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Ideal fluid jets
Inr. J. Engng Sci.
Vol. 6, pp. 3 17-328.
Pergamon Press 1968.
IDEAL
FLUID
Printed in Great Britain
JETS
A. E. GREEN and N. LAWS School of Mathematics, University of Newcastle upon Tyne. England (Communicated
by P. M. NAGHDI)
Abstract-Constitutive equations for one dimensional incompressible ideal fluid jets are established based on a general theory given by Green and Laws[ I]. We discuss the special kinematics of a jet in motionalong a fixed curve. It is shown that a circular jet in the atmosphere can move with uniform speed in a straight line and twist in either sense about its axis.
I. INTRODUCTION
paper a jet is like a thin (not necessarily straight) column of water moving through air. We consider the motion of an ideal fluid jet when the jet is surrounded by a continuum (called the atmosphere) which exerts a constant pressure on the surfaces of the jet. A typical example of the motion of such a jet is obtained by visualizing the flow of a fluid after it has issued from a Borda tube. It is well known that it is very difficult to analyse these ideal fluid flows from three dimensional considerations even when the flow is assumed to be irrotational, as in the case of fluid emerging from a Borda tube. Rotational problems in which pressure is prescribed on the surface of the jet are even more difficult. In a series of lectures published in 1960 Truesdell[21 discussed oriented materials with special reference to the theory of elastic rods. In the course of his lecture on rods he stated that “We could try to formulate a theory of one dimensional twisted liquid jets. If we could set this up right so as to express the concept of fluidity, we might even be able to predict the twisting that everybody sees in the jet from his garden hose”. Such a theory could, of course only reflect the main characteristics of such a flow and not the finer details. However, three dimensional analysis for rotational flows of this type does not seem to be available so it is of some interest to consider alternative approaches. The main purpose of the present paper is to formulate such a one dimensional theory and to demonstrate the existence of a solution of the basic equations of the type conjectured by Truesdell. The theory is based on a paper by Green and Laws [ 11 in which a general thermodynamical framework was developed which could be used for any kind of material, solid or fluid. The necessary kinematics is given in section 2 and the basic equations are summarized in section 3. Appropriate constitutive equations for ideal incompressible fluid jets are also established in section 3. Motion of a jet along a curve fixed in space is discussed in section 4. Section 5 is devoted to three dimensional considerations. Whilst the theory of Green and Laws[l] is based on one dimensional concepts alone, Green, Laws and Naghdi [31 have shown how the one dimensional theory may be obtained by an approximation to a three dimensional theory. We emphasize that these latter authors have not proved that the theory of the former authors is precisely related to the three dimensional theory. We particularly need the results of Green, Laws and Naghdi[3] in order to IN THIS
317 1J.E.S. Vol6.6-A
318
A. E. GREEN
and N. LAWS
pose boundary conditions which are meanin~ul in a three dimensional context. For completeness we quote the main results which are relevant to our immediate needs. With the help of the results contained in section 5 we go on to discuss Truesdell’s problem of a straight jet of uniform circular cross section. We show that such a jet may twist in either sense about its axis as well as move with uniform speed along its axis. A motion is also established in which the jet moves with uniform speed and rotates about its axis as a rigid body and this motion is compared with the three dimensional motion since this is one problem which has an exact three dimensional solution. 2. KINEMATICS
Let a curve c, embedded in Euclidean 3-space. be defined by the equation 1:= r(8, t) where r is the position vector, relative to a fixed origin, of a point on c and t denotes the time. We regard 0 as a convected coordinate defining points on the curve. We consider a curve at each point of which there are two assigned directors. At time t, these directors are denoted by a, (a = 1,2), and the motion of the whole system is given by r = r(O,
a,=
f).
a,(@,f).
(2.1)
The vector a,, which is tangent to the curve c, is defined by a, = a,(O, t) = 5
(2.2)
and it is assumed that 1t is convenient
to introduce the kinematical quantities aij, clii, Kij, i3i .
a,i =
n =
Njj,
~;j
through
det cl+ (2.3)
aaj Kij
=
23, ’ s,
where 8;. is the Kronecker delta and Latin indices have the values 1.2,3 and repeated indices are summed. We shall also adopt the notation that Greek indices have the values I, 2 and the usual summation convention still applies. We note that atlij Kij+
Kji
=
(2.4)
-. afl
A superposed dot is used to denote the material time derivative t, holding 19fixed. Hence the velocity of a point on c is given by
with respect to 12.5)
V-;i
and the director velocities are defined by w, = i,. We put y = uiai = viai,
(2.6) W, =
14’&li.
(2.7)
By taking the material derivative of (2.2) and making use of (2.3) we find that
(2.8)
319
Ideal fluidjets
Thus, if 271~= a*
’
aj+&
’
(2.9)
$
then
2na3 =
W,3
+
$
dV3 z--
7733 =
-
K;I”V,
(2.12)
K@,.
From (2.6), we observe that
aw, D -=a0
uaa
(2.13)
a0
Dt
where D/Dt denotes the material derivative, and from this result, we may show that?
.
awaR
‘GUS=
- a*
. Ku3
aWa3
=--
-
ai3
W,K;
+
(2.14)
WpsK:
(2.15)
W,Ki’+
We note that the element of length along the curve c is ds = V$233) de. 3. BASIC
The local equation of mass conservation
THEORY
is
~(P~U33) =0
(3.1)
where p is the mass per unit length of c, so we put pda33 = k.
where k is a function of 8 but is independent
$+ 7pR -
#3 +
oft. The equations of motion are
K.jnr+ kf i =
qR(’ + p”3K;e
(3.2)
&.a’,
(3.3)
pYRK;p- PYaK$R = -
paPKd
-
,f
=
0
0
(3.4) (3.5)
where the six quantities fl’ are defined by+ ,,d
=
kq*i +
+!$?+K:par.
