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The physics of welding
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1984 Physics in Technology 15 73
(http://iopscience.iop.org/0305-4624/15/2/I05)
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Phys. Technol.. Vol. 15. 1984. Printed in Northern Ireland 
THE PHYSICS OF WELDING 
J F Lancaster 
Greater understanding of the physics of 
welding is leading to improved application 
and control of welding processes. Further 
gains in welding productivity could follow 
030~624/84/020073+07$02.25 0 1984 The Institute of Physlcs 
Welding is an ancient art, and has been practised 
ever since man first learned to extract and refine 
iron. Until about the beginning of this century, the 
method of welding was the same as that used in 
Roman times, and still employed in the black- 
smith's forge today. The two pieces of metal to be 
joined are heated and then hammered or pressed 
together, so as to squeeze out slag and oxide and 
allow the surfaces to bond together. This is forge 
welding, and is an example of a solid-phase 
welding process. 
The alternative technique is fusion welding. Here 
the edges of the two pieces of metal to be joined 
are melted and fused together. In order to melt the 
metal locally in this way an intense heat source is 
required, and it is largely in the provision and use 
of such energy sources that physical problems arise. 
In the case of a surface heat source the minimum 
rate of energy release per unit area q" required to 
maintain a molten weld pool of radius r is 
approximately 
q" = AkT,/r 
where k is thermal conductivity, T, is melting 
temperature and A is a factor dependent on 
welding speed, weld size and thermal diffusivity. 
The weld pool size is limited by practical 
considerations: it must be manageable, on the one 
hand, and it must be large enough to fuse the edges 
of a weld preparation, on the other. Manual welds 
in steel are usually 10-20" in width giving a 
required power density of the order of lo7 W m-'. 
The electric arc, which is the most generally used 
heat source in fusion welding, generates such a 
power density. Figure 1 shows the energy densities 
for various types of welding heat source. 
The temperature distribution relative to a point 
heat source of power q on the surface of a 
semi-infinite medium moving at velocity U in the x 
direction is 
73 
1013 
10 Ion 1' t Vaprisptior conductton and meltin (keyholing7 
Radial 
mduction 
dominated 
with meliing 
Arc processes 
Figure 1 Power density for various welding processes 
(from Lancaster 1984) 
T = (qi2;rkr) exp [- u(r - x)/2cu] 
where 2 = x2 + y 2 + z2 and (Y is thermal diffusivity. 
When T = T, and z = 0 this equation gives the 
theoretical boundary of the molten weld pool. 
Measurements of weld pool width and length have 
been made over a wide range of heat input rates 
and for different materials and processes (Christen- 
sen et a1 1965). Assuming that the thermal 
properties of the liquid metal are the same as for 
the solid, the calculated width and length of the 
pool are, in almost every case, smaller than the 
measured values. This is consistent with a high 
effective thermal conductivity in the liquid metal, 
and indicates that (as would be expected) heat 
transfer in the weld pool is partly convectional. In 
general, however, the simple formulae for tempera- 
ture distribution that have been derived assuming 
point or line sources of heat, and ignoring 
variations of thermal properties with temperature, 
agree reasonably well with the results obtained by 
experiment. 
The manner in which heat is transferred to 
the weld pool depends on the character of the 
welding process. Most arc welding is done with a 
consumable electrode, so that heat generated at 
both poles of the arc is absorbed by the workpiece. 
When a non-consumable electrode is used, and the 
workpiece forms the positive electrode of the arc, 
heat is generated by the condensation of electrons 
and by conduction and radiation from the arc 
plasma across the boundary layer at the metal 
surface. If the workpiece is the negative pole, then 
a non-thermionic cathode is formed. The mechan- 
ism of such non-thermionic cathodes, which is 
described later. implies that there is substantial 
power generation in a thin layer (1 nm to 1 km) at 
the surface. Part of this power is absorbed in the 
evaporation of electrons and in ionisation, but 
some is transferred to the metal. Normally the heat 
input per ampere of arc current is greater at the 
positive electrode, where the electrons give up their 
heat of condensation, than at the negative 
electrode, where heat is absorbed by electron 
emission. 
The electric arc in welding: the cathode 
The tungsten inert gas (TIG) welding process 
employs a thoriated or zirconiated non-consumable 
tungsten electrode, and the arc operates inside an 
argon or. less frequently, a helium gas shield. 
