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Experiments on the breakup of a liquid bridge of magnetic fluid
Journal of Magnetism and Magnetic Materials 201 (1999) 324}327
Experiments on the breakup of a liquid bridge of magnetic #uid Alexander Rothert, Reinhard Richter* Institut fu( r Experimentelle Physik, Abteilung Nichtlineare Pha( nomene, Otto-von-Guericke Universita( t, Postfach 4120, D-39016 Magdeburg, Germany Received 25 May 1998; received in revised form 10 August 1998
Abstract Under variation of an external magnetic "eld a liquid bridge of magnetic #uid disintegrates. We measure the temporal evolution of the minimal neck diameter immediately before the breakup, and compare it with a universal theory. Moreover, we report on the dynamics of satellite drops emanating from the liquid bridge. 1999 Elsevier Science B.V. All rights reserved. Keywords: Magnetic liquids; Drops and bubbles
Introduction Surface-tension-driven #ows and their tendency to decay spontaneously into drops have been investigated for more than 150 years [1]. Linear stability theory governs the onset of breakup and was developed by Rayleigh [2], Plateau, and Maxwell. However, only recently has attention turned to the nonlinear behaviour in the immediate vicinity of the singular point where a drop separates. Introducing an inner length scale l and the correspondJ ing time scale t (for de"nition see Table 1) and J starting from the Navier}Stokes equations for an axisymmetric column Eggers derived a simple onedimensional description of the last stage of jet
* Corresponding author. Fax: ##391-6718108. E-mail address: [email protected] (R. Richter)
pinching [3]. With the pinch}o! radius as the expansion parameter he was able to reduce the problem to a coupled set of ordinary di!erential equations for two scaling functions (m),t(m) of a similarity variable m(l , t , z!z , t !t) which J J measures the distance from the breakup point (z , t ). The two universal functions and t de"ne the "nal shape and velocity of the pinch}o! region. As an important result the "nal shape scales only with the coe$cients l and t but is independent of J J the macroscopic initial conditions. These scaling laws have been investigated in a number of recent experiments analysing dripping faucets, the decay of periodically driven jets, and the breakup of liquid bridges e.g. Refs. [4}6]. However, up to our knowledge, the separation singularity has not yet been looked at for a magnetic #uid (MF). Thus it is not clear at all if the universal scaling laws survive in the presence of a magnetic "eld. Here the magnetic forces could become more important for the pinch}o! process than the
0304-8853/99/$ } see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 9 9 ) 0 0 0 8 2 - 7
A. Rothert, R. Richter / Journal of Magnetism and Magnetic Materials 201 (1999) 324}327
325
surface tension and inertia, especially because the lines of the magnetic "eld are guided by the shape of the liquid bridge and thus change in time. In the following we tackle this question by investigating the scaling law for the minimum neck diameter d "a < (t !t) (1)
J of standard, non-magnetic #uids [3,5], with the parameters a "0.0608 and < "l /t . J J J 2. Experimental setup We have investigated the breakup of a liquid bridge of ferro#uid in two di!erent experimental arrangements shown in Fig. 1. In setup A a liquid bridge of MF (volume 45 ll) was suspended in}between the pole shoes of two electro}magnets. This simple arrangement o!ers an elegant way to control the stability of the liquid bridge solely by magnetic forces. For the selected distance of the pole shoes the "eld is not homogeneous, but increases towards the edges of the pole shoes. As a consequence upon increase of the exciting current the MF is pulled towards both magnets and thus the neck diameter decreases. If a critical current is surpassed the liquid bridge disintegrates. Setup B consists of a pair of microscope slides which are mounted in parallel. They have been oriented horizontally in the homogeneous magnetic "eld of a pair of Helmholtz coils. A drop of magnetic liquid (120 ll) is placed on the lower slide. Upon increase of the exciting current the drop develops a cusp and elongates. After further increase of the current it "nally touches the upper
Fig. 1. The two di!erent setups used to investigate the decay of a liquid bridge of ferro#uid. In setup A the liquid bridge is suspended between two pole shoes of diameter 4.5 mm positioned 4.5 mm apart. In setup B the bridge is formed between two microscope slides mounted 5 mm apart in}between a pair of Helmholtz coils.
slide. The liquid bridge formed this way disintegrates when decreasing the current. In both setups we record the fast collapse of the liquid bridge by means of a high speed CCD}camera (Kodak Ektapro Hi}Spec Motion Analyzer). Table 1 gives the #uid parameters for the investigated MFs together with those for water and Glycerol. According to Eq. (1) the "nal velocity of the pinch}o! is scaled by < , which is presented in line J "ve of Table 1.