(3.6)
tPerhaps the easiest way of obtaining (2.14) and (2.15) is to use (2.6) together with equation (2.22) of Green and Laws[ I]. SThe quantities @’ used here differ from the corresponding quantities in the work of Green and Laws[ I] by a factor d/(as3).
320
Ideal fluid jets
Also, d are force components and pai director force components along the vectors ai, fi are components of assigned force per unit mass and q”’ are components of the
difference between assigned director force and director inertia terms. We taket cxptip . a’
qpi = P-
Cpnot summed)
(3.7)
where (Y~,(Yeare functions of 13but are independent oft and Pi are components assigned director force per unit mass. The energy equation is -ti-k(TS+Sf)+kr+
of the
(n3--p”3K~)r/33+t(nP-pa3K~)TJ~3
where A is the Helmholtz free energy per unit mass. Y is the heat supply function per unit mass per unit time, h is the flux of heat along c per unit time, S is the entropy per unit mass and T is the (positive) temperature. The entropy production inequality is (3.9) From (3.8) and (3.9) we obtain -k(k+s?)
+
(n”--p*3Kg)~33+2(na-p*“K~)r)a3 h
+$(?7*‘+
da-pypK;a
-pYaK;P)
Qp
aT
+paikai-~~
2
0.
(3.10)
We wish to emphasize that the preceding work does not depend upon the choice of constitutive equations. We now obtain the constitutive equations for what we call an ideal incompressible fluid jet. To do so, we first define a jet to be incompressible if it is susceptible to only those motions for which* V(a) = [a,a,a,] is independent
oft. Hence, for an incompressible
jet we have
nij7)ij= 0 that is (3.11)
uaDv),p+ 2aD3r)p3+ Cz33r/33 = 0.
We multiply the left hand side of (3.11) by ~V\/(U~~) and add the resulting expression (which is zero) to (3.10): - k(k + s ?-) + (n3 -
pa3Kg
+
733-t 2 ( np - Paz&f + paP3du33) %33
pL7332/U,3)
+ paif&+ 4 (Tao + i+ -
pypK;a
-
pYaK;P
We are now led to define an ideal incompressible . equations: A =/I(T),
S=S(T),
+
2pUaP&33)
7),@
- f
g
2
0.
(3.12)
fluid jet by the following constitutive h = h(T,dT/dO)
(3.13)
tThis means that we take the yuBof Green and Laws [ I] to have the values y” = u,. y12= y” = 0. yzs = LY?. and this is consistent with the discussion of section 5. $See the discussion of section 5.
321
Ideal fluidjets
and that the coefficients of rf33, qB3, qap, ri,$ in (3.12) depend only upon T. Next we choose p so that the coefficient of r/33 in (3.12) vanishes. The remaining independent variables vp3. nn8, ri,i, ? can then be chosen arbitrarily. Hence (3.14)
np= -paR3V/a33, n3 =
- pa33Vla33
(3.16)
-h$20.
(3.17)
The function p is an arbitrary scalar which represents the resultant a section of the jet. The energy equation (3.8) now reduces to kr-kT+=
across
(3.18)
0.
Since p”‘= 0, we collect the simplified forms of the equations and (3.5): rr”B= 7@“,
pressure
of motion (3.4)
@3=&3
(3.19)
(/3 not summed).
(3.20)
where we now have @i = k/Pi - karRGR . ai
With the help of (3.16) and (3.20), equations (3.3), (3.1.5)e and (3.19) provide nine equations of motion for the determination of the position of the curve c and the two directors a,. Also equations (3.14) and (3.18) provide a single equation for the temperature. A rather obvious result of our constitutive assumptions is that the thermal and mechanical effects are distinct. In the following sections we consider only the mechanical problem. 4. KINEMATICS
OF MOTION
ALONG
A FIXED
CURVE
In ffuid dynamics it is customary to formulate the theory and boundary-value problems in a spatiai as opposed to a material description. In this section we shall discuss this transition in the case when the curve c is in motion along a fixed curve 4. We have delayed this discussion until now for the earlier part of this paper is valid whether or not the motion has any simplifying features. We introduce rectangular Cartesian coordinates x, and use upper case Latin indices to denote Cartesian components. Thus, let the fixed curve V be given by
r = r(5) = xx@) e, where eK are the unit Cartesian base vectors
and 5 is a fixed parameter
(4-I) defining %?.
322
A. E. GREEN
and N. LAWS
Now the material point with convected coordinate 8 occupies various places at different times and, since .$ parametrizes %?, we may denote the place occupied at time r, by the material point with convected coordinate 8, by 6 = f(8, t). The Cartesian components
(4.2)
of the velocity of a point of c are (4.3)
But we have already denoted the velocity by v, hence
ax, D,$ V=U~&K=-~Ka[ Dt
where we have used (2.2). Since v = uiai
we have (4.4) Any variable associated fixed coordinates. If
with the motion can be written in terms of convected
or
then DF -=-+-- af Dt at
af Df
ag Dt
(4.5)
and (4.6) If we now choose the convected ordinate 6 at time t, then
coordinate aF af -=--, ae ag
8 so that it coincides
v32&
with the fixed co-
(4.7)
With this choice of 8, there is no ambiguity if we do not indicate the arguments of functions at time t, since they will always be 4 and t. It is apparent that the preceding caicuiation is similar to that used by Oldroyd[4]. Next we deduce the reduced forms of the acceleration, the director velocities and the director accelerations. These are the only kinematical quantities which are needed
323
Ideal fluid jets
in the theory of the ideal incompressible fluid jet. The acceleration of the material point which occupies the place with fixed coordinate 5 at time t is given by
av
3av v=ar+vag 2+v3 au3 ( .
-a3 at
+ v3K’1 3 r
(4.8)
* >
Also the director velocities w, are found to be w, =
and the director accelerations
2 +v3K;a,
(4.9)
become
In classical fluid dynamics it is customary to speak of steady motion as being motion for which the velocity of the successive particles at a fixed spatial point is constant in time. Here we adopt a slight variation of this concept and define a motion to be steady if the vectors af and the velocity v are functions off alone: v=
ai = ai(
(4.11)
v(l).
