Wolfgang von Ohnesorge
Gareth H. McKinley and Michael Renardy 
 
Citation: Physics of Fluids (1994-present) 23, 127101 (2011); doi: 10.1063/1.3663616 
View online: http://dx.doi.org/10.1063/1.3663616 
View Table of Contents: http://scitation.aip.org/content/aip/journal/pof2/23/12?ver=pdfcov 
Published by the AIP Publishing 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to  IP:
128.173.125.76 On: Wed, 20 Nov 2013 20:05:03
Wolfgang von Ohnesorge
Gareth H. McKinley1,a) and Michael Renardy2,b)
1Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge,
Massachusetts 02139, USA
2Department of Mathematics, Virginia Tech, Blacksburg, Virginia 24061-0123, USA
(Received 13 September 2011; accepted 27 October 2011; published online 7 December 2011)
This manuscript got started when one of us (G.H.M.) presented a lecture at the Institute of
Mathematics and its Applications at the University of Minnesota. The presentation included a
photograph of Rayleigh and made frequent mention of the Ohnesorge number. When the other of
us (M.R.) enquired about a picture of Ohnesorge, we found out that none were readily available on
the web. Indeed, little about Ohnesorge is available from easily accessible public sources. A good
part of the reason is certainly that, unlike other ?numbermen? of fluid mechanics, Ohnesorge did
not pursue an academic career. The purpose of this article is to fill the gap and shed some light on
the life of Wolfgang von Ohnesorge. We shall discuss the highlights of his biography, his scientific
contributions, their physical significance, and their impact today. VC 2011 American Institute of
Physics. [doi:10.1063/1.3663616]
I. BIOGRAPHYOF WOLFGANG VON OHNESORGE
A. Family background and early years
Wolfgang von Ohnesorge was born on September 8,
1901 in Potsdam. His full name was Wolfgang Feodor Her-
mann Alfred Wilhelm.1 His family had extensive land hold-
ings in the area of Poznan, then part of Prussia. His father,
Feodor von Ohnesorge, was a career military officer. As
mentioned in his brother?s wedding announcement,2 Feodor
was a great-great-grandson of Gebhard Leberecht von
Blu?cher, the Prussian general who, in conjunction with an
allied army under the Duke of Wellington, defeated
Napoleon at Waterloo.36 Feodor was decorated in World
War I and retired in 1925 at the rank of Major General. He
died only a year and a half later. Wolfgang attended the
Augusta-Viktoria Gymnasium in Poznan and subsequently
the Klosterschule Ro?leben in Thuringia, where he grad-
uated in 1921. The ?Klosterschule? is a prestigious private
boarding school which was once affiliated with a monastery.
The monastery was dissolved during the reformation, but the
school continued and kept its name.
B. Student years
After graduating from high school, Wolfgang was
admitted to the University of Freiburg, intending to study
music and art history. Although he retained a love of music
throughout his life, he changed his mind about pursuing it as
a career. He did not attend the University of Freiburg. After
working as a trainee in the mining and steel industries, he en-
rolled as a student of mechanical engineering at the Techni-
cal University (then called Technische Hochschule) in Berlin
in the fall of 1922. He graduated with a diploma in April
1927. After a brief return to industry, he became an assistant
at the Technical University in 1928, in the institute directed
by Hermann Fo?ttinger. He retained this position until 1933.
Wolfgang later commented to his family that the experi-
ments done for his dissertation were tedious and difficult,
and there were many initial failures. He submitted his doc-
toral thesis3 on November 14, 1935 and was awarded a doc-
torate the following year. He presented a lecture based on his
doctoral research on September 25, 1936 at the GAMM (Ge-
sellschaft fuer Angewandte Mathematik und Mechanik) con-
ference in Dresden, and a paper appeared in the proceedings.4
Other participants at the GAMM conference included Prandtl,
Schlichting, Tollmien, and Weber.5 Web of Science shows
more than 90 citations of Ohnesorge?s paper since 1975. A
more condensed version of the paper was published in 1937
(Fig. 1) in the Zeitschrift des Vereins deutscher Ingenieure.6
Wolfgang married Antonie von Stolberg-Wernigerode
on April 20, 1929. A daughter, Gisela, was born in 1930.