3. Experimental results Fig. 2 gives three representative snapshots for the experiments with setup A. From left to right one clearly observes the formation of a neck (a), a long "lament (b) and "nally a satellite drop (c). The further fate of the satellite drop is shown in a series of enlargements in Fig. 3. The satellite drop is
Table 1 The #uid parameters for water, glycerol, APG J12, APG S11. The values are quoted from Ref. [3] and from the speci"cation sheets of Ferro#uidics Company. The internal scales l , t and the velocity < are calculated from the kinematic viscosity l and from the ratio of J J J surface tension p density o. Quantity
Water
Glycerol
APG J12
APG S11
l (cm/s) p/o (cm/s) l "lo/p (cm) J t "lo/p (s) J < "l /t (cm/s) J J J
0.01 72.9 1.38;10\ 1.91;10\ 7.2;10
11.8 50.3 2.79 0.652 4.28
0.36 30.63 4.24;10\ 4.99;10\ 85
0.609 30.43 1.22;10\ 2.44;10\ 50
326
A. Rothert, R. Richter / Journal of Magnetism and Magnetic Materials 201 (1999) 324}327
Fig. 2. The decay of a liquid bridge of ferro#uid (APG J12). The frames are taken at t"0 ms (a), 2 ms (b), and 3 ms (c).
Fig. 3. The formation of a satellite drop after the decay of a liquid bridge of ferro#uid (APG J12). The time span between the consecutive frames is one ms.
#oating for about 5 ms at the vertical position z"0, i.e. in mid range between the two pole shoes until it is moving towards the upper magnet. The temporal dynamics has been plotted in Fig. 4 and can be well described by the function z(t)" a exp(bt). The exponential movement of the droplet is characteristic for the start in the neighbourhood of an unstable node or saddlepoint. Similar dynamics in downward direction have also been observed. From both we conclude that the satellite drop is formed in the immediate vicinity of the saddlepoint, i.e. slightly above or under its location. Next, we investigate the evolution of the minimum neck diameter as extracted from the images (for details see Ref. [7]). In Fig. 5 the diameter of the MF APG S11 has been plotted versus the scaled time. The full circles mark the neck diameter for setup A, the open squares the one for setup B. Despite the di!erent initial conditions in the two setups both curves converge in the immediate
Fig. 4. The vertical position of the satellite drop versus time. The solid line displays a "t with the function z(t)"a exp(bt), where a"7;10\ mm and b"0.369 ms\.
vicinity d (0.24 mm of the pinch}o! with a com mon shrink velocity for the minimum neck diameter. The range of constant shrink velocity is even larger than the pinch}o! region d (l "0.12 mm
J
A. Rothert, R. Richter / Journal of Magnetism and Magnetic Materials 201 (1999) 324}327
327
newtonian, non-magnetic #uids. The shrink velocity of the MF APG S11 is about 2.2 times larger than predicted. If this discrepancy has its origin in a modi"cation of the #uid parameters under in#uence of the applied magnetic "eld or is simply a consequence of magnetic forces remains to be investigated by further experiments.
Acknowledgements Fig. 5. The minimal neck diameter plotted as a function of the scaled time t< ,< "0.5 m/s for APG S11. The full circles (open J J squares) mark the neck diameter for setup A (B), respectively. The inset displays a zoom of the range near the origin. The dashed dotted line displays the decrease of the minimal neck diameter according to Eq. (1). The solid (dashed) line gives the "t according to Eq. (1) for the slope a of the minimal neck diameter for setup A (B), respectively. For setup A we obtain a"0.1313, t "!0.5 ms and for setup B a"0.1359, t "!0.52 ms, respectively.
demanded by Eggers for the validity of his approximation. The solid and the dashed line in Fig. 5 gives the "t according to Eq. (1) for setup A and B, respectively. For comparison the dashed dotted line provides the shrink velocity given by Eq. (1) for
The authors thank Thomas M. Bock and Ingo Rehberg for helpful advice. Financial support by the &Deutsche Forschungsgemeinschaft' through grant En278/2-1 is gratefully acknowledged.
References [1] [2] [3] [4]
F. Savart, Annal. Chim. 53 (1833) 337, plates in vol. 54. J.W.S. Rayleigh, Proc. Lond. Math. Soc. 10 (1878) 4. J. Eggers, Rev. Mod. Phys. 69 (1997) 865. X.D. Shi, M.P. Brenner, S.R. Nagel, Science 265 (1994) 219. [5] T.A. Kowalewski, Fluid Dyn. Res. 17 (1996) 121. [6] M. Tjahjadi, H.A. Stone, J.M. Ottino, J. Fluid. Mech. 243 (1992) 297. [7] A. Rothert, R. Richter, in preparation.