We use this definition of steady only for convenience and note that certain motions which are steady in the three dimensional sense are not steady in the sense used here. It is apparent that many of the kinematical results of this section are much simplified in steady motion.
5. THREE
DIMENSIONAL
CONSIDERATIONS
The preceding theory employs only one dimensional concepts and makes no attempt to say anything about a full three dimensional theory. However, in some recent work, Green, Laws and Naghdi[3] have been able to shed some light on the relationship between the general theory of Green and Laws[ 11 and the classical three dimensional theory of continuum mechanics. In this section we outline some of the results of Green, Laws and Naghdi[3] which are important in the discussion of specific problems. Let the points of a three dimensional continuum be defined by a general convected coordinate system 8’. Also let the covariant and contravariant base vectors at points of the continuum at time t be denoted by g,, gi with corresponding metric tensors gij, g”. Thus if R is the position vector of a typical particle 0’ at time t, then dR
g,=ae” gij=
The stress vectort
gi'giv
R=R(81,02.83,t), gij
=
gi
. g,
i &.gi=@.
(5.1)
1
across a surface whose unit normal is u can be put in the form t=;;)
(5.2)
324
A. E. GREEN
and N. LAWS
where U =
Ti = +g$(g).
Uig’,
I: = det gij
(5.3)
and +j are the components of the symmetric contravariant stress tensor. When given surface forces P are applied at the boundary surface of the continuum then t= P.
(5.4)
The three dimensional body force per unit mass is denoted by F and the three dimensional density by (T. The parametric equations f? = 0
(5.5)
define a curve c in space, at time t which we take to be the ‘line of centroids’. This curve may be identified with the curve c used previously so that the equation of c is 8” G 0.
r = r(0. t) = R(O,O, 8”, t);
(5.6)
We assume that the region of space occupied by the three dimensional some neighbourhood of c which is bounded by a surface?
continuum is
f(8’. e’) = 0.
(5.7)
The work of Green, Laws and Naghdi[3] shows, amongst other things, that the directors a, are just the values of the base vectors g, on the curve c: a, = a&d, t) = %(O,O, f13.t).
(5.8)
Hence the directors supply some limited information about the deformation of the cross section. Also the quantities which we have designated mass per unit length of c, assigned force per unit mass, f, and assigned director force per unit mass. I*, may be identified by the formulae ~v$I&
pfX+zas) = j j crFv(&dB’d82+$
P(u’dBZ-2dB’)
= ~~crFX&)dB’d0*+$ pP~((a&
(5.9)
= j j- aV?g)dO’d@
= j j cr~F~(~)d01d02+
q/(n)
(TId@-T2d01)
(5.10)
$ POU(lclde”- zc”dO’)V\/(R) (5.11)
= IIa~~F~~a)d8’de’+~fP(T,dHL--T,dB’)
where 14 are defined in (5.3). The double integrals are over any section f13= constant and the line integrals are taken along the curve 8” = constant.
f(0’. f3’) = 0.
In addition the vectors n and p* may be associated system by n= II
with the three dimensional
(5.12)
T:,dO’d@
p* = j j &T,dO’dB” +Note that (5.7) does not necessarily
imply that the continuum
stress
has constant
(5.13) cross section.
ldeal fluid jets
32s
the integrals again being over a section @ = constant. The director inertia coefficients q, ar, (see (3.7)) are given by PCY&“(~~~)= f s crV?g) (B8)2dB’d~2.
(5.14)
In the three dimensional theory it is usual to assume that V(R) =
lkg,g31 '0
for all values of 6’. In particular it is true for 8” = 0 so we have l/(n)
(5.15)
= [afa2a3] > 0.
If the three dimensional continuum is incompressible then d(g) is constant in time. Since Q is the value of g on the curve c we are led to a definition of an incompressible jet by d(a) = constant in time. (5.16) The problems discussed in the later part of this paper are concerned with the motion of a jet through the atmosphere which exerts a constant pressure p. on the surfaces of the jet. In the absence of three dimensional body forces, we see from (5.3). (5. IO) and (5.1 1) that ,efV?a,,) pla~(a,~)
= $ (T~d~~-T~d~l)
(5.17)
= 4; P(T1d6z-T2d@1}
(5.18)
where Ti = -pOgi”gjV&). 6. A STRAIGHT
CIRCULAR
(5.19) JET
We consider the motion of an ideal incompressible jet with constant circular cross section, and whose centre line c is in motion along a fixed straight line %‘.We use the theory developed earlier to characterize the main features of such a motion. We make use of the notation of section 4 whereby xX denote rectangular cartesian coordinates and e,- is the standard orthonorma1 basis. It is convenient to choose the convected coordinates to coincide with the fixed coordinates 5’ at time c where?
where c(I is independent given by
of el. t2, &. We assume that the lateral surface of the jet is
so that b is the radius of the (uniform) cross section of the jet. From (5.1) and (6.1) we have, at time c, tWe especially choose this ‘twisted’ coordinate system to exhibit a solution which corresponds to a twisted jet.
326
A. E. GREEN g, =
and N. LAWS
e, cos $15+ e2 sin $<
g,=-ee,sin$~+e2cos@ g3=-1cl(&
(6.2)
sin95+e2
1
cos J15)eI +$J(& cos J15-t2 sin$t)e,+e,
and with the help of (5.8) g,= a,,
g3= a3+JI(51a2-52aI)
(6.3)
with a3 = e3.
(6.4)
Hence gij
=
1
0
0
1
$61
L-442
441
1 +dJ2t4:+m
-
$52
1-t
g’j = 1 ’
vJ”5E
-q12.f& i
952
$62
-$25152
I +$A$; -!@I
--$4, 1
g=a=l
I
(6.5
and a..LJ= & = 8..1.l.
(6.6
These kinematical formulae enable us to show from (5.9) and (5.14) that k=p=mrb2,
,+za2=fb2
(6.7
kf=o,
kl” = -p,7rb’a*.