The marriage ended in divorce in 1934.
C. Post-graduation years, second marriage,
and World War II
Wolfgang?s original post-dissertation plan had been to
work for Borsig, at the time the leading manufacturer of loco-
motives in Germany. However, the Great Depression derailed
this plan. Wolfgang took up a position in the Eichverwaltung
(Bureau of Standards). In this position he worked first in Ber-
lin and then in Reichenberg (Liberec) in Bohemia.7
On September 2, 1939, Wolfgang married Sigrid von
Bu?nau. Sigrid?s mother Hildegard was from the Crevese
branch of the Bismarck family.8,37 The marriage led to five
children:1 Johannes-Leopold (born 1940), Reinhild (born
1942), Elisabeth (born 1943), Wolfgang (born 1949), and
Sigrid (born 1953).
Wolfgang remained in his position in the Eichverwal-
tung for a part of the war but was eventually drafted into the
army. He served first in France and then in Russia, where he
was wounded. Because of this circumstance and his rela-
tively advanced age, he was assigned,9 for three months in
a)Electronic mail: gareth@mit.edu.
b)Electronic mail: mrenardy@math.vt.edu.
1070-6631/2011/23(12)/127101/6/$30.00 VC 2011 American Institute of Physics23, 127101-1
PHYSICS OF FLUIDS 23, 127101 (2011)
 This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to  IP:
128.173.125.76 On: Wed, 20 Nov 2013 20:05:03
1944, to guard duty at the Plaszow concentration camp near
Cracow.38 He eventually succeeded in getting transferred
back to regular military service. At the end of the war, he
held the rank of lieutenant, was taken prisoner of war by the
Soviets, but managed to escape.7 He reunited with his family
and moved west.
D. Work on behalf of the Johanniter order
After the war, Wolfgang and his family settled in Co-
logne. From 1951 to 1966, he was the director of the Lande-
seichdirektion (Bureau of Standards) of the newly created
state of Nordrhein-Westfalen.7 During this period, the Johan-
niter order39 became a very important part of his life. The
Johanniter are a Lutheran offshoot of the Knights Hospitaller
with extensive historic connections to the Prussian royal
family and the Prussian state. In modern times, they are
mostly known for their sponsorship of a number of charities.
The most visible one is the Johanniter-Unfallhilfe, which
assists victims of traffic accidents (there is an analogue
called the St. John Ambulance in English speaking coun-
tries). Wolfgang von Ohnesorge became a member of the
order in 1951 (his father had also been a member), and in
1952, he cofounded and became the first president of the
Johanniter-Hilfsgemeinschaft. The goal of this organization
was to assist families in need; in the post war years, this
meant primarily those affected by the war. Wolfgang chaired
the organization until 1958. In recognition of his merits, he
was named Ehrenkommendator. A ?Kommendator? is a re-
gional leader of the order; an Ehrenkommendator is an hon-
orary rank meant to be equivalent but without the duties of
an active Kommendator. In later years, Wolfgang became
leader of the Subkommende (local district) of the Johanniter
in Cologne and represented the order on the board of direc-
tors of an affiliated hospital (Fig. 2).
Wolfgang von Ohnesorge died in Cologne on May 26,
1976.
II. OHNESORGE?S WORK AND ITS SIGNIFICANCE
A. Research contribution of Wolfgang von Ohnesorge
Ohnesorge?s thesis was entitled ?Application of a cine-
matographic high frequency apparatus with mechanical con-
trol of exposure for photographing the formation of drops
and the breakup of liquid jets? and was carried out under the
guidance of Professors Fo?ttinger and Stenger at the Techni-
sche Hochschule Berlin (now TU Berlin). The key technical
contribution was a sophisticated spark flash timing and vari-
able exposure system that could be used to take magnified
images of dripping and jetting phenomena with high tempo-
ral resolution. The quality of the images and the temporal
resolution is impressive even by modern standards. A repre-
sentative sequence of images from the thesis is shown in
Figure 3 at an imaging frequency of 300 Hz. His imaging sys-
tem was also able to resolve the dynamics associated with
more complex phenomena such as jetting as well as quasiperi-
odic transitions close to the onset of jetting such as ?double
dripping? (see, for example, Abbildung 30, p. 67 of Ref. 3).