Except for welding aluminium, the electrode forms 
the cathodic pole of the arc. Various cathode 
modes have been described, but in most cases 
welding is performed with a pointed electrode at 
the tip of which a well defined thermionic cathode 
spot is formed. The current density of the cathode 
does not vary much with electrode material (e.g. 
zirconium instead of tungsten) and is probably 
governed by conditions in the arc column. This is 
fortunate since with pure tungsten electrodes the 
tip of the electrode melts to form a small sphere. 
and at low currents the cathode spot wanders over 
the surface of the sphere, causing the arc column 
to wander and the welder to lose control. The 
addition of small amounts of zirconia or thoria to 
the tungsten reduces the work function. and allows 
thermionic emission at the required current density 
to occur at a lower temperature. For thoriated and 
zirconiated material this temperature is below the 
melting point, and the geometry of the electrode tip 
is maintained, thus stabilising the arc. 
Other metals that are encountered as electrodes 
in welding. such as steel and aluminium, boil at 
temperatures below that required for a sufficiently 
high thermionic current density level. The cathode 
that forms on these metals is not fixed in any one 
position, but moves rapidly over the surface in a 
random fashion. Understanding of the mechanism 
and behaviour of such non-thermionic cathodes has 
made a significant advance following recent work 
by Guile (1979). There are at least three types of 
non-thermionic cathode: the vapour type, which 
forms on unfilmed metal; the tunnelling type, which 
forms on metal having a thin oxide film (less than 
about 10 nm); and the switching type, which forms 
on thicker oxide films. Guile suggests that positive 
ions originating from the arc plasma condense on 
the oxide surface and set up a high electric field. In 
the case of thin films electrons may 'tunnel' through 
the film and generate an emitting site; for thicker 
74 
films a phenomenon known as switching makes the 
film locally conductive. Such mechanisms allow 
relatively large currents to flow in filamentary 
channels through the oxide. Individual emitting 
sites are 1 nm to 1 pm in diameter, and have a 
lifetime of 1 ns to 1 ps. Examination of the cathodic 
afea by the scanning electron microscope after 
arcing for a short duration such as 1 ps shows a 
pattern of craters, as in figure 2. Thus, the 
non-thermionic cathode operates by the formation 
and decay of numbers of small emitting sites, and 
removes the oxide film from thecathodic area. 
This effect is put to good use in the TIG welding 
of aluminium. Aluminium oxide persists as a solid 
film after the metal has melted, and may cause 
discontinuities in the completed weld. Welding with 
the workpiece negative results in the oxide being 
stripped by cathodic action, and the weld is free 
from oxide film. In practice alternating current is 
used to avoid overheating the electrode. 
In other cases the formation of a thick oxide film 
is employed to control the extent of movement 
of the cathode spot. Metal inert gas (MIG) welding, 
in which a consuniable bare wire electrode is 
protected by an inert gas shield, can normally be 
operated with electrode positive only. With 
electrode negative the cathode wanders up and 
down the electrode, making the process uncontroll- 
able. Coating the electrode surface with a relatively 
thick oxide, however, confines the arc root to the 
tip of the rod, giving a symmetrical and controllable 
arc. In welding steel by the same process with 
electrode positive and a pure argon gas shield, the 
cathode spots may wander over the plate to an 
extent that makes the process unstable. Adding a 
few per cent oxygen to the argon forms a thicker 
oxide film on the metal surface and restricts 
movement of the cathode to an acceptable degree. 
The arc column 
The column or gaseous portion of the electric arc is 
characterised by two features: high temperature, 
such that the gas is sufficiently ionised to be 
conductive; and high flow velocity, the direction of 
which is, under welding conditions, from electrode 
to workpiece. The temperature is maintained by 
ohmic heating, which balances losses by conduc- 
tion, convection and radiation. The proportion of 
energy lost from a TIG arc by radiation increases 
with current and at 100A is about 20% of the 
total column energy. The relative importance of 
conduction and convection may be assessed from 
the Peclet number Pe 
Pe = puLC,/k 
where p is density, U is flow velocity, L is a typical 
dimension, C,, is specific heat and k thermal 
Figure 2 Mild steel cathode with 2.5 nm oxide film. 
Surface damage caused by a 4.5 A arc of 30 ns duration. 
