close-up views at the rear and the front perimeter of the contact area are shown respectively in b. and c.

3.4. Dynamics: Sliding Shaped drops on an inclined plane

Figure 3.14: The Schematic view of the setup to obtain the contact line of the ball by using an array of LEDs. a) Front view, b) side view of the setup.

line of the largest slope start point

V

β

F(V)

φ

a

_b

Figure 3.15: Left: superposition of 48 images of a half-disc drop as it passes between the rows of LEDs. The white line is the perimeter we extract from this image. Right: perimeter extracted from the previous image. The forces acting on the advancing portion of the perimeter and the receding portion are symbolized by respectively the red and the blue arrows. Bottom: The total force is not collinear with the velocity, they make an angle ϕ. Note that the axis of symmetry of the drop makes an angle β with the velocity.

of the magnet h and the total mass of the drop m (which is in direct connection with h). One may argue that within the margin of errors values of Ψa, Ψr and/or Ψa + Ψr are constant with respect to the velocity and elevation h for the same magnet geometry.

While the drop shaped with a half disk magnet is moving down the plate it preserves the angles ϕ and β and does not rotate around the center of mass. Therefore, the total moment of the resisting forces on the contact line with respect to the center of mass should be zero. We examined this fact in our experiments by finding the point at which the total moment of the friction forces at the perimeter is zero. Fig. 3.17 shows that this point matches the center of mass to within the experimental accuracy.

3.5

Conclusion and perspectives

In this chapter, we investigated a situation in which a liquid object, submitted to its weight and viscous forces, adopts a different direction than the steepest descent. The main reason is a left/right asymmetry of our systems. An asymmetric perimeter leads to a total viscous force that is not collinear with the velocity, this effect arises from the simple dynamics we have proposed. But, as we have seen with the half-disc geometry, even symmetric shapes can deviate from the steepest descent (even if, in average, the line of the largest slope remains a symmetry axis). Another issue has then to be accounted for: the total torque of the viscous forces has to play a stabilizing role for the drop. In our case, the half-disc has to be tilted. We have shown that imposing the shape of a contact line (or a drop perimeter) can dramatically influence its dynamics. Up till now, studies in dynamic wetting mainly focused on forcing the dynamics (imposed slope, imposed velocity. . . ), we suggest that our approach offers new tools. More systematic measurements of the force acting on a moving contact line, while the velocity is not normal, could be performed. The ferrofluid we used, made with an organic carrier, did not allow us to observe a Landau-Levitch transition. The critical speed is here very slow, even on a fluorinated substrate. Using well controlled aqueous ferrofluids could make possible studies on the Landau-Levitch transition, by forcing the contact line to be tilted and/or corrugated.

3.5. Conclusion and perspectives

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.2 1.4 1.6 1.8 2.0 2.2 2.4 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 sin( ) a r a r P a . s (a) sin( ) disk, l=2 mm, d= 8 mm, h =1.0 mm disk, l=1 mm, d=6 mm, h =1.82 mm disk, l=1 mm, d=10 mm, h = 1.83 mm half disk, l=1 mm, d=10 mm, h =1.68 mm a r ( P a . s ) (b) P a . s a r (hal f disk) a (hal f disk) r (hal f disk) a r (disk) h (mm) (c) a r (hal f disk) a (hal f disk) r (hal f disk) a r (disk) P a . s m/m mag

Figure 3.16: a) Ψa, Ψr and Ψa + Ψr for the half disk geometry with l = 1 mm and d = 10 mm and total mass m = 0.51 g, plotted versus the sine of the tilt angle. b) Ψa+ Ψr for disk and half disk geometries. () m = 1.00 ± 0.02 g, (•) m = 0.475 ± 0.01 g, (N) m = 1.216 ± 0.02 g, (H) m = 0.51 ± 0.01 g. c) Ψa, Ψr and Ψa+ Ψras a function of h for the same half disk and constant plate tilt angle θ = 4.19◦_{. The disk magnet dimensions l = 1 mm, d = 10 mm. d) The same values} in (c) versus the total mass of the drop over the mass of the magnet, mmag.

54 56 58 60 62 64 24 22 20 18 16 14 center of mass

Figure 3.17: The black crosses show the contact line of the shaped drop with a half disk magnet of l = 1 mm, d = 10 mm. 150 µL of fluid is added to the magnet. The drop is moving on a plate with 4.19◦inclination. The red circle shows the center of mass and the blue one is showing the point at which the total torque on the drop is zero.