Many Body Collisions under an External Force
A A Agbormbai
Department of Aeronautics
Imperial College of Science, Technology and Medicine
Prince Consort Road
London SW7 2BY, England
Abstract. Whereas rarefied gas dynamics has traditionally assumed a dilute gas assumption, in which the densities are so
low that only binary collisions and single-body gas surface interactions occur, expressions for many-body collision rates
and for many-body gas surface interaction (GSI) rates appear to suggest that at lesser heights the dilute gas assumption is
not valid. In particular, at heights close to the transition regime into continuum flow, three- and four-body collisions
become important. In this paper I consider the effect of an external force on the scattering behaviour of particles
undergoing many body collisions. This complements a previous work on the effects of an external force on the energy
exchange process of many body collisions. I formulate the effects on two-body collisions and then I extend the results to
three- and four-body collisions. In particular, I present deflection angle formulae for two-, three- and four-body collisions
under an external force. The philosophy of the approach is to reduce a many body collision to the dynamics of a number
of reduced particles in addition to the motion of the centre of mass of the system. I also describe how to handle external
forces in Direct Simulation Monte Carlo (DSMC) computations. Basically, an external force affects both the collision
and convection phases of the DSMC method. The DSMC method is the standard approach for computing rarefied gas
phenomena.
I. INTRODUCTION
In Ref. 1 I discussed the problem of many body collisions in which the effects of an external force were limited to its
influence on the energy exchange process. An external force was accounted for by including the centre of mass
energy in the energy exchange process, the argument being that the centre of mass accelerates through the collision
and cannot therefore be cancelled out of the reciprocity equation. This represents only part of the story, when
considering the effects of an external force. An external force also influences the scattering behaviour of the
particles. In this paper I show that the effect of an external force on the scattering process is to create further
deflections in the particles. I demonstrate the procedure for binary collisions and then I extend the results to threeand four-body collisions. I present deflection angle formulae for the scattering of the reduced particles.
II. RESOLVING POTENTIALS AND SEPARATIONS
In formulating deflection angle formulae we find that we have to resolve a potential function into in-plane and outof-plane directions. 'In-plane' means in the collision plane and 'out-of-plane' means in the normal direction to the
collision plane. Since the collision plane changes through the interaction we use the initial collision plane as our
reference collision plane. There are two in-plane directions: in-line and transverse. The in-line direction is along the
intermolecular separation of the two particles that are used to form the reduced particle whose motion is being
considered. The transverse direction is normal to this. Whereas the in-plane potential induces in-plane deflections
the out-of-plane potential induces out-of-plane deflections. The original potential is assumed to depend only on r,
where this may be the intermolecular separation between one of the two particles lying in the in-line direction and
some other particle of the system. On the other hand, r may be the magnitude of the position vector of a particle on
which an external force is acting. We assume that the in-line direction is jc, the transverse in-plane direction is y, and
the out-of-plane direction isz. Thus x andj lie in the initial collision plane whereas z lies perpendicular to this plane.
CP585, Rarefied Gas Dynamics: 22nd International Symposium, edited by T. J. Bartel and M. A. Gallis
© 2001 American Institute of Physics 0-7354-0025-3/01/$18.00
Note that (*;, y, z) is thus used to denote either the separations or the position co-ordinates of the particles. No
conflict arises as long as the context is kept in mind.
Given the original potential:
V = V(r)9
where r2 = x2 + y2 + z2, suchthat
Fr = ———
dr
we want to resolve the potential along x, y, andz, such that:
dV
Fx = -—±,
ox
dVv
Fy = -—y-,
dy
dV
Fz=-—^
dz
where
F = (FX,FVy,FZ)
is the force generated by V.
Knowing that:
dV(r) __ x dV(r)
dx
r dr
dV(r) ^
dy
y
y dV(r)
r dr
z
dV(r) __ z dV(r)
dz
r dr
we get:
dr
/
I
y
J r I dr
o V
I
/
J r I dr
o V
If we assume that along x we have 7 = 0, z = 0; along y we have ;c = 0, z = 0; and along z we have x = 0, y = 0 then
the preceding relations simplify to:
K, = V(x\
Vy = V(y\
Vz = V(z)
Note that the potentials depend only on distances, not directions. This means that the potentials are spherically
symmetric.
To resolve separations and other vectors along the three orthogonal directions we need the unit vectors along
these directions. For the in-line direction we can use the separation vector along that direction to construct the unit
vector. For the out-of-plane direction we use the angular momentum vector to construct the unit vector. Crossmultiplying these unit vectors gives the unit vector in the transverse in-plane direction. The results are:
In-line:
—,
Out-of-plane: p^——4,
Transverse:
— x-r^——4
where r07- is the separation vector and g . is the relative velocity.
III. TWO BODY COLLISIONS
The effects of an external force on a many body collision are similar to the effects on a binary collision. Therefore,
we start by examining the effects on binary collisions and then extend the results into many body collisions.
