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An extended theory for incompressible viscous fluid flow
lmn~d
ELSEVIER
J. Non-Newtonian Fluid Mech., 66 (1996) 233-255
An extended theory for incompressible viscous fluid flow A . E . G r e e n ~'*, P . M . N a g h d i b't ~Mathematical Institute, 24-29 St. Giles, Oxford, OXI 3LB, UK bDepartment of Mechanical Engineering, University of California, Berkeley