(6.8)
and from (5.17)-( 5.19) that
The preceding calculations only serve the purpose of specifying the geometry of the jet at time f. which in turn gives us the position of the directors and ‘surface loading’ at the current time. We must now use this information in the basic theory of section 3 to investigate the motion of the directors and the centre line of the jet. This theory will not supply a complete solution to the full three dimensional problem. We first look for solutions of the field equations in steady motion so that we take $ = constant.
(6.9)
From (6.2) and (6.3) we see that
and that the only non-zero components
of
K 12 =
Kij -
are K2,
=
$J.
(6.10)
We assume that v3 = constant = V
(say)
Since the motion is steady, we obtain from (4.9) and (4.10) w, = VJla,,
w2 = - V$a,
(6.12)
i, = - V2J12a,,
Gt, = - Vf!J2a2.
(6.13)
Also, because a, is a unit vector, V is the speed of the centre line of the jet.
327
Idea1 fluid jets
With the help of (6.7) and (6.8) and the definitions (3.20) we may show that +I = &?z= - porb2 -t-4kb2V24fr2
(6.14)
+2=q==fl~3=qr23=2:(),
(6.15)
In this example, the constitutive
equations (3.15) and (3.16) yield np = 0 . *ll
=
792-t
r2?
$=-p
(6.16)
=
-p
(6.17)
7121 z
().
(6.18)
From (6.15) we see that (6.18) is automati~aIly satisfied by the assumed deformation field. Those equations of motion (3.3) and (3.19) which are not trivially satisfied show that we need n3 = -p = constant p = porb2 - $kb=V’%j?.
(6.19)
Ifp is the average pressure in the jet so that p = jhrb2
(6.20)
then (6.21) Thus if p. = ji, JI = 0 and the jet suffers no rotation. It is perhaps, instructive to explain the significance of these results whent p. > p. The solution (6.21) tells us that an observer situated at a fixed point on the line %’sees that the directors at that point do not vary in time. However, an observer situated on a moving particle sees that the directors attached to that particle describe a circle with constant angular velocity Q, where
~(Po-P)
a2=
ub2
’
We therefore say that, provided p. > p, an elementary exact solution predicts that the jet twists and that the twist may be in either sense. We think it worthy of note that TruesdelI~2~ has conjectured that a theory of the type proposed here would enable one to obtain this sort of result. Let us now consider a motion which is unsteady in the sense of section 4. We make use of (6.1) to (6.8) but put g=0 and look for a solution in which v3 = V = constant a, = ra2,
a, = -Ta,
tin order that the pressure be positive everywhere in the jet, p must be positive. This is only a necessary condition but is likely to be sufficient if~~2~~z/p~ is su~cientiy small.
328
A. E. GREEN and N. LAWS
where r is a constant. It is easy to verify that such a deformation of motion and that I’ is given by
satisfies the equations
(6.23) This solution corresponds to the rigid rotation and translation of the jet. It is noteworthy that this particular problem may be solved from the full three dimensional equations of an ideal incompressible fluid. The result (6.23) agrees with that obtained from three dimensions. [l] [?I (31 [4f
REFERENCES A. E. GREEN andN. LAWS.Pvot. R. Sot. A293,145 (1966). C. TRUESDELL, Pn’rzciptes ~~Con~j~~u~ ~e~~~~~ics. p. 3 19. Socony Mobil Oil Co, Dallas, (I 960). A. E. GREEN. N. LAWS and P. M. NAGHDI, Proc. Cnmh. Phil. Sot. To be published. J. G. OLDROYD, Prot. R. Sot. AZOO,523 i 19501. (Received
2 Jnttuctry
1968)
RCsumb-On &die les iquations de constitution d’un jet de fluide. ideal, unidimensionnel et incompressible et se basant sur une thCorie gCn&ale don&e par Green et Laws [ 11.On examine la cinematique particuli6re d’un jet, en mouvement le long d’une courbe dCterminCe. II est montrC qu’un jet circulaire, dans I’atmosph&re. peut se diplacer en ligne droite, B une vitesse uniforme et se tordre dans les deux sens autour de son axe. Zussammenfasung- Auf der Grundlage einer von Green und Laws aufgestellten Theorie[l] werden die Konstitutivg~eichungen fir aus einer i~komprimierbaren. idealen Fliissigkeit gebildete Strahlen bestimmt und die speziellan kinematis~hen Gesetzm~ssigkeiten eines sich entlang einer festgelegten Kurve bewegenden Strahfs erijrtert. Es wird nachgewiesen. dass sich ein kreisfdrmiger Strahl in der Atmosphere mit gleichfdrmiger Geschwindigkeit bewegen und in beiden Richtungen urn seine Achse verdrehen kann. Sommario-Si impostano equazioni costitutive per getti ideali di fluide incomprimibiii monodimensionali, in base a una teoria generale avanzata da Green e Laws[ I]. Si discute le cinematica speciale di un getto in moto lungo una curva fissa e si dimostra the un getto circolare nell’atmosfera si pub spostare con velocitB uniforme lunge una retta e &are in entrambe le direzioni sul proprio asse. ,&CT~KT--BblBOnRTCR KOHCTHTyTHBHble ypaBHeHMRnna OAHOpalMepHblXCTpyfi HeCXGitvlaeMOti HReaIibHOfi x~~KOCTW,Ha ocnoaarimi o&ue& -reopua co3naHHOB PPHHOM M flOyCOM [I]. 06cyxQaeTcn CneuIiailbHaR KMneMaTuKacTpyH, nponalzraloluetica no ycTaHosneHwoR ~pa~0ir. .QoKa3bmaeTca, 410 Kpyrslas crpyn MO)l(eT“pOJlBMraTbCfiB aTMOC+epe C paBHOMepHOi% CKOpOCTbtOn0 npaMOi? JIMHMHM CKpyqH5aTbCa B G60MXHanpaaneHwnx BOKpyf CBOeiiocn.
