The equations of balance of momentum and energy usually are formulated under the assumption of conservation of mass. However, mass is not conserved when sources of mass are present or when the equations of balance are applied to a non-material volume. Mass then is said to be variable for the system under consideration. It is the scope of the present contribution to review the mechanical equations of balance for variable-mass systems. Our review remains within the framework of the classical, non-relativistic continuum mechanics of solids and fluids. We present general formulations and refer to various fields of applications, such as astronomy, machine dynamics, biomechanics, rocketry, or fluid dynamics. Also discussed are the equations for a single constituent of a multiphase mixture. The present review thus might be of interest to workers in the field of heterogeneous media as well. We first summarize the general balance law and review the Reynolds transport theorem for a non-material volume. Then the latter general formulations are used to derive and to review the equations of balance of mass and linear momentum in the presence of sources of mass in the interior of a material volume. We also discuss the appropriate modeling of such sources of mass. Subsequently, we treat the equations of balance of mass and linear momentum when mass is flowing through the surface of a non-material volume in the absence of sources of mass in the interior, and we point out some analogies to the previously presented relations. A strong emphasis is given in this article to the historical evolution of the balance equations and the physical situations to which they have been applied. The equations of balance of angular momentum and energy for variable-mass systems will be treated separately, in a second part of the review, to be published later. This review article cites 96 references.

1.
Pauli W (2000), Relativita¨tstheorie, Neu herausgegeben und kommentiert von D Giulini, Springer, Berlin (First published in Encyklopa¨die der Mathematischen Wissenschaften, Vol V, A Sommerfeld (ed), BG Teubner, Leipzig, 1921).
2.
Shames IH (1992), Mechanics of Fluids, 3rd Edition, McGraw-Hill, New York (first edition published in 1962).
3.
White FM (1999), Fluid dynamics, Fundamentals of Fluid Mechanics, JA Schetz and AE Fuhs (eds), John Wiley, New York, 8–32.
4.
Truesdell C and Toupin R (1960), The classical field theories, Handbuch der Physik, Band III/1: Prinzipien der Klassischen Mechanik und Feldtheorie, S Flu¨gge (Hsg), Berlin, Springer-Verlag, 226–793.
5.
Reynolds O (1903), The Sub-Mechanics of the Universe, Cambridge Univ Press, distributed for the Royal Society in London, Cambridge.
6.
Hamel G (1927), Die Axiome der Mechanik, Handbuch der Physik, Band V: Grundlagen der Mechanik, Mechanik der Punkte und starren Ko¨rper, H Geiger and K Scheel (eds), Verlag von Julius Springer, Berlin, 1–41.
7.
Prigogine I (1947), Etude Thermodynamique des Phenomenes Irreversibles, Dunod-Desoer, Paris.
8.
H
(
1952
),
Statistical mechanics, thermodynamics, and fluid dynamics of systems with an arbitrary number of integrals
,
Comm. Pure Appl. Math.
5
,
455
494
.
9.
Ericksen JL (1960), Tensor fields, Handbuch der Physik, Band III/1: Prinzipien der Klassischen Mechanik und Feldtheorie, S Flu¨gge (Hsg), Berlin, Springer-Verlag, 794–858.
10.
Dugas R (1988), A History of Mechanics, Dover, New York (first published in 1955).
11.
Warsi ZUA (1999), Fluid Dynamics, Theoretical and Computational Approaches, 2nd Edition, CRC Press, Boca Raton (first edition published in 1993).
12.
Ziegler F (1998), Mechanics of Solids and Fluids, 2nd English Edition, corrected 2nd printing, Springer, New York (first published as Technische Mechanik der festen und flu¨ssigen Ko¨rper, Springer-Verlag, Vienna, 1985).
13.
Jaumann G (1905), Die Grundlagen der Bewegungslehre von einem modernen Standpunkte aus, JA Barth, Leipzig.
14.
Spielrein J (1916), Lehrbuch der Vektorrechnung nach den Bedu¨rfnissen in der Technischen Mechanik und Elektrizita¨tslehre, Verlag von K Wittwer, Stuttgart (second edition 1926).
15.
Thompson PA (1972), Compressible-Fluid Dynamics, McGraw-Hill, New York.
16.
Ziegler
F
(
1998
),
Didaktische Aspekte in mechanischen Erhaltungssa¨tzen
,
Mitt. Ges. Angew. Math. Mech.
1
,
61
72
.
17.
Thomas
TY
(
1949
),
The fundamental hydrodynamical equations and shock conditions for gases
,
Math. Mag.
22
,
169
189
.
18.
Maugin GA (1988), Continuum Mechanics of Electromagnetic Solids, Amsterdam, North-Holland.
19.
Bednarczyk
H
(
1967
),
Zur Gestalt der Grundgleichungen der Kontuinuumsmechanik an Unstetigkeitsfla¨chen
,
Acta Mech.