Consider two particles undergoing a binary collision. Denote the position vector of a particle by r, the
intermolecular force by f, and the external force by F. By writing down and manipulating the equations of motion
for each particle we can re-express these equations in terms of the motion of the centre of mass as well as of a single
reduced particle, as follows:
ft = (/HO +ml)R= (m0 + ml)G=¥0 + ¥l
(1)
=—*i-—*o-toi
ml
m0
where R and G are the position and velocity of the centre of mass and:
r
oi=ri~ro>
=
~
—+ —
ju m
m
or
l
0
This result shows that the centre of mass of the system accelerates through the collision because of the action of
external forces. The internal force does not contribute to this motion.
We can resolve the external forces into in-plane (in-line and transverse) and out-of-plane components. The inplane components will enhance the in-plane deflection of the reduced particle, while the out-of-plane component
will introduce out-of-plane deflections in the reduced particle. Assuming that the external forces are conservative
(i.e. can be expressed in terms of a potential) we can express the deflection angle components using the general
form:
dp
,*) = *-2*J^
J
»
1
c/
'** \f >
^
±t
/2\
where A is the signed coefficient of the potential; and j,k are integers.
For the specific forms of the in-plane and out-of-plane deflection components these symbols are given appropriate
meaning below. The in-plane deflection is as follows:
% = %Ql + %Q + %i, with Xjk an<^ Xj being given by (2) where the symbols are : b = b , for
U
LL
niQ
where
r 1><g
qj = U/ + y] Y2, */ = r- •^l, y- = r, • ^-x
°
r
r
[
koiXg|J
where : bx = in -plane impact parameter, r01 = separation vector,
MI
7 being an integer
Y}- = position co-ordinate
r = 70! = intermolecular separation,
rm = separation at point of closest approach
Xj = in -line component of position vector, Xjm = component at point of closestapproach
yj = transverse component of position vector, yjm = component at point of closestapproach
qj = resultant of jcyandjy, qjm = resultant at point of closest approach
Vjk = intermolecular potential,
Vj = external potential corresponding to external force Fy
The external potentials are assumed to depend only on the magnitude of the position vector of the particle on which
they act. Note that the signed coefficients of the external potentials have been derived from the equations of motion
for the reduced particle (i.e. from (1) - a negative force implies a positive potential and vice versa).
The out-of-plane deflection is:
£ = £0 + £1? with BJ being given by (2) where the symbols are : b = b£, for £0 : p = z0, V = VQ, A = —;
for EI : p = z1? V = Fi, A = ——;
where
where : b£ = out - of - plane impact parameter,
Zj = r •
[ LiJrm x g .
r ——• ,
r01 = separation vector,
7 being an integer
ry = position co-ordinate
Zj = out - o f - plane component of position vector, zjm = component at point of closest approach
Vf = external potential corresponding to external force F
In writing down these expressions for the deflection angles we have used a similar reasoning to our handling of the
extra intermolecular forces that are generated by the presence of additional bodies in a many body collision (Ref. 1).
IV. EXTENSION TO MANY BODY COLLISIONS
Many body collisions are handled by taking two bodies at a time and formulating the motion of their reduced
particle. An TV-body problem reduces to the motion ofN-l reduced particles in addition to the motion of the centre of
mass. Particle 0 is considered to be the reference particle, relative to which each other particle moves. Therefore the
two-body results can be generalised to N bodies by combining particle 0 with each of the other particles, to form a
reduced particle. Using this logic we can immediately write down deflection angle formulae for three- and four-body
collisions by extending the formulae for collisions without external forces (i.e. the formulae in Ref. 1).
V. IN-PLANE DEFLECTION
Note that in the subsequent formulae we use the following symbols:
#(00
=
deflection angle for reduced particle /%•
XE = deflection angle due to external forces,
Xi = deflection angle due to internal forces
%Qi = primary deflection angle, the rest being secondary
bXQj = impact parameter for in- plane scattering of reduced particle £%
rt = intermolecular separation for reduced particle £%,
rim = separation at point of closest approach
Xj = in -line component of position vector, Xjm = component at point of closestapproach
yj = transveise component of position vector, yjm = component at point of closest approach
qj = resultant of X j a n d y j ,
qjm = resultant at point of closest approach
x.k = in - line component of separation vector,
xjkm = component at point of closestapproach
yjk = transveise component of separation vector, yjkm = component at point of closestapproach
qjk = resultant of Xjkandyjk,
Vjk = intermolecular potential,
qjkm = resultant at point of closest approach
Vj = external potential corresponding to external force Fy
go/ = initial relative speed for reduced particle /X0/
g'^ = initial relative velocity for reduced particle ^i%
The in-plane deflection angles for three body collisions are:
(a) First reduced particle.
#(0i) = XE + Xi 9 where Xi is given in Ref. 1 and XE = #o + #1? w^h Xj being given by (2) where the symbols are :
b=bxQl, // = M)i, g' = g'oi, andfor;^: p = qQ, V = K0, A = ^2L; far Xi ' P = q\, V = Vl9 A = -^2L;
mQ
ml
where
qf = uc/ + y / / 2 , xf = r, • — , yf =rf • — x-r-^—^L I j being an integer
'01
l/oi F o i X f t l J
(b) Second reduced particle.