Magnification X3000 (photograph courtesy A E Guile) 
conductivity. For the TIG arc at atmospheric 
pressure Pe is about 10, at which value convection 
dominates. At low pressures Pe may fall below 1, 
where heat flow is primarily by conduction and the 
arc column becomes spherical in form, whilst at 
high pressures Pe increases above the atmospheric 
value. 
Most of the measurements of temperature 
distribution have been for the argon-shielded TIG 
arc, illustrated in figure 3. The visible boundary 
represents an isotherm, probably about 1 x lo4 K. 
The measured temperature level varies quite 
significantly from one observer to another; earlier 
investigators found temperatures of about 2 x lo4 K 
near the cathode, whereas others obtained values of 
1 X 104K in the same location. Arcs between iron 
electrodes have a column temperature of about 
6 X lo3 K, presumably because of the higher 
conductivity of iron vapour at lower temperatures. 
Mass flow in the arc column 
Mass flow in the arc plasma may result from 
chemical reactions, such as the breakdown of an 
electrode coating, or it may be externally imposed, 
as in plasma welding. The plasma torch is similar to 
that used in TIG welding, but with a constricted 
nozzle so as to direct a jet of hot plasma on to the 
metal surface. The primary interest here, however, 
lies with the electromagnetically induced jets that 
are observed in TIG and MIG welding. 
In virtually all arc welding operations the current 
flow is between a point-like electrode and an 
approximately flat plate. The current streamlines 
therefore spread outwards from the electrode. 
Because of this configuration, the interaction of the 
current and its self-induced magnetic field results in 
forces that induce flow from the electrode towards 
the plate. The flow is jet-like and the axial velocity 
is of the order of hundreds of metres a second. 
Increasing the ambient pressure (as may occur in 
75 
underwater welding) causes the jet to become more 
intense, and vice versa. 
Calculations of mass and heat flow have been 
attempted for simple cases like the TIG arc. A 
complete analysis requires the simultaneous solu- 
tion of the equations for conservation of mass, 
energy, momentum and electric charge, together 
with Ohm’s law and Maxwell’s equations for 
magnetic fields. Such an analysis is possible using 
numerical methods, and promising results have 
been obtained (Lancaster 1984). 
Axial flow in the welding arc column is desirable 
in two ways. In using coated electrodes the gas flow 
produced by decomposition of the coating protects 
the molten metal from contamination by atmos- 
pheric nitrogen and oxygen. In TIG and MIG 
welding the electromagnetically induced flow gives 
the arc the quality of ‘stiffness’; it may be directed 
as required and is resistant to deflection by external 
forces such as stray magnetic fields. The flow is 
converted to a stagnation pressure where it 
impinges on the weld pool. This pressure generates 
the ‘arc force’, which may have useful effects in 
ensuring good penetration of the molten weld into 
the workpiece. On the other hand, if the flow, and 
the resulting arc force, are too high, instabilities 
may occur and in extreme cases the molten metal 
may be blown out of its proper location. 
Metal transfer 
In welding with a consumable electrode, the 
electrode is at one and tlie same time a conductor 
for the arc current, thereby providing a heat 
source, and a source of liquid filler metal for 
the joint to be welded. It is essential for good 
welding that the major part of this liquid metal is 
transferred to the weld pool, and not dispersed as 
spatter over the surrounding plate 
There are two ways in which a smooth transfer 
may be effected. Where the flight path of droplets 
detaching from the electrode tip is erratic, as in gas 
metal arc (GMA)~ welding with a CO2 shield, the 
arc is kept short and the drop contacts the weld 
pool before it detaches, causing a short circuit until 
the liquid metal is drawn into the pool by surface 
tension and electromagnetic forces. When the flight 
of such droplets is directed in line with the 
electrode axis, on the other hand, it is possible to 
operate in a ‘free flight’ mode. This is the case with 
argon-shielded gas metal arc welding. We will here 
be primarily concerned with the free flight mode of 
metal transfer. 
t GMA welding is similar to MIG, but the shielding gas 
may be wholly or in part chemically active. Likewise 
gas tungsten arc (GTA) may employ a chemically active 
gas shield. 