4
,
122
127
.
20.
Bednarczyk
H
(
1968
),
Einige dynamische Kompatibilita¨tsbedingungen bei der Wechselwirkung senkrechter Unstetigkeitsfla¨chen
,
Acta Mech.
6
,
117
139
.
21.
Kluwick
A
(
1971
),
Zur Ausbreitung schwacher Sto¨ße in dreidimensionalen instationa¨ren Stro¨mungen
,
Z. Angew. Math. Mech.
51
,
225
232
.
22.
Kelly
PD
(
1964
),
A reacting continuum
,
Int. J. Eng. Sci.
2
,
129
153
.
23.
Eringen
AC
and
Ingram
JD
(
1965
),
A continuum theory of chemically reacting media—I
,
Int. J. Eng. Sci.
3
,
197
212
.
24.
Hutter
K
,
Jo¨hnk
K
, and
Svendsen
R
(
1994
),
On interfacial transition conditions in two phase gravity flow
,
Z. Angew. Math. Phys.
45
,
746
762
.
25.
Morland
LW
, and
Sellers
S
(
2001
),
Multiphase mixtures and singular surfaces
,
Int. J. Non-Linear Mech.
36
,
131
146
.
26.
Irschik
H
(
2003
),
On the necessity of surface growth terms for the consistency of jump relations at a singular surface
,
Acta Mech.
162
,
195
211
.
27.
Tait PG (1895), Dynamics, Adam and Charles Black, London.
28.
Ambrosi
D
, and
Mollica
F
(
2002
),
On the mechanics of a growing tumor
,
Int. J. Eng. Sci.
40
,
1297
1316
.
29.
Meshchersky IV (1949), Works on the Mechanics of Bodies with Variable Mass (in Russian), with an introduction by AA Kosmodemyansky, GITTL, Moscow, Leningrad.
30.
Routh EJ (1960), The Elementary Part of a Treatise on the Dynamics of a System of Rigid Bodies, Dover, New York (first edition published in 1905).
31.
Poeschl Th (1927), Technische Anwendungen der Stereomechanik, Handbuch der Physik, Band V: Grundlagen der Mechanik, Mechanik der Punkte und starren Ko¨rper, H Geiger and K Scheel (eds), Verlag von Julius Springer, Berlin, 484–577.
32.
J
(
1967
),
Secular variation of mass and the evolution of binary systems
,
5
,
131
188
.
33.
Plastino
AR
, and
Muzzio
JC
(
1992
),
On the use and abuse of Newton’s second law for variable mass problems
,
Celestial Mechanics and Dynamical Astronomy
53
,
227
232
.
34.
Cveticanin L (1998), Dynamics of Machines with Variable Mass, Gordon and Breach Science Publishers, Amsterdam.
35.
Lamb H (1997), Hydrodynamics, Cambridge Univ Press (reprint of the 6th edition 1932, first published as Treatise on the Mathematical Theory of the Motion of Fluids in 1879).
36.
Oswatitsch K (1959), Physikalische Grundlagen der Stro¨mungslehre, Handbuch der Physik, S Flu¨gge (ed), Band VIII/1: Stro¨mungsmechanik I, C Truesdell (ed), Springer-Verlag, Berlin, 1–124.
37.
Truesdell
C
(
1957
),
Sulle basi della termomeccanica
,
Atti Accad. Naz. Lincei, Cl. Sci. Fis., Mat. Nat., Rend.
22
,
33
38
, 158–166.
38.
Lagally M (1927), Ideale Flu¨ssigkeiten, Handbuch der Physik, Vol VII: Mechanik der flu¨ssigen und gasfo¨rmigen Ko¨rper, H Geiger and K Scheel (eds), Verlag von Julius Springer, Berlin, 1–91.
39.
Arrighi
G
(
1933
),
Una generalizzazione dell’ equazione di continuita
,
Atti Accad. Naz. Lincei, Cl. Sci. Fis., Mat. Nat., Rend.
XVIII
,
302
307
.
40.
Lubarda
VA
and
Hoger
A
(
2002
),
On the mechanics of solids with a growing mass
,
Int. J. Solids Struct.
39
,
4672
4664
.
41.
Stefan
J
(
1871
),
U¨ber das Gleichgewicht und die Bewegung, insbesondere die Diffusion von Gasmengen
,
Sitzungsberichte Akademie der Wissenschaften in Wien
63.2
,
63
124
.
42.
de Boer R (2000), Theory of Porous Media: Highlights in the Historical Development and Current State, Springer-Verlag, Berlin.
43.
de Boer
R
(
2000
),
Contemporary progress in porous media theory
,
Appl. Mech. Rev.
53
,
323
369
.
44.
Morland
LW
(
1992
),
Flow of viscous fluid through a porous matrix
,
Surv. Geophys.