By varying the physical properties of the fluid exiting
from the nozzle (water, aniline, glycerin, and two hydrocar-
bon oils were studied in the thesis) as well as the speed of
the exiting fluid stream, Ohnesorge showed that there were
four important regimes, which were labeled 0?III in the the-
sis and described as follows:
(0) Slow dripping from the nozzle under gravity with no
formation of a jet.
(I) Breakup of a cylindrical jet by axisymmetric perturba-
tions of the surface (according to Rayleigh10,11).
(II) Breakup by screw-like perturbations of the jet (wavy
breakup according to Weber-Haenlein12,13).40
(III) Atomization of the jet.
An extensive discussion of dimensional analysis forms a
large part of the thesis, and Ohnesorge investigated the rela-
tive importance of fluid inertia, viscosity, and surface tension
in controlling the transitions between the different docu-
mented modes of jet breakup in terms of the Reynolds num-
ber for the jet Re?qVd/g, the Weber number We? qV2d/r,
as well as several other nondimensional groupings. The
FIG. 2. Wolfgang von Ohnesorge on the occasion of his retirement from the
Landeseichdirektion in 1966.
FIG. 1. Wolfgang von Ohnesorge in 1937.
127101-2 G. H. McKinley and M. Renardy Phys. Fluids 23, 127101 (2011)
 This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to  IP:
128.173.125.76 On: Wed, 20 Nov 2013 20:05:03
thesis concludes with a demonstration that the clearest way
of delineating the boundaries of the distinct operating
regimes for the jet breakup problem is by defining a new
Kennzahl or dimensionless group given by
Z ? gffiffiffiffiffiffiffiffi
qrd
p : (1)
Here g is the shear viscosity of the fluid, r is the surface ten-
sion, q is the density, d is the jet diameter, and V is the fluid
speed. This is the original definition of what is now referred
to commonly as the Ohnesorge number. The central findings
of the thesis were published in the ZAMM (Zeitschrift fuer
Angewandte Mathematik und Mechanik) article of 1936
(Ref. 4) that featured a slightly expanded operating diagram
(reproduced here in Figure 4) with data for two additional
fluids (?gas oil,? i.e., diesel or heating oil and ?ricinus,? i.e.,
castor oil) beyond those studied in the thesis.
B. Physical interpretation
Representing the experimental results on an operating dia-
gram of the form in Figure 4 clearly delineates the transitions
between different modes of breakup, and it is immediately
apparent that there appears to be a simple power
law relationship between the critical Reynolds number and
the corresponding value of the dimensionless number Z,
although Ohnesorge never gave such an expression (as is
discussed further below). The dimensionless grouping of
variables captured in the parameter Z can be best under-
stood as a ratio of two time scales, the Rayleigh timescale
for breakup of an inviscid fluid jet, tR 
ffiffiffiffiffiffiffiffiffiffiffiffi
qd3=r
p
and the
viscocapillary time scale tvisc gd/r that characterizes the
thinning dynamics of a viscously dominated thread:14,15
Z ? tvisc
tR
? gd=rffiffiffiffiffiffiffiffiffiffiffiffi
qd3=r
p : (2)
The Ohnesorge number, thus, provides a ratio of how large
each of these timescales is for a fluid thread or jet of diame-
ter d, given knowledge of the fluid viscosity, density, and
surface tension. In typical jets (with d 1 mm) of low vis-
cosity fluids (such as water or aniline), the Ohnesorge num-
ber is very small, Z  1; in viscous liquids such as glycerin
or machine oils, the Ohnesorge number can exceed unity.
FIG. 3. ?Static? drop breakup associated with slow dripping of a viscous fluid from a nozzle. The nozzle diameter (d? 2r) is expressed in terms of the ratio r/
a? 0.52, where a ?