76 
Figure 3 Argon-shielded arc with tungsten cathode. The 
arc column is typically bell-shaped (photograph courtesy 
The Welding Institute) 
Details of the transfer process are visible in high 
speed motion pictures of the arc, which may if 
necessary by correlated with oscillographic records 
of arc current and voltage. Transfer from coated 
electrodes has been examined by radiography, and 
in this way the movement of metal may be 
distinguished from that of the slag. When the 
electrodes are fully deoxidised, a depression forms 
at the root of the arc, distorting the drop at the 
electrode tip and eventually resulting in detach- 
ment either by short circuit or by the pinching-off 
of droplets. In electrodes that are not fully 
deoxidised a bubble of CO forms inside the drop; 
eventually the bubble bursts and a spray of fine 
drops is projected towards the electrode. 
Transfer from steel electrodes in gas metal arc 
welding with argon, argon-oxygen or argon-COz 
shielding is altogether more regular in character. 
At low currents (below 200A with a 1.2”diameter wire) the drops form as oblate spheroids, 
elongated in line with the axis of the electrode. 
These detach with an initial velocity and accelera- 
tion at fairly regular intervals. Above 200A a 
conical tip appears at the end of the electrode, and 
droplets form and detach - again in a regular 
fashion - from the tip of the cone. At still higher 
currents (about 250 A) the conical tip transforms 
into a relatively long cylinder of liquid metal from 
the end of which a stream of fine drops is projected 
(streaming transfer). Further increase of current 
causes the cylinder to transform into a rotating 
spiral (rotating transfer). Applying a longitudinal 
magnetic field to streaming transfer causes the 
transition to a rotating spiral to occur at a lower 
current (Lancaster 1984). 
Metals of higher thermal and electrical conduc- 
tivity, such as aluminium and copper, do not show 
the same transitions of metal transfer mode as 
steel. Metal is detached in the form of drops, as for 
steel below 200A. Typical figures for the rate of 
drop detachment at 200 A are 10 dropsis for steel, 
20dropsls for copper and 170drops/s for alumin- 
ium. 
The regular behaviour of transferring drops in 
gas metal arc welding has encouraged various 
investigators to attempt a quantitative analysis of 
the phenomenon. One method of approach has 
been to assess the forces to which the drop at the 
electrode tip is subject. Those tending to detach the 
drop are gravity (assuming downward welding). the 
drag force due to the shielding gas flow, and 
(usually) the electromagnetic force, whilst surface 
tension acts in the opposite sense. These forces 
have been measured for steel in the range 0-220 A . 
The electromagnetic force on the drop at the tip of 
a cylindrical electrode may be calculated assuming 
that there is no internal flow. The magnitude and 
direction of the force so calculated depends on the 
relationship between the diameter of the electrode 
and that of the anode spot. If the diameter of the 
anode spot is smaller than that of the electrode, 
then the force acts towards the electrode: when it is 
larger it acts in the opposite direction. It was found 
that at low currents, corresponding to a small 
anode spot size, the electromagnetic force did 
indeed act towards the electrode, whilst at higher 
values up to 160A there was good agreement 
between calculated and measured values (Waszink 
and Graat 1983). 
The dynamics of metal transfer have also been 
explored using the linear approximation employed 
by Lord Rayleigh (1879) for investigating the 
stability of a liquid cylinder or jet. This analysis was 
extended to the case of a cylinder carrying an 
electric current by Murty (1961. see also Alfven 
and Falthammar 1963). Various modes of instability 
are possible for the current-carrying cylinder. 
The simplest case is the pinch, varicose or 
sausage-type instability. when the surface of the 
cylinder is deformed so that its longitudinal section 
has a sinusoidal form. causing it eventually to 
disperse into drops. Analysis of the electromagnetic, 
surface tension and inertia forces associated with 
this mode of deformation shows that there is a 
critical wavelength above which the system is 
unstable. It has been suggested that the drop at the 
tip of the electrode in GMA welding will grow until 
its length is about equal to the critical wavelength, 
after which it becomes unstable and may be 
pinched off (Lancaster 1984). 
The analysis also yields a time constant from 
which the initial velocity. acceleration and detach- 
ment time of drops may be estimated as a function 
of current for various electrode diameters and 
materials. The values so obtained are consistent 
with experimental results (Lancaster 1984). 
In the presence of a longitudinal magnetic field a 
higher unstable mode may appear. This is the kink 
unstability, when the liquid cylinder collapses into 
an expanding spiral. Figure 4 shows metal transfer 
in the case of high current plasma-MIG welding. 