Experiments on the breakup of a liquid bridge of magnetic #uid Alexander Rothert, Reinhard Richter* Institut fu( r Experimentelle Physik, Abteilung Nichtlineare Pha( nomene, Otto-von-Guericke Universita( t, Postfach 4120, D-39016 Magdeburg, Germany Received 25 May 1998; received in revised form 10 August 1998
Abstract Under variation of an external magnetic "eld a liquid bridge of magnetic #uid disintegrates. We measure the temporal evolution of the minimal neck diameter immediately before the breakup, and compare it with a universal theory. Moreover, we report on the dynamics of satellite drops emanating from the liquid bridge. 1999 Elsevier Science B.V. All rights reserved. Keywords: Magnetic liquids; Drops and bubbles
Introduction Surface-tension-driven #ows and their tendency to decay spontaneously into drops have been investigated for more than 150 years [1]. Linear stability theory governs the onset of breakup and was developed by Rayleigh [2], Plateau, and Maxwell. However, only recently has attention turned to the nonlinear behaviour in the immediate vicinity of the singular point where a drop separates. Introducing an inner length scale l and the correspondJ ing time scale t (for de"nition see Table 1) and J starting from the Navier}Stokes equations for an axisymmetric column Eggers derived a simple onedimensional description of the last stage of jet
* Corresponding author. Fax: ##391-6718108. E-mail address: [email protected] (R. Richter)
pinching [3]. With the pinch}o! radius as the expansion parameter he was able to reduce the problem to a coupled set of ordinary di!erential equations for two scaling functions (m),t(m) of a similarity variable m(l , t , z!z , t !t) which J J measures the distance from the breakup point (z , t ). The two universal functions and t de"ne the "nal shape and velocity of the pinch}o! region. As an important result the "nal shape scales only with the coe$cients l and t but is independent of J J the macroscopic initial conditions. These scaling laws have been investigated in a number of recent experiments analysing dripping faucets, the decay of periodically driven jets, and the breakup of liquid bridges e.g. Refs. [4}6]. However, up to our knowledge, the separation singularity has not yet been looked at for a magnetic #uid (MF). Thus it is not clear at all if the universal scaling laws survive in the presence of a magnetic "eld. Here the magnetic forces could become more important for the pinch}o! process than the
0304-8853/99/$ } see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 9 9 ) 0 0 0 8 2 - 7
A. Rothert, R. Richter / Journal of Magnetism and Magnetic Materials 201 (1999) 324}327
325
surface tension and inertia, especially because the lines of the magnetic "eld are guided by the shape of the liquid bridge and thus change in time. In the following we tackle this question by investigating the scaling law for the minimum neck diameter d "a < (t !t) (1)
J of standard, non-magnetic #uids [3,5], with the parameters a "0.0608 and < "l /t . J J J 2. Experimental setup We have investigated the breakup of a liquid bridge of ferro#uid in two di!erent experimental arrangements shown in Fig. 1. In setup A a liquid bridge of MF (volume 45 ll) was suspended in}between the pole shoes of two electro}magnets. This simple arrangement o!ers an elegant way to control the stability of the liquid bridge solely by magnetic forces. For the selected distance of the pole shoes the "eld is not homogeneous, but increases towards the edges of the pole shoes. As a consequence upon increase of the exciting current the MF is pulled towards both magnets and thus the neck diameter decreases. If a critical current is surpassed the liquid bridge disintegrates. Setup B consists of a pair of microscope slides which are mounted in parallel. They have been oriented horizontally in the homogeneous magnetic "eld of a pair of Helmholtz coils. A drop of magnetic liquid (120 ll) is placed on the lower slide. Upon increase of the exciting current the drop develops a cusp and elongates. After further increase of the current it "nally touches the upper
Fig. 1. The two di!erent setups used to investigate the decay of a liquid bridge of ferro#uid. In setup A the liquid bridge is suspended between two pole shoes of diameter 4.5 mm positioned 4.5 mm apart. In setup B the bridge is formed between two microscope slides mounted 5 mm apart in}between a pair of Helmholtz coils.
slide. The liquid bridge formed this way disintegrates when decreasing the current. In both setups we record the fast collapse of the liquid bridge by means of a high speed CCD}camera (Kodak Ektapro Hi}Spec Motion Analyzer). Table 1 gives the #uid parameters for the investigated MFs together with those for water and Glycerol. According to Eq. (1) the "nal velocity of the pinch}o! is scaled by < , which is presented in line J "ve of Table 1.