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
r=?qg?
p
is the capillary length (or ?Laplace constant? as Ohnesorge refers to it) of the dripping fluid stream. The sequence of image
frames plays from right to left as was customary in German hydrodynamic literature of the era. Reproduced from W. v. Ohnesorge, Anwendung eines kinema-
tographischen Hochfrequenzapparates mit mechanischer Regelung der Belichtung zur Aufnahme der Tropfenbildung und des Zerfalls flu?ssiger Strahlen. Copy-
rightVC 1937 by Konrad Triltsch (Abbildung 26, p. 66).
FIG. 4. The operating diagram devel-
oped by Ohnesorge in his thesis to distin-
guish between the critical conditions for
transition between different modes of
breakup for a cylindrical jet exiting from
an orifice. The dimensionless number or
Kennzahl on the ordinate axis is now
referred to as the Ohnesorge number.
Reprinted with permission from W. v.
Ohnesorge, Z. Angew. Math. Mech. 16,
355 (1936). Copyright 1936, Wiley
Interscience.
127101-3 Wolfgang von Ohnesorge Phys. Fluids 23, 127101 (2011)
 This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to  IP:
128.173.125.76 On: Wed, 20 Nov 2013 20:05:03
Another very useful way to consider the physical signifi-
cance of this grouping is to recognize that an appropriate
Reynolds number for self-similar breakup of a fluid thread
(in which there is no external forcing scale such as an
imposed jet velocity V) is to use the capillary thinning veloc-
ity Vcap d/tvisc as the relevant velocity scale. This leads to
an effective Reynolds number
Re ? q?r=g?d
g
? qrd
g2
? Z2: (3)
Typically, as the characteristic length scale reduces, this
effective Reynolds number decreases and viscous effects
become increasingly important and the self-thinning process
crosses over into a ?universal regime? in which surface ten-
sion, viscosity, and inertial effects are all equally impor-
tant.14,16 However for special choices such as liquid mercury,
inertially dominated pinchoff (corresponding to Z  1) can
be observed even at nanometer length scales.17 For further
discussion of the resulting similarity solutions that govern
breakup when Re  1 and Re  1, see Eggers.18
C. Subsequent confusion and rediscovery
Ohnesorge?s paper was published in one of the leading
mechanics journals of its time and also presented at the
GAMM conference in 1936. Nevertheless, the results were
not as widely appreciated or uniformly incorporated into the
broader literature as they might have been, and this has led
to some subsequent confusion and rediscovery. The reasons
for this may be related to the complex geopolitical issues of
the late 1930s.
A shortcoming of Ohnesorge?s paper is that he did not
provide a quantitative expression for the functional form
Z? Z(Re) that is immediately apparent from Figure 4.
Richardson19 nevertheless attributes one to him. However,
the expression given by Richardson, Z ? 2000Re4=3 for the
transition from regime I (Rayleigh breakup) to regime II
(screw symmetric breakup), is inconsistent with Ohnesorge?s
figure. A factor 200 gives a reasonable fit, so this could be a
typo. Becher20 notes the error and also the incorrect attribu-
tion to Ohnesorge. He comments that although the paper is
written in ?the turgid academic German of its period,?
Richardson should have noticed the absence of the equation
he attributes to it. This remark may be more of a sarcastic
jab at Richardson than an actual comment on Ohnesorge?s
paper. Becher?s paper does not give a corrected formula, but
a later erratum suggests Z ? 50Re4=3. This is also inconsis-
tent with Ohnesorge?s plot. Richardson and Becher are far
from alone. It has been shown in a recent study21 that the lit-
erature is confused by several other reports with wildly vary-
ing positioning of the transition lines, all inconsistent with
Ohnesorge?s plot, but frequently attributed to him. Close
inspection of the lines drawn by Ohnesorge in Figure 4
shows that they are actually not very well described by a
power-law slope of 4/3 but instead by an exponent closer
to 5/4. Fitting the data for the transition from region I to
region II, we obtain a numerical relationship Z ? 125Re5=4.
A qualitatively similar functional form governs the onset of
splashing in drop on demand printing applications as we dis-
cuss further below. Additional numerical confusion in
describing these boundaries can also easily arise depending
on the choice of radius or diameter d? 2r in the characteris-
tic scales for the Reynolds number and the Rayleigh time-
scale. Great care must be taken in comparing values from
different literature.