In this process an arc is maintained between 
an auxiliary tungsten electrode and the workpiece 
and the consumable electrode, which also carries a 
current, passes through the plasma so formed. In 
this instance the kink instability has developed first, 
but the pinch mode is visible at the end of the 
electrode, which eventually disperses into drops. In 
the normal MIG or GMA process the appearance of 
rotating transfer is similar to that shown in figure 4. 
In neither case is there an imposed magnetic field, 
but of course the spiral formation will generate its 
own longitudinal field. Thus the character of metal 
transfer in GMA welding is consistent, at least 
qualitatively, with the theory of instability of liquid 
cylinderst. 
In submerged arc welding drops are directed 
towards the weld pool in a different manner. This 
process uses a bare wire electrode and the arc and 
weld pool are protected from the atmosphere by a 
powdered flux. Flux melts around the arc, forming 
an expanding bubble that periodically bursts and 
then reforms. Drops are detached from the 
electrode tip in random directions, but those that 
fly outwards are trapped by the bubble of molten 
flux and therby directed into the weld pool. 
Flow in the weld pool 
If current enters a hemisphere of liquid in a 
symmetrical manner from the plane surface, the 
electromagnetically induced flow should be toroidal 
in form. Such toroidal flow is rarely, if ever, 
observed in weld pools. The closest approach to an 
ideal geometry is in TIG welding at low current, 
when the weld pool is almost hemispherical. This 
type of weld pool tends to rotate. The rotation may 
take the form of a double circulation or, more 
t Recent work (including that of Murty 1961) on the 
stability of cylindrical systems relates mainly to problems 
in cosmology and nuclear fusion devices (Alfven and 
Falthammar 1963. Chandrasekhar 1961). It may be of 
interest theretore to note a possible application in 
welding. The quantities concerned are of course rather 
different. In GMA welding the time for development of an 
electromagnetic pinch instability is of the order s; in 
cosmology Chandrasekhar (1961) instances the case of a 
cylinder 250parsecs (7.7 x 10I8m) in diameter and of 
density 2 x kg r C 3 , for which the characteristic 
time of break-up due to gravitational instability is 
10'years. 
77 
Figure 4 Metal transfer in high-current plasma-MIG 
welding. A rotating spiral of liquid metal eventually 
disperses into drops (photograph courtesy Philips, 
Eindhoven) 
commonly, the pool rotates as a whole. The 
direction of rotation may be dhanged by changing 
the location of the earth return, and it would 
appear that the magnetic field due to assymmetric 
current flow in the workpiece is sufficient to cause 
rotation. 
In low current TIG welding the weld pool surface 
is flat or slightly raised, but in many welding 
operations a depression forms in the liquid metal 
below the electrode, due to the stagnation pressure 
generated by gas flow or to the impingement of 
liquid drops or both. Metal that is melted at the 
front of the weld pool is accelerated through the 
restricted cross-section around the depression. 
Thus there is a circulation along the bottom of the 
weld pool from front to back and along the surface 
from back to front (figure 5) . This circulation 
convects heat backwards along the weld axis, and 
causes the weld pool to be more elongated than 
would be calculated assuming isotropic thermal 
properties. For weld pools generally the Peclet 
number lies within the range 10-5 X lo3, so that 
heat flow is predominantly convectional. 
Some investigators have suggested that flow may 
be induced in a weld pool by surface tension 
gradients. Such flows, which are toroidal, have 
beendemonstrated in liquid pools of paraffin wax 
78 
and other organic substances. Attempts to generate 
flow due to surface tension gradients in mercury 
were however unsuccessful, except under high 
vacuum. It is thought that under normal atmos- 
pheric exposure the metal surface becomes 
contaminated with surface-active material, and that 
stresses generated by temperature gradients are 
nullified by a redistribution of the surface-active 
agents. 
High energy density welding 
The electron beam and laser welding processes are 
both capable of producing very narrow, deep- 
penetration welds, such as that illustrated in figure 
6. This capability is particularly attractive for the 
welding of machined components such as gears and 
aero-engine components, since the volume occu- 
pied by the weld is much smaller than for normal 
fusion welding, and distortion is correspondingly 
reduced. 
Such deep penetration welds are made by 
producing a cavity, known as a 'keyhole', and 
traversing this along the joint. In electron beam 
and laser welding this cavity is maintained by the 
vaporisation of metal. The internal pressure so 
generated is balanced by the stress due to surface 
tension y in the film of liquid metal surrounding the 
cavity 
p = yfr 
where r is the radius of the keyhole. The tendency 
for a cylindrical cavity of this type to collapse 
inwards is counteracted by increased evaporation at 
the point of collapse, so that a stable configuration 
can be maintained. 