Vol. 6, pp. 3 17-328.
Pergamon Press 1968.
IDEAL
FLUID
Printed in Great Britain
JETS
A. E. GREEN and N. LAWS School of Mathematics, University of Newcastle upon Tyne. England (Communicated
by P. M. NAGHDI)
Abstract-Constitutive equations for one dimensional incompressible ideal fluid jets are established based on a general theory given by Green and Laws[ I]. We discuss the special kinematics of a jet in motionalong a fixed curve. It is shown that a circular jet in the atmosphere can move with uniform speed in a straight line and twist in either sense about its axis.
I. INTRODUCTION
paper a jet is like a thin (not necessarily straight) column of water moving through air. We consider the motion of an ideal fluid jet when the jet is surrounded by a continuum (called the atmosphere) which exerts a constant pressure on the surfaces of the jet. A typical example of the motion of such a jet is obtained by visualizing the flow of a fluid after it has issued from a Borda tube. It is well known that it is very difficult to analyse these ideal fluid flows from three dimensional considerations even when the flow is assumed to be irrotational, as in the case of fluid emerging from a Borda tube. Rotational problems in which pressure is prescribed on the surface of the jet are even more difficult. In a series of lectures published in 1960 Truesdell[21 discussed oriented materials with special reference to the theory of elastic rods. In the course of his lecture on rods he stated that “We could try to formulate a theory of one dimensional twisted liquid jets. If we could set this up right so as to express the concept of fluidity, we might even be able to predict the twisting that everybody sees in the jet from his garden hose”. Such a theory could, of course only reflect the main characteristics of such a flow and not the finer details. However, three dimensional analysis for rotational flows of this type does not seem to be available so it is of some interest to consider alternative approaches. The main purpose of the present paper is to formulate such a one dimensional theory and to demonstrate the existence of a solution of the basic equations of the type conjectured by Truesdell. The theory is based on a paper by Green and Laws [ 11 in which a general thermodynamical framework was developed which could be used for any kind of material, solid or fluid. The necessary kinematics is given in section 2 and the basic equations are summarized in section 3. Appropriate constitutive equations for ideal incompressible fluid jets are also established in section 3. Motion of a jet along a curve fixed in space is discussed in section 4. Section 5 is devoted to three dimensional considerations. Whilst the theory of Green and Laws[l] is based on one dimensional concepts alone, Green, Laws and Naghdi [31 have shown how the one dimensional theory may be obtained by an approximation to a three dimensional theory. We emphasize that these latter authors have not proved that the theory of the former authors is precisely related to the three dimensional theory. We particularly need the results of Green, Laws and Naghdi[3] in order to IN THIS
317 1J.E.S. Vol6.6-A
318
A. E. GREEN
and N. LAWS
pose boundary conditions which are meanin~ul in a three dimensional context. For completeness we quote the main results which are relevant to our immediate needs. With the help of the results contained in section 5 we go on to discuss Truesdell’s problem of a straight jet of uniform circular cross section. We show that such a jet may twist in either sense about its axis as well as move with uniform speed along its axis. A motion is also established in which the jet moves with uniform speed and rotates about its axis as a rigid body and this motion is compared with the three dimensional motion since this is one problem which has an exact three dimensional solution. 2. KINEMATICS
Let a curve c, embedded in Euclidean 3-space. be defined by the equation 1:= r(8, t) where r is the position vector, relative to a fixed origin, of a point on c and t denotes the time. We regard 0 as a convected coordinate defining points on the curve. We consider a curve at each point of which there are two assigned directors. At time t, these directors are denoted by a, (a = 1,2), and the motion of the whole system is given by r = r(O,
a,=
f).
a,(@,f).
(2.1)
The vector a,, which is tangent to the curve c, is defined by a, = a,(O, t) = 5
(2.2)
and it is assumed that 1t is convenient
to introduce the kinematical quantities aij, clii, Kij, i3i .
a,i =
n =
Njj,
~;j
through
det cl+ (2.3)
aaj Kij
=
23, ’ s,
where 8;. is the Kronecker delta and Latin indices have the values 1.2,3 and repeated indices are summed. We shall also adopt the notation that Greek indices have the values I, 2 and the usual summation convention still applies. We note that atlij Kij+
Kji
=
(2.4)
-. afl
A superposed dot is used to denote the material time derivative t, holding 19fixed. Hence the velocity of a point on c is given by
with respect to 12.5)
V-;i
and the director velocities are defined by w, = i,. We put y = uiai = viai,
(2.6) W, =
14’&li.
(2.7)
By taking the material derivative of (2.2) and making use of (2.3) we find that
(2.8)
319
Ideal fluidjets
Thus, if 271~= a*
’
aj+&
’
(2.9)
$
then
2na3 =
W,3
+
$
dV3 z--
7733 =
-
K;I”V,
(2.12)
K@,.
From (2.6), we observe that
aw, D -=a0
uaa
(2.13)
a0
Dt
where D/Dt denotes the material derivative, and from this result, we may show that?
.
awaR
‘GUS=
- a*
. Ku3
aWa3
=--
-
ai3
W,K;
+
(2.14)
WpsK:
(2.15)
W,Ki’+
We note that the element of length along the curve c is ds = V$233) de. 3. BASIC
The local equation of mass conservation
THEORY
is
~(P~U33) =0
(3.1)
where p is the mass per unit length of c, so we put pda33 = k.
where k is a function of 8 but is independent
$+ 7pR -
#3 +
oft. The equations of motion are
K.jnr+ kf i =
qR(’ + p”3K;e
(3.2)
&.a’,
(3.3)
pYRK;p- PYaK$R = -
paPKd
-
,f
=
0
0
(3.4) (3.5)
where the six quantities fl’ are defined by+ ,,d
=
kq*i +
+!$?+K:par.
(3.6)
tPerhaps the easiest way of obtaining (2.14) and (2.15) is to use (2.6) together with equation (2.22) of Green and Laws[ I]. SThe quantities @’ used here differ from the corresponding quantities in the work of Green and Laws[ I] by a factor d/(as3).