When looking up Ohnesorge?s work in Web of Science,
he suffers from a ?nobleman?s curse.? His paper4 appears in
the Web of Science separately under both the entry
?Vonohnesorge W? and ?Ohnesorge WV,? while the shorter
second paper6 appears under the entry ?von Ohnesorge W.?
As a humorous aside, ?Zerfall flu?ssiger Strahlen,? which
means ?breakup of liquid jets,? is translated by Web of Sci-
ence as ?decomposition of liquid irradiance.? It seems that
automatic translation software still needs a little fine tuning.
Overall, his 1936 ZAMM article has had approximately 90
citations since 1975.
The Laplace number is a quantity closely related to
the Ohnesorge number, specifically, La?Z2 (see Eq. (3)).
Laplace, in conjunction with Young, is honored as one of the
pioneers in the field of surface tension and capillary phenom-
ena;22,23 however, his work does not specifically involve
dimensionless constants. The Laplace number figures promi-
nently in Weber?s 1931 paper;12 for instance, the abscissa on
his Figure 14 is equal to 2/9 La (or, equivalently, 2/9 Z2).
Weber does not name this dimensionless combination or
introduce a symbol for it. Of course Weber?s name is already
attached to a different dimensionless number. In a very
widely cited work on liquid-liquid dispersion processes,
Hinze24 defines the same grouping of variables as Eq. (1)
simply as a ?viscosity group.?
In the broader literature, the Laplace number has also
been named the Suratman number. The review article of
Boucher and Alves25 cites Riley26 as the source. Riley?s pa-
per does not explain the origin of the term. Elsewhere in the
paper, Riley cites a 1955 publication,27 in which one of the
authors is named Suratman. Indeed, this is the only paper in
the field which we were able to find and which has an author
by that name. However, this paper does not introduce the
?Suratman number? per se. Compilations of dimensionless
variables, see, e.g., Refs. 25 and 28, now often give defini-
tions of both the Suratman number and Ohnesorge number.
Broadly speaking, it appears that in the jet breakup and
atomization literature, the name of Ohnesorge is more recog-
nized, whereas in the emulsification and droplet breakup lit-
erature, the Suratman name is more familiar.
D. Present day applications
In present usage, the dimensionless grouping given by
Eq. (1) is often referred to as the Ohnesorge number and
given the symbol Oh, rather than Z. It provides a convenient
way of capturing the relative magnitudes of inertial, viscous,
and capillary effects in any free surface fluid mechanics
problem. With the rapid explosion in drop-on-demand and
continuous ink-jet printing processes, understanding the
relative balance of time scales captured by the Ohnesorge
number Oh (or equivalently the relative magnitude of forces
127101-4 G. H. McKinley and M. Renardy Phys. Fluids 23, 127101 (2011)
 This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to  IP:
128.173.125.76 On: Wed, 20 Nov 2013 20:05:03
in the equation of motion given by Oh2) is of central impor-
tance in understanding the dynamic processes controlling
breakup as well as the shape and size of the droplets that are
formed.18
Of particular importance is the fact that the Ohnesorge
number is independent of the external forcing dynamics
(e.g., the flow rate Q or jet velocity V). It is solely a reflection
of the thermophysical properties of the fluid and the size of
the nozzle. Experiments with a given fluid and given geome-
try thus correspond to constant values of Oh, i.e., horizontal
trajectories through an operating space (which may be
unknown a priori) such as the one sketched originally by
Ohnesorge. By contrast, numerical simulations of such proc-
esses typically employ more familiar dynamical scalings in
terms of the Reynolds number and the Weber number, which
both vary with the dynamical forcing (V). Experiments and
simulations for a specific fluid thus correspond to fixed val-
ues of the ratio
ffiffiffiffiffiffi
We
p
=Re ? Oh (as recognized for example
by Kroesser and Middleman29). By keeping this ratio con-
stant, numerical simulations can be used to systematically
explore transitions between different drop pinchoff regimes,
drop sizes, and formation of satellite droplets; see, for exam-
ple Refs. 30 and 31. Successful operation of drop on demand
inkjet printing operations becomes increasingly difficult as
the fluid becomes more viscous, or the droplet size becomes
smaller so that the Ohnesorge number exceeds unity. The
Ohnesorge number also plays an important role in droplet
deposition processes when fluid droplets impact the substrate
on which they are being printed/deposited. A recent review
of this field has been presented by Derby32,41 and the existing
knowledge of relevant transitions in terms of critical values
of We, Re, and Oh can again be succinctly summarized in
terms of an operating diagram reminiscent of Ohnesorge?s
figure above (Fig. 4). If the axes are selected to be the Reyn-
olds number and the Ohnesorge number, this operating dia-
gram takes the form shown in Figure 5. In Ohnesorge?s
terms, the lower diagonal line marks the boundary between
regimes (0) and (I) (not plotted by Ohnesorge), while the
second diagonal line is of the same functional form as the
boundaries separating regimes (I), (II), and (III) in Fig. 4. In
this respect, both operating diagrams capture a number of the
key physical boundaries that constrain the operation of a par-
ticular commercial fluid dynamical process.