13
,
209
268
.
45.
Levi-Civita
T
(
1928
),
Sul moto di un corpo di massa variabile
,
Atti Accad. Naz. Lincei, Cl. Sci. Fis., Mat. Nat., Rend.
VIII
,
329
333
.
46.
Levi-Civita
T
(
1928
),
Aggiunta alla Nota: Sul moto di un corpo di massa variabile
Atti Accad. Naz. Lincei, Cl. Sci. Fis., Mat. Nat., Rend.
VIII
,
621
622
.
47.
Gylden
H
(
1884
),
Die Bahnbewegung in einem Systeme von zwei Ko¨rpern in dem Falle, dass die Massen Vera¨nderungen unterworfen sind
,
Astron. Nachr.
109
,
2593
94
.
48.
Seeliger H (1890), U¨ber Zusammensto¨sse und Theilungen planetarischer Massen, Abh der Ko¨nigl Bayer Akademie der Wiss, Cl II, Bd XVII, Abth II, 459–490.
49.
Szabo I (1987), Geschichte der mechanischen Prinzipien und ihrer wichtigsten Anwendungen, 3rd Edition, Birkha¨user Verlag (first edition published in 1977).
50.
Holl
HJ
,
Belyaev
AK
, and
Irschik
H
(
1999
),
Simulation of the Duffing-oscillator with time-varying mass by a BEM in time
,
Comput. Struct.
73
,
177
186
.
51.
Messerschmid E and Fasoulas S (2000), Raumfahrtsysteme, Springer, Berlin.
52.
Routh EJ (1960), A Treatise on Dynamics of a Particle, Dover, New York (first edition published in 1898).
53.
Cayley
A
(
1857
),
On a class of dynamical problems
,
Proc. R. Soc. London
VIII
,
506
511
.
54.
Irschik H and Cojocaru (2003), Continuum Mechanics Based Formulations for the Cayley Class of Continuous-Impact Dynamical Problems, [in publication].
55.
Wittenbauer
F
(
1905
),
Die Bewegungsgesetze der vera¨nderlichen Masse
,
Z. Angew. Math. Phys.
52
,
150
164
.
56.
Jose JV and Saletan EJ (1998), Classical Dynamics: A Contemprorary Approach, Cambridge Univ Press, Cambridge.
57.
Synge JL (1960), Classical dynamics, Handbuch der Physik, Band III/1: Prinzipien der Klassischen Mechanik und Feldtheorie, S Flu¨gge (Hsg) Berlin, Springer-Verlag, 1–225.
58.
Agostinelli
C
(
1935
–1936),
Sui sistemi di masse variabili
,
Atti Accad. Naz. Lincei, Cl. Sci. Fis., Mat. Nat., Rend.
71
,
254
272
.
59.
Pars LA (1965), A Treatise on Analytical Dynamics, Heinemann, London.
60.
Ge
Z-M
(
1984
),
The equations of motion of nonlinear nonholonomic variable mass system with applications
,
ASME J. Appl. Mech.
51
,
435
437
.
61.
Luo
S-K
and
Mei
F-X
(
1992
),
The principles of least action of variable mass nonholonomic nonconservative system in noninertial reference frames
,
Appl. Math. Mech.
13
,
851
859
.
62.
Ge
Z-M
and
Cheng
Y-H
(
1982
),
Extended Kane’s equations for non-holonomic variable mass systems
,
ASME J. Appl. Mech.
49
,
429
431
.
63.
Zhang
Y
and
Qiao
Y
(
1995
),
Kane’s equations for percussion motion of variable mass nonholonomic mechanical systems
,
Appl. Math. Mech.
16
,
839
850
.
64.
Musicki
D
(
1999
),
General energy change law for systems with variable mass
,
Eur. J. Mech. A/Solids
18
,
719
730
.
65.
Federhofer K (1922), Dynamik sich a¨ndernder Massen, Mitteilungen des Deutschen Ingenieur-Vereines in Ma¨hren, Hauptvereines deutscher Ingenieure in der tschechoslow Republik 11(H.6), 83–86, (H.7), 115–118.
66.
Serrin J (1959), Mathematical principles of classical fluid mechanics, Handbuch der Physik, S Flu¨gge (ed), Band VIII/1: Stro¨mungsmechanik I, C Truesdell (ed), Springer-Verlag, Berlin, 125–263.
67.
Prandtl L and Tietjens O (1929), Hydro- und Aeromechanik, Vol 1, Verlag von Julius Springer, Berlin (English Edition published as Fundamentals of Hydro- and Aeromechanics, McGraw-Hill, New York, 1934).
68.
Oswatitsch K (1952), Gasdynamik, Springer-Verlag, Wien.
69.
Oswatitsch K (1976), Grundlagen der Gasdynamik, Springer-Verlag, Wien.