320
Ideal fluid jets
Also, d are force components and pai director force components along the vectors ai, fi are components of assigned force per unit mass and q”’ are components of the
difference between assigned director force and director inertia terms. We taket cxptip . a’
qpi = P-
Cpnot summed)
(3.7)
where (Y~,(Yeare functions of 13but are independent oft and Pi are components assigned director force per unit mass. The energy equation is -ti-k(TS+Sf)+kr+
of the
(n3--p”3K~)r/33+t(nP-pa3K~)TJ~3
where A is the Helmholtz free energy per unit mass. Y is the heat supply function per unit mass per unit time, h is the flux of heat along c per unit time, S is the entropy per unit mass and T is the (positive) temperature. The entropy production inequality is (3.9) From (3.8) and (3.9) we obtain -k(k+s?)
+
(n”--p*3Kg)~33+2(na-p*“K~)r)a3 h
+$(?7*‘+
da-pypK;a
-pYaK;P)
Qp
aT
+paikai-~~
2
0.
(3.10)
We wish to emphasize that the preceding work does not depend upon the choice of constitutive equations. We now obtain the constitutive equations for what we call an ideal incompressible fluid jet. To do so, we first define a jet to be incompressible if it is susceptible to only those motions for which* V(a) = [a,a,a,] is independent
oft. Hence, for an incompressible
jet we have
nij7)ij= 0 that is (3.11)
uaDv),p+ 2aD3r)p3+ Cz33r/33 = 0.
We multiply the left hand side of (3.11) by ~V\/(U~~) and add the resulting expression (which is zero) to (3.10): - k(k + s ?-) + (n3 -
pa3Kg
+
733-t 2 ( np - Paz&f + paP3du33) %33
pL7332/U,3)
+ paif&+ 4 (Tao + i+ -
pypK;a
-
pYaK;P
We are now led to define an ideal incompressible . equations: A =/I(T),
S=S(T),
+
2pUaP&33)
7),@
- f
g
2
0.
(3.12)
fluid jet by the following constitutive h = h(T,dT/dO)
(3.13)
tThis means that we take the yuBof Green and Laws [ I] to have the values y” = u,. y12= y” = 0. yzs = LY?. and this is consistent with the discussion of section 5. $See the discussion of section 5.
321
Ideal fluidjets
and that the coefficients of rf33, qB3, qap, ri,$ in (3.12) depend only upon T. Next we choose p so that the coefficient of r/33 in (3.12) vanishes. The remaining independent variables vp3. nn8, ri,i, ? can then be chosen arbitrarily. Hence (3.14)
np= -paR3V/a33, n3 =
- pa33Vla33
(3.16)
-h$20.
(3.17)
The function p is an arbitrary scalar which represents the resultant a section of the jet. The energy equation (3.8) now reduces to kr-kT+=
across
(3.18)
0.
Since p”‘= 0, we collect the simplified forms of the equations and (3.5): rr”B= 7@“,
pressure
of motion (3.4)
@3=&3
(3.19)
(/3 not summed).
(3.20)
where we now have @i = k/Pi - karRGR . ai
With the help of (3.16) and (3.20), equations (3.3), (3.1.5)e and (3.19) provide nine equations of motion for the determination of the position of the curve c and the two directors a,. Also equations (3.14) and (3.18) provide a single equation for the temperature. A rather obvious result of our constitutive assumptions is that the thermal and mechanical effects are distinct. In the following sections we consider only the mechanical problem. 4. KINEMATICS
OF MOTION
ALONG
A FIXED
CURVE
In ffuid dynamics it is customary to formulate the theory and boundary-value problems in a spatiai as opposed to a material description. In this section we shall discuss this transition in the case when the curve c is in motion along a fixed curve 4. We have delayed this discussion until now for the earlier part of this paper is valid whether or not the motion has any simplifying features. We introduce rectangular Cartesian coordinates x, and use upper case Latin indices to denote Cartesian components. Thus, let the fixed curve V be given by
r = r(5) = xx@) e, where eK are the unit Cartesian base vectors
and 5 is a fixed parameter
(4-I) defining %?.
322
A. E. GREEN
and N. LAWS
Now the material point with convected coordinate 8 occupies various places at different times and, since .$ parametrizes %?, we may denote the place occupied at time r, by the material point with convected coordinate 8, by 6 = f(8, t). The Cartesian components
(4.2)
of the velocity of a point of c are (4.3)
But we have already denoted the velocity by v, hence
ax, D,$ V=U~&K=-~Ka[ Dt
where we have used (2.2). Since v = uiai
we have (4.4) Any variable associated fixed coordinates. If
with the motion can be written in terms of convected
or
then DF -=-+-- af Dt at
af Df
ag Dt
(4.5)
and (4.6) If we now choose the convected ordinate 6 at time t, then
coordinate aF af -=--, ae ag
8 so that it coincides
v32&
with the fixed co-
(4.7)
With this choice of 8, there is no ambiguity if we do not indicate the arguments of functions at time t, since they will always be 4 and t. It is apparent that the preceding caicuiation is similar to that used by Oldroyd[4]. Next we deduce the reduced forms of the acceleration, the director velocities and the director accelerations. These are the only kinematical quantities which are needed
323
Ideal fluid jets
in the theory of the ideal incompressible fluid jet. The acceleration of the material point which occupies the place with fixed coordinate 5 at time t is given by
av
3av v=ar+vag 2+v3 au3 ( .
-a3 at
+ v3K’1 3 r
(4.8)
* >
Also the director velocities w, are found to be w, =
and the director accelerations
2 +v3K;a,
(4.9)
become
In classical fluid dynamics it is customary to speak of steady motion as being motion for which the velocity of the successive particles at a fixed spatial point is constant in time. Here we adopt a slight variation of this concept and define a motion to be steady if the vectors af and the velocity v are functions off alone: v=
ai = ai(
(4.11)
v(l).