As inkjet printing processes become increasingly sophis-
ticated and fluids with more complex rheology (e.g., biologi-
cal materials, polymer solutions, or colloidal dispersions) are
deployed; additional dimensionless groupings must also
become important in fully defining the operating conditions
for a particular process. Following Ohnesorge?s lead, it makes
physical sense to isolate dynamical effects into a single
dimensionless variable (e.g., a Reynolds number, Re or if pre-
ferred a Weber number or capillary number) and then group
the remaining material properties in terms of ratios of rele-
vant time scales; for example, in drop pinchoff and jetting
of polymeric fluids, the ratio of the polymer relaxation time
to the Rayleigh time gives rise to a Deborah number
De ? k=
ffiffiffiffiffiffiffiffiffiffiffiffi
qd3=r
p
which plays an analogous role to the Ohne-
sorge number in controlling which dynamical processes dom-
inate the dripping and jetting of the fluid being considered.33
In this short note, we hope to have provided some inter-
esting historical background on Wolfgang von Ohnesorge
and also clarified his specific contributions to the research lit-
erature on atomization and jet breakup. The continuous
growth in the importance of inkjet printing processes34,35 in
a wide variety of commercial and manufacturing fields is
likely to keep his name relevant for the foreseeable future.
ACKNOWLEDGMENTS
We are grateful to Wolfgang?s children, Sigrid Vierck,
and Leopold von Ohnesorge for valuable information and for
the photographs. We thank Philip Threlfall-Holmes for valu-
able comments. Research in jetting and spraying at MIT is
supported by AkzoNobel Research Development and Inno-
vation, and research at VT is supported by the National Sci-
ence Foundation under Grant DMS-1008426.
1Genealogisches Handbuch des Adels: Adelige Ha?user B XVI (C. A. Starke,
Limburg 1985).
2Natalie B. Conkling. ?Conkling married to Baron Johannes L. von
Ohnesorge,? The New York Times, January 28 (1898).
3W. v. Ohnesorge, Anwendung eines kinematographischen Hochfrequen-
zapparates mit mechanischer Regelung der Belichtung zur Aufnahme der
Tropfenbildung und des Zerfalls flu?ssiger Strahlen (Konrad Triltsch,
Wu?rzburg, 1937).
4W. v. Ohnesorge, ?Die Bildung von Tropfen an Du?sen und die Auflo?sung
flu?ssiger Strahlen,? Z. Angew. Math. Mech. 16, 355 (1936).
5Willers, ?Nachrichten: Gesellschaft fu?r angewandte Mathematik und
Mechanik, Hauptversammlung in Dresden,? Z. Angew. Math. Mech. 16,
320 (1936).
6W. v. Ohnesorge, ?Die Bildung von Tropfen aus Du?sen beim Zerfall flu?ssiger
Strahlen,? Zeitschrift des Vereines deutscher Ingenieure. 81, 465 (1937).