70.
Cisotti
U
(
1917
),
Sulle azione dinamiche di masse fluide continue
,
Rend Lombardo
50
,
502
515
.
71.
von Mises R (1908), Theorie der Wasserra¨der, BG Teubner, Leipzig (reprinted from Zeitschrift fu¨r Mathematik und Physik, 57).
72.
Mu¨ller
W
(
1933
),
U¨ber den Impulssatz der Hydromechanik fu¨r bewegte Gefa¨ßwa¨nde und die Berechnung der Reaktionskra¨fte der Flu¨ssigkeit
,
Ann. Phys. (Paris)
5
(
16
),
489
512
.
73.
Rosser JB, Newton RR, and Gross GL (1947), Mathematical Theory of Rocket Flight, McGraw-Hill, New York.
74.
Rankin
RA
(
1948
),
The mathematical theory of the motion of rotated and unrotated rockets
,
Philos. Trans. R. Soc. London
241
,
457
538
.
75.
Gantmakher FR and Levin LM (1964), The Flight of Uncontrolled Rockets, Macmillan, New York.
76.
Meirowitch
L
(
1970
),
General motion of a variable mass flexible rocket with internal flow
,
J. Spacecr. Rockets
7
,
186
195
.
77.
Meirovitch L (1970), Methods of Analytical Dynamics, McGraw-Hill, New York.
78.
Djerassi
S
(
1998
),
Algorithm for simulation of motions of variable mass systems
,
J. Guid. Control Dyn.
21
,
427
434
.
79.
Grubin
C
(
1963
),
Mechanics of variable mass systems
,
J. Franklin Inst.
276
,
305
312
.
80.
81.
Eke
FO
and
Wang
S-M
(
1994
),
Equations of motion of two-phase variable mass systems with solid base
,
ASME J. Appl. Mech.
61
,
855
860
.
82.
Thorpe
JF
(
1962
),
On the momentum theorem for a continuous system of variable mass
,
Am. J. Phys.
30
,
637
640
.
83.
Leitman
G
(
1957
),
On the equation of rocket motion
,
J. Br. Interplanet. Soc.
16
,
141
147
.
84.
Thomson
WT
(
1966
),
Equations of motion of the variable mass system
,
AIAA J.
4
,
766
768
.
85.
Belknap
SB
(
1972
),
A general transport rule for variable mass dynamics
,
AIAA J.
10
,
1137
1138
(Full paper referenced as N72-21570, Natl Tech Information Service, VA).
86.
Kapoulitsas
G
(
1987
),
The mass-centre motion of a continuously variable system of particles, Part 2
,
Ingenieur-Archiv
57
,
91
98
.
87.
Parkus H (1995), Mechanik der festen Ko¨rper, 2nd Edition, 5th reprint, Springer-Verlag, Wien (first edition published in 1960).
88.
Riemer M (1993), Technische Kontinuumsmechanik, synthetische und analytische Darstellung, BI Wissenschaftsverlag, Mannheim.
89.
Wauer J (1976), Querschwingungen bewegter eindimensionaler Kontinua vera¨nderlicher La¨nge, Fortschritt-Berichte der VDI Zeitschriften, Reihe 11, Nr 26, VDI-Verlag, Du¨sseldorf.
90.
Riemer
M
and
Wauer
J
(
1988
),
Zur Behandlung von Schubgelenken in Mehrko¨rpersystemen mit verformbaren Teilstrukturen
,
Z. Angew. Math. Mech.
68
,
T111–T113
T111–T113
.
91.
Hamel G (1949), Theoretische Mechanik, Vol LVII of “Die Grundlehren der Mathematischen Wissenschaften,” W Blaschke et al. (eds), Springer-Verlag, Berlin.
92.
Szabo I (1975), Einfu¨hrung in die Techische Mechanik, 8th Edition, Springer-Verlag, Berlin (first edition published in 1957).
93.
Steiner
W
and
Troger
H
(
1995
),
On the equations of motion of an inextensible string
,
Z. Angew. Math. Phys.
46
,
960
979
.
94.
Crellin
EB
,
Janssens
F
,
Poelaert
D
,
Steiner
W
, and
Troger
H
(
1997
),
On balance and variational formulations of the equation of motion of a body deploying along a cable
,
ASME J. Appl. Mech.
64
,
369
374
.
95.
Irschik
H
and
Holl
H
(
2002
),
The equations of Lagrange written for a non-material volume
,
Acta Mech.
153
,
231
248
.
96.
Kluwick A (2000), Zur Bedeutung der Prandtl’schen Untersuchungen u¨ber die dissipative Struktur von Verdichtungssto¨ßen, Ludwig Prandtl, ein Fu¨hrer durch die Stro¨mungslehre, GEA Meier (ed), Vieweg, Braunschweig, 139–146.