We use this definition of steady only for convenience and note that certain motions which are steady in the three dimensional sense are not steady in the sense used here. It is apparent that many of the kinematical results of this section are much simplified in steady motion.
5. THREE
DIMENSIONAL
CONSIDERATIONS
The preceding theory employs only one dimensional concepts and makes no attempt to say anything about a full three dimensional theory. However, in some recent work, Green, Laws and Naghdi[3] have been able to shed some light on the relationship between the general theory of Green and Laws[ 11 and the classical three dimensional theory of continuum mechanics. In this section we outline some of the results of Green, Laws and Naghdi[3] which are important in the discussion of specific problems. Let the points of a three dimensional continuum be defined by a general convected coordinate system 8’. Also let the covariant and contravariant base vectors at points of the continuum at time t be denoted by g,, gi with corresponding metric tensors gij, g”. Thus if R is the position vector of a typical particle 0’ at time t, then dR
g,=ae” gij=
The stress vectort
gi'giv
R=R(81,02.83,t), gij
=
gi
. g,
i &.gi=@.
(5.1)
1
across a surface whose unit normal is u can be put in the form t=;;)
(5.2)
324
A. E. GREEN
and N. LAWS
where U =
Ti = +g$(g).
Uig’,
I: = det gij
(5.3)
and +j are the components of the symmetric contravariant stress tensor. When given surface forces P are applied at the boundary surface of the continuum then t= P.
(5.4)
The three dimensional body force per unit mass is denoted by F and the three dimensional density by (T. The parametric equations f? = 0
(5.5)
define a curve c in space, at time t which we take to be the ‘line of centroids’. This curve may be identified with the curve c used previously so that the equation of c is 8” G 0.
r = r(0. t) = R(O,O, 8”, t);
(5.6)
We assume that the region of space occupied by the three dimensional some neighbourhood of c which is bounded by a surface?
continuum is
f(8’. e’) = 0.
(5.7)
The work of Green, Laws and Naghdi[3] shows, amongst other things, that the directors a, are just the values of the base vectors g, on the curve c: a, = a&d, t) = %(O,O, f13.t).
(5.8)
Hence the directors supply some limited information about the deformation of the cross section. Also the quantities which we have designated mass per unit length of c, assigned force per unit mass, f, and assigned director force per unit mass. I*, may be identified by the formulae ~v$I&
pfX+zas) = j j crFv(&dB’d82+$
P(u’dBZ-2dB’)
= ~~crFX&)dB’d0*+$ pP~((a&
(5.9)
= j j- aV?g)dO’d@
= j j cr~F~(~)d01d02+
q/(n)
(TId@-T2d01)
(5.10)
$ POU(lclde”- zc”dO’)V\/(R) (5.11)
= IIa~~F~~a)d8’de’+~fP(T,dHL--T,dB’)
where 14 are defined in (5.3). The double integrals are over any section f13= constant and the line integrals are taken along the curve 8” = constant.
f(0’. f3’) = 0.
In addition the vectors n and p* may be associated system by n= II
with the three dimensional
(5.12)
T:,dO’d@
p* = j j &T,dO’dB” +Note that (5.7) does not necessarily
imply that the continuum
stress
has constant
(5.13) cross section.
ldeal fluid jets
32s
the integrals again being over a section @ = constant. The director inertia coefficients q, ar, (see (3.7)) are given by PCY&“(~~~)= f s crV?g) (B8)2dB’d~2.
(5.14)
In the three dimensional theory it is usual to assume that V(R) =
lkg,g31 '0
for all values of 6’. In particular it is true for 8” = 0 so we have l/(n)
(5.15)
= [afa2a3] > 0.
If the three dimensional continuum is incompressible then d(g) is constant in time. Since Q is the value of g on the curve c we are led to a definition of an incompressible jet by d(a) = constant in time. (5.16) The problems discussed in the later part of this paper are concerned with the motion of a jet through the atmosphere which exerts a constant pressure p. on the surfaces of the jet. In the absence of three dimensional body forces, we see from (5.3). (5. IO) and (5.1 1) that ,efV?a,,) pla~(a,~)
= $ (T~d~~-T~d~l)
(5.17)
= 4; P(T1d6z-T2d@1}
(5.18)
where Ti = -pOgi”gjV&). 6. A STRAIGHT
CIRCULAR
(5.19) JET
We consider the motion of an ideal incompressible jet with constant circular cross section, and whose centre line c is in motion along a fixed straight line %‘.We use the theory developed earlier to characterize the main features of such a motion. We make use of the notation of section 4 whereby xX denote rectangular cartesian coordinates and e,- is the standard orthonorma1 basis. It is convenient to choose the convected coordinates to coincide with the fixed coordinates 5’ at time c where?
where c(I is independent given by
of el. t2, &. We assume that the lateral surface of the jet is
so that b is the radius of the (uniform) cross section of the jet. From (5.1) and (6.1) we have, at time c, tWe especially choose this ‘twisted’ coordinate system to exhibit a solution which corresponds to a twisted jet.
326
A. E. GREEN g, =
and N. LAWS
e, cos $15+ e2 sin $<
g,=-ee,sin$~+e2cos@ g3=-1cl(&
(6.2)
sin95+e2
1
cos J15)eI +$J(& cos J15-t2 sin$t)e,+e,
and with the help of (5.8) g,= a,,
g3= a3+JI(51a2-52aI)
(6.3)
with a3 = e3.
(6.4)
Hence gij
=
1
0
0
1
$61
L-442
441
1 +dJ2t4:+m
-
$52
1-t
g’j = 1 ’
vJ”5E
-q12.f& i
952
$62
-$25152
I +$A$; -!@I
--$4, 1
g=a=l
I
(6.5
and a..LJ= & = 8..1.l.
(6.6
These kinematical formulae enable us to show from (5.9) and (5.14) that k=p=mrb2,
,+za2=fb2
(6.7
kf=o,
kl” = -p,7rb’a*.