#(02)
=
XE + Xi > wnere Xi is given ™ Ref- l and XE = #o + #2 » with Xj being givenbY (2) wnere tne symbols are
b=bzQ2, ^ = M o 2 ? g/ = go2> andfor^ 0 : p = qQ, V = F"0, A = ^2; for^ 2 : p = q2, V = V2, A = -• ^-;
mQ
m2
where
q . = \xt + yW2 , ^ , = r , » - ^ - , y- = r- • -21 Xp^—21.1
r*»
02
I »»
1 r02
I*»N/<T||
| r 02 X g2| I
j being an integer
The in-plane deflection angles for four body collisions are:
(a) First reduced particle.
=
yf(oi)
X E + Xi •> where Xi *s givenin Ref. 1 and %E is given as for three body collisions.
(b) Second reduced particle.
=
#(02)
XE + Xi 9 wherej/ is givenin Ref.l and %E is given as for threebody collisions.
(c) Third reduced particle.
= XE + Xi> where Xi is givenin Ref. 1 and XE = XQ + #3' w*m Xj being given by (2) where the symbols are
j£
6=6,03,
6,03, M = A)3,
where
g =go3, andfor^: p = q0, V = V0, A =
; for^ : p = q3, V = V3, A = -21;
*/ = r, •^i, 7, = r, • ^-x-^—^T I j being an integer
qf = be/ +7/P,
r*•
I r»»
03
^ 03
li»r
V r» I
I
| 03 X g3|J
VI. COLLISION PLANE CALCULATION
The following set of symbols are used for the out-of-plane deflection:
eQf) = deflection angle for reduced particle /%
£E = deflection angle due to external forces,
£/ = deflection angle due to internal forces
b£0i = impact parameter for out -of -plane scattering of reduced particle j%
Zj = out -of- plane component of position vector, zjm = component at point of closest approach
Zjk = out-of-plane component of Fj k ,
Zjkm = component at point of closest approach
The out-of-plane deflection angles for three body collisions are:
(c) First reduced particle.
e
(oi)
= £
E + £ />
where e7 is givenin Ref. 1 andeE = £0 +sl , with e- being given by (2) where the symbols are :
b=be0l, AI = /IOI, g' = goi,
where
r01
,
zy = r • I -————lw»
NX r* I I
J
Foi x 8i
andfor£ 0 : ^ = z0, V = K0, A = ^-;
j being an integer
yi
(d) Second reduced particle.
e
(02) = £E +£i>
where £7 is givenin Ref. 1 and£^ = £0 + e2, with e. being given by (2) where the symbols are
b = b£Q2, H — ^02? g' — gQ2> and for £0 : p = z0, V = F0, A = —22-; for£ 2 : p = z2, V =V2, A = ——02.;
where
x
z, = r, • p^——^- ,
/ being an integer
The out-of-plane deflection angles for four body collisions are:
(d) First reduced particle.
Z(oi) = XE + Xi> where Xi *s givenin Ref. 1 and XE *s givenas for three body collisions.
(e) Second reduced particle.
=
X(QT>
XE +Xi> where Xi is givenin Ref. 1 and XE ® given as for three body collisions
(f) Third reduced particle.
£
= £E + £7 ?
where £7 is givenin Ref. 1 and SE = £0 + £3, with e. being given by (2) where the symbols are :
6=6,03, M = Mo3> S'=8w andfore 0 : p = z0, F = F0, A = ^; for£ 3 : p = z3, V = V3, A = -^2i;
where
z,J =
I
r 0 3 xg 3 1
,
, L
r
j being an integer
03 X 83
VII.
ENERGY CONSIDERATIONS
An external force accelerates the particles between collisions and influences the particles during a collision. During
collisions the external force creates further deflections of the particles while accelerating the centre of mass of the
system. This acceleration of the centre of mass means that the centre of mass energy must be included in energy
exchange formulations. These formulations relate the pre-collision energies of the reduced particles and centre of
mass to their post collision energies. Therefore the reduced particles not only exchange energy with one another, but
also exchange energy with the centre of mass. The appropriate energy exchange models have been formulated in
Ref. 2 using reciprocity modelling.
VIII. DSMC COMPUTATIONS
A DSMC computation comprises two phases: a MOVE phase and a COLLISION phase. In the move phase the
particles translate over a time interval, while in the collision phase the particles undergo a collision. A move phase is
calculated and then a collision phase, and so on. In the absence of external forces the velocities at the end of the
move phase are the same as the velocities at the start of the move phase, i.e. no acceleration takes place. On the
other hand, external forces accelerate the particles during the move phase. Thus, an external force will affect both
the move phase and the collision phase.
IX. REFERENCES
. A. Agbormbai, Dynamical and Statistical Modelling of Many Body Collisions I: Scattering, submitted to Rarefied Gas
Dynamics 22nd symp.
. A. Agbormbai, Dynamical and Statistical Modelling of Many Body Collisions II: Energy Exchange, submitted to Rarefied
Gas Dynamics 22nd symp.