3. Experimental results Fig. 2 gives three representative snapshots for the experiments with setup A. From left to right one clearly observes the formation of a neck (a), a long "lament (b) and "nally a satellite drop (c). The further fate of the satellite drop is shown in a series of enlargements in Fig. 3. The satellite drop is
Table 1 The #uid parameters for water, glycerol, APG J12, APG S11. The values are quoted from Ref. [3] and from the speci"cation sheets of Ferro#uidics Company. The internal scales l , t and the velocity < are calculated from the kinematic viscosity l and from the ratio of J J J surface tension p density o. Quantity
Water
Glycerol
APG J12
APG S11
l (cm/s) p/o (cm/s) l "lo/p (cm) J t "lo/p (s) J < "l /t (cm/s) J J J
0.01 72.9 1.38;10\ 1.91;10\ 7.2;10
11.8 50.3 2.79 0.652 4.28
0.36 30.63 4.24;10\ 4.99;10\ 85
0.609 30.43 1.22;10\ 2.44;10\ 50
326
A. Rothert, R. Richter / Journal of Magnetism and Magnetic Materials 201 (1999) 324}327
Fig. 2. The decay of a liquid bridge of ferro#uid (APG J12). The frames are taken at t"0 ms (a), 2 ms (b), and 3 ms (c).
Fig. 3. The formation of a satellite drop after the decay of a liquid bridge of ferro#uid (APG J12). The time span between the consecutive frames is one ms.
#oating for about 5 ms at the vertical position z"0, i.e. in mid range between the two pole shoes until it is moving towards the upper magnet. The temporal dynamics has been plotted in Fig. 4 and can be well described by the function z(t)" a exp(bt). The exponential movement of the droplet is characteristic for the start in the neighbourhood of an unstable node or saddlepoint. Similar dynamics in downward direction have also been observed. From both we conclude that the satellite drop is formed in the immediate vicinity of the saddlepoint, i.e. slightly above or under its location. Next, we investigate the evolution of the minimum neck diameter as extracted from the images (for details see Ref. [7]). In Fig. 5 the diameter of the MF APG S11 has been plotted versus the scaled time. The full circles mark the neck diameter for setup A, the open squares the one for setup B. Despite the di!erent initial conditions in the two setups both curves converge in the immediate
Fig. 4. The vertical position of the satellite drop versus time. The solid line displays a "t with the function z(t)"a exp(bt), where a"7;10\ mm and b"0.369 ms\.
vicinity d (0.24 mm of the pinch}o! with a com mon shrink velocity for the minimum neck diameter. The range of constant shrink velocity is even larger than the pinch}o! region d (l "0.12 mm
J
A. Rothert, R. Richter / Journal of Magnetism and Magnetic Materials 201 (1999) 324}327
327
newtonian, non-magnetic #uids. The shrink velocity of the MF APG S11 is about 2.2 times larger than predicted. If this discrepancy has its origin in a modi"cation of the #uid parameters under in#uence of the applied magnetic "eld or is simply a consequence of magnetic forces remains to be investigated by further experiments.
Acknowledgements Fig. 5. The minimal neck diameter plotted as a function of the scaled time t< ,< "0.5 m/s for APG S11. The full circles (open J J squares) mark the neck diameter for setup A (B), respectively. The inset displays a zoom of the range near the origin. The dashed dotted line displays the decrease of the minimal neck diameter according to Eq. (1). The solid (dashed) line gives the "t according to Eq. (1) for the slope a of the minimal neck diameter for setup A (B), respectively. For setup A we obtain a"0.1313, t "!0.5 ms and for setup B a"0.1359, t "!0.52 ms, respectively.
demanded by Eggers for the validity of his approximation. The solid and the dashed line in Fig. 5 gives the "t according to Eq. (1) for setup A and B, respectively. For comparison the dashed dotted line provides the shrink velocity given by Eq. (1) for
The authors thank Thomas M. Bock and Ingo Rehberg for helpful advice. Financial support by the &Deutsche Forschungsgemeinschaft' through grant En278/2-1 is gratefully acknowledged.
References [1] [2] [3] [4]
F. Savart, Annal. Chim. 53 (1833) 337, plates in vol. 54. J.W.S. Rayleigh, Proc. Lond. Math. Soc. 10 (1878) 4. J. Eggers, Rev. Mod. Phys. 69 (1997) 865. X.D. Shi, M.P. Brenner, S.R. Nagel, Science 265 (1994) 219. [5] T.A. Kowalewski, Fluid Dyn. Res. 17 (1996) 121. [6] M. Tjahjadi, H.A. Stone, J.M. Ottino, J. Fluid. Mech. 243 (1992) 297. [7] A. Rothert, R. Richter, in preparation.