(6.8)
and from (5.17)-( 5.19) that
The preceding calculations only serve the purpose of specifying the geometry of the jet at time f. which in turn gives us the position of the directors and ‘surface loading’ at the current time. We must now use this information in the basic theory of section 3 to investigate the motion of the directors and the centre line of the jet. This theory will not supply a complete solution to the full three dimensional problem. We first look for solutions of the field equations in steady motion so that we take $ = constant.
(6.9)
From (6.2) and (6.3) we see that
and that the only non-zero components
of
K 12 =
Kij -
are K2,
=
$J.
(6.10)
We assume that v3 = constant = V
(say)
Since the motion is steady, we obtain from (4.9) and (4.10) w, = VJla,,
w2 = - V$a,
(6.12)
i, = - V2J12a,,
Gt, = - Vf!J2a2.
(6.13)
Also, because a, is a unit vector, V is the speed of the centre line of the jet.
327
Idea1 fluid jets
With the help of (6.7) and (6.8) and the definitions (3.20) we may show that +I = &?z= - porb2 -t-4kb2V24fr2
(6.14)
+2=q==fl~3=qr23=2:(),
(6.15)
In this example, the constitutive
equations (3.15) and (3.16) yield np = 0 . *ll
=
792-t
r2?
$=-p
(6.16)
=
-p
(6.17)
7121 z
().
(6.18)
From (6.15) we see that (6.18) is automati~aIly satisfied by the assumed deformation field. Those equations of motion (3.3) and (3.19) which are not trivially satisfied show that we need n3 = -p = constant p = porb2 - $kb=V’%j?.
(6.19)
Ifp is the average pressure in the jet so that p = jhrb2
(6.20)
then (6.21) Thus if p. = ji, JI = 0 and the jet suffers no rotation. It is perhaps, instructive to explain the significance of these results whent p. > p. The solution (6.21) tells us that an observer situated at a fixed point on the line %’sees that the directors at that point do not vary in time. However, an observer situated on a moving particle sees that the directors attached to that particle describe a circle with constant angular velocity Q, where
~(Po-P)
a2=
ub2
’
We therefore say that, provided p. > p, an elementary exact solution predicts that the jet twists and that the twist may be in either sense. We think it worthy of note that TruesdelI~2~ has conjectured that a theory of the type proposed here would enable one to obtain this sort of result. Let us now consider a motion which is unsteady in the sense of section 4. We make use of (6.1) to (6.8) but put g=0 and look for a solution in which v3 = V = constant a, = ra2,
a, = -Ta,
tin order that the pressure be positive everywhere in the jet, p must be positive. This is only a necessary condition but is likely to be sufficient if~~2~~z/p~ is su~cientiy small.
328
A. E. GREEN and N. LAWS
where r is a constant. It is easy to verify that such a deformation of motion and that I’ is given by
satisfies the equations
(6.23) This solution corresponds to the rigid rotation and translation of the jet. It is noteworthy that this particular problem may be solved from the full three dimensional equations of an ideal incompressible fluid. The result (6.23) agrees with that obtained from three dimensions. [l] [?I (31 [4f
REFERENCES A. E. GREEN andN. LAWS.Pvot. R. Sot. A293,145 (1966). C. TRUESDELL, Pn’rzciptes ~~Con~j~~u~ ~e~~~~~ics. p. 3 19. Socony Mobil Oil Co, Dallas, (I 960). A. E. GREEN. N. LAWS and P. M. NAGHDI, Proc. Cnmh. Phil. Sot. To be published. J. G. OLDROYD, Prot. R. Sot. AZOO,523 i 19501. (Received
2 Jnttuctry
1968)
RCsumb-On &die les iquations de constitution d’un jet de fluide. ideal, unidimensionnel et incompressible et se basant sur une thCorie gCn&ale don&e par Green et Laws [ 11.On examine la cinematique particuli6re d’un jet, en mouvement le long d’une courbe dCterminCe. II est montrC qu’un jet circulaire, dans I’atmosph&re. peut se diplacer en ligne droite, B une vitesse uniforme et se tordre dans les deux sens autour de son axe. Zussammenfasung- Auf der Grundlage einer von Green und Laws aufgestellten Theorie[l] werden die Konstitutivg~eichungen fir aus einer i~komprimierbaren. idealen Fliissigkeit gebildete Strahlen bestimmt und die speziellan kinematis~hen Gesetzm~ssigkeiten eines sich entlang einer festgelegten Kurve bewegenden Strahfs erijrtert. Es wird nachgewiesen. dass sich ein kreisfdrmiger Strahl in der Atmosphere mit gleichfdrmiger Geschwindigkeit bewegen und in beiden Richtungen urn seine Achse verdrehen kann. Sommario-Si impostano equazioni costitutive per getti ideali di fluide incomprimibiii monodimensionali, in base a una teoria generale avanzata da Green e Laws[ I]. Si discute le cinematica speciale di un getto in moto lungo una curva fissa e si dimostra the un getto circolare nell’atmosfera si pub spostare con velocitB uniforme lunge una retta e &are in entrambe le direzioni sul proprio asse. ,&CT~KT--BblBOnRTCR KOHCTHTyTHBHble ypaBHeHMRnna OAHOpalMepHblXCTpyfi HeCXGitvlaeMOti HReaIibHOfi x~~KOCTW,Ha ocnoaarimi o&ue& -reopua co3naHHOB PPHHOM M flOyCOM [I]. 06cyxQaeTcn CneuIiailbHaR KMneMaTuKacTpyH, nponalzraloluetica no ycTaHosneHwoR ~pa~0ir. .QoKa3bmaeTca, 410 Kpyrslas crpyn MO)l(eT“pOJlBMraTbCfiB aTMOC+epe C paBHOMepHOi% CKOpOCTbtOn0 npaMOi? JIMHMHM CKpyqH5aTbCa B G60MXHanpaaneHwnx BOKpyf CBOeiiocn.