7W. Threde and T. v. Bonin, Johanniter im Spannungsfeld an Weichsel und
Warthe (Posen-Westpreussische Genossenschaft des Johanniterordens, Ars
Una Verlags-Gesellschaft, Neuried, 1998), pp. 164?165.
FIG. 5. (Color online) A schematic diagram showing the operating regime
for stable operation of drop-on-demand inkjet printing. The diagram is
redrawn from Ref. 32 using the Ohnesorge number as the ordinate axis in
place of the Weber number We? (ReOh)2 . The criterion for a drop to pos-
sess sufficient kinetic energy to be ejected from the nozzle is given by Derby
as Wecrit 4 or Re 2/Oh. The criterion for onset of splashing following
impact is given by Derby32 as OhRe5/4 50.
127101-5 Wolfgang von Ohnesorge Phys. Fluids 23, 127101 (2011)
 This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to  IP:
128.173.125.76 On: Wed, 20 Nov 2013 20:05:03
8G. L. M. Strauss, Men who have Made the New German Empire (Tinsley
Brothers, London, 1875).
9W. Benz and B. Distel, Der Ort des Terrors: Geschichte der nationalsozia-
listischen Konzentrationslager (Beck Verlag, Munich, 2008), Vol. 8, p. 247.
10Lord Rayleigh, ?On the instability of jets,? Proc. London Math. Soc. 10, 4
(1879).
11Lord Rayleigh, ?On the capillary phenomena of jets,? Proc. R. Soc. Lon-
don Ser. A 29, 71 (1879).
12C. Weber, ?Zum Zerfall eines Flu?ssigkeitsstrahles,? Z. Angew. Math.
Mech. 11, 136 (1931).
13A. Haenlein, ?U?ber den Zerfall eines Flu?ssigkeitsstrahles,? Forsch. Ingen-
ieurwes. 2, 139 (1931).
14J. Eggers, ?Universal pinching of 3d axisymmetric free-surface flow,?
Phys. Rev. Lett. 71, 3458 (1993).
15D. T. Papageorgiou, ?On the breakup of viscous liquid threads,? Phys.
Fluids 7, 1529 (1995).
16M. P. Brenner, J. R. Lister, and H. A. Stone, ?Pinching threads, singular-
ities and the number 0.0304?,? Phys. Fluids 8, 2827 (1996).
17J. C. Burton, J. E. Rutledge, and P. Taborek, ?Fluid pinch-off dynamics at
nanometer length scales,? Phys. Rev. Lett. 92, 244505 (2004).
18J. Eggers and E. Villermaux, ?Physics of liquid jets,? Rep. Prog. Phys. 71,
036601 (2008).
19E. G. Richardson, ?The formation and flow of emulsion,? J. Colloid Sci. 5,
404 (1950).
20P. Becher, ?The so-called Ohnesorge equation,? J. Colloid. Interface. Sci.
140, 300 (1990) (Erratum: 147, 541 (1991)).
21P. Threlfall-Holmes, ?Spray Dryer Modelling,? Ph.D. dissertation (Heriot-
Watt University, 2009).
22P. S. de Laplace, Me?canique Celeste (Courcier, Paris, 1805), Vol. X
(supplement).
23T. Young, ?An essay on the cohesion of fluids,? Philos. Trans. R. Soc.
London 95, 65 (1805).
24J. O. Hinze, ?Fundamentals of the hydrodynamic mechanism of splitting
in dispersion processes,? AIChE J. 1, 289 (1955).
25D. F. Boucher and G. E. Alves, ?Dimensionless numbers,? Chem. Eng.
Prog. 55, 55 (1959) and 59, 75 (1963).
26D. J. Riley, ?Review: The thermodynamic and mechanical interaction of water
globules and steam in the wet steam turbine,? Int. J. Mech. Sci. 4, 447 (1962).
27P. van der Leeden, L. D. Nio, and P. C. Suratman, ?The velocity of free
falling droplets,? Appl. Sci. Res. 5, 338 (1955).
28J. P. Catchpole and G. Fulford, ?Dimensionless groups,? Ind. Eng. Chem.
58, 46 (1966).