As in conventional welding processes the width 
of the completed weld is determined by practical 
considerations. A very narrow weld may require 
excessive accuracy in preparation and positioning 
of the joint, whilst if the weld is too wide it may 
show protrusion at the root and sink-away at the 
surface. Typical weld widths lie within the range 
0.75-3". Now in heat flow generated by a line 
Figure 5 Flow pattern in a submerged arc welding pool. 
For clarity the electrode and arc have been omitted 
(from Lancaster 1984) 
source, such as the deep penetration electron 
beam, at least half the heat is absorbed into the 
solid metal, the remainder being used to melt the 
weld metal. Assuming this minimum value, the 
total power of the line source is 
q = 2wdvp C,T, 
where w is weld width, d is weld depth, v is 
velocity, p density and C, specific heat. This heat 
must be generated within the keyhole, which may 
reasonably be expected to have a radius about 
one-half that of the weld pool, i.e. w14. Then the 
surface power density must be at least 
q” = 16q/nw2 
which for a 5 mm weld leads to a minimum power 
density of 1.5 X lo1’ Wm-’. This is indeed at the 
bottom end of the spectrum for deep penetration 
electron beam welding. 
Plasma welding may be operated in the keyhole 
mode, but the power density is lower than for 
electron beam welding, and the cavity is 
maintained by pressure from the plasma jet. 
Vaporisation of the,metal does not occur to any 
significant extent in plasma welding. 
Both electron beam and plasma welding torches 
may be operated as surface heat sources, as in 
electric arc welding, and for certain applications 
this may be advantageous. But the keyhole mode is 
the most important, since it makes possible novel 
joint configurations, reduces distortion and per- 
mits the welding of some’ of the more difficult of 
metals and alloys. 
Future developments 
At present the emphasis in development work is on 
the improved application and control of existing 
processes, rather than the introduction of new 
methods. The major use of arc fusion welding is in 
the construction Of process plant, bridges, steel 
buildings, ships, machine frames and the like, and 
the processes most commonly employed here are 
manual welding with coated electrodes, submerged 
arc welding and gas metal arc welding, whilst TIG or 
GTA welding is used for thinner sections and special 
materials. There is a requirement for improved 
quality, and at the Same time for improved 
productivity through automation and the use of 
robots. To this end numbers Of Current investiga- 
tors are working on diagnostic techniques, with the 
object of providing feedback control and improving 
the consistency of the welding operation. 
The use of electron beam (EB) and laser welding 
continues to be limited by high capital cost and, in 
the case of EB welding, by the need, in most cases, 
to evacuate the welding chamber. Nevertheless, the 
inherent advantages of a deep penetration keyhole 
Figure 6 Typical deep-penetration weld produced by the 
electron beam process (photograph courtesy The Welding 
znstitute) 
weld are such that much effort goes into methods 
for applying the EB process to the welding of thick 
sections. Success in this activity could lead to a 
major advance in welding productivity, always 
provided that the limitations of the vacuum 
chamber can be overcome. 
References 
A,fvCn H and Fllthammar C-G 1963 Cosmical Electroe 
dynamics (Oxford: Oxford University Press) 
Chandrasekhar S 1961 Hydrodynamic and Hydromagnetic 
Stability (New York: Dover) 
Christensen N, Davies V de L and Gjermundsen K 1965 
‘Distribution of temperature in arc welding’ Brit. 
we1ding J . U 54-74 
Guile A E 1979 ‘Processes at arc cathode roots on 
non-refractory metals having films of their own oxide’ 
in Arc Physics and Weld Pool Behaviour (Cambridge: 
The Welding Institute) 
Lancaster J F (ed) 1984 The Physics of Welding (to be 
published Oxford: Pergamon Press for the International 
Institute of Welding) 
Murty G S 1961 ‘Instability of a conducting fluid cylinder 
in the presence of an axial current’ Ark. F. Fys. 19 483 
Rayleigh Lord 1879 ‘On the instability of jets’ Proc. 
Land, Math. soc, 1o 4-13 
Waszink J H and Graat L H J 1983 GExperimental 
investigation of the forces acting on a drop of weld 
metal’ Welding J. 62 108-S-16-s 
79