29F. W. Kroesser and S. Middleman, ?Viscoelastic jet stability,? AIChE J.
15, 383 (1969).
30J. E. Fromm, ?Numerical calculation of the fluid dynamics of drop-on-
demand jets,? IBM J. Res. Dev. 28, 322 (1984).
31B. Ambravaneswaran, E. D. Wilkes, and O. A. Basaran, ?Drop formation
from a capillary tube: Comparison of one-dimensional and two-dimensional
analyses and occurrence of satellite drops,? Phys. Fluids 14, 2606 (2002).
32B. Derby, ?Inkjet printing of functional and structural materials: Fluid
property requirements, feature stability and resolution,? Annu. Rev. Mater.
Res. 40, 395 (2010).
33G. H. McKinley, ?Dimensionless groups for understanding free surface
flows of complex fluids,? SOR Bull. 74 (2), 6 (2005).
34O. Basaran, ?Small-scale free surface flows with breakup: Drop formation
and emerging applications,? AIChE J. 48, 1842 (2002).
35I. M. Hutchings, Ink-jet printing in micro-manufacturing: opportunities
and limitations, in 4M, ICOMM (Professional Engineering, Westminster,
2009), pp. 47?57.
36Feodor?s brother, Baron Johannes Leopold von Ohnesorge, was a lieuten-
ant in the German Army and married in New York City in 1898 to Miss
Natalie Conkling, the daughter of the pastor of the Rutgers Presbyterian
Church. After the wedding, Johannes returned with his bride (now Baron-
ess von Ohnesorge) to live in Saxe-Weimar, Germany. It does not appear
that Feodor joined him on the trip to New York. The announcement men-
tions that the engagement ring worn by the bride was a family heirloom
that had once been given to Blu?cher?s wife.
37The ancestry of the Bismarck family is extremely well documented. For
instance, the Genealogisches Handbuch des Adels (Ref. 1) provides genea-
logical information; however, the Bismarck family is scattered over sev-
eral volumes. A more convenient reference on the web was provided by
Brigitte Gastel Lloyd. The web site (http://worldroots.com/brigitte/fa-
mous/h/herbordbismarckdesc1280-index.htm) seems to be no longer
active but can still be accessed through the web archive (http://www.archi-
ve.org). The earliest known ancestor is Herbord von Bismarck
(1200?1280). Friedrich ?the Permutator? von Bismarck (1513?1589) got
his nickname because of a land trade with the Elector of Brandenburg
(Ref. 8). Crevese and Scho?nhausen were among the villages the Bismarck
family obtained in this trade. The Bismarcks living today are descendants
of one of two sons of Friedrich: Pantaleon (the ?Crevese? line) and
Ludolph (the ?Scho?nhausen? line). The most well-known Bismarck (Otto
von Bismarck) was from the Scho?nhausen line.
38This is the camp where Schindler recruited his work force.
39Or, more formally, ?The Bailiwick of Brandenburg of the Chivalric Order
of Saint John of the Hospital at Jerusalem.?
40The name of A. Haenlein is now largely forgotten. In 1931, he published
an extensive photographic study of jet breakup for a range of viscous fluids
which motivated the parallel theoretical analysis of Weber. These experi-
ments documented the dramatic decrease in the breakup length of a jet that
accompanies transition from Rayleigh mode breakup to screw-symmetric
or wavy perturbations in which aerodynamic effects play a significant role.
41In this review article (Ref. 32), Derby defines an Ohnesorge number as
Oh ? g=
ffiffiffiffiffiffiffiffi
qrd
p
(this is consistent with our Eq. (1), but he then also
denotes this as Oh? 1/Z (resulting in his parameter Z being the inverse of
the original definition of Ohnesorge). The source of this additional confu-
sion is attributed to Fromm (Ref. 30), but in fact Fromm does not introduce
any dimensionless parameter Z or Oh, but presents his results directly in
terms of ratios of the square root of the Weber number and Reynolds
number.
127101-6 G. H. McKinley and M. Renardy Phys. Fluids 23, 127101 (2011)
 This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to  IP:
128.173.125.76 On: Wed, 20 Nov 2013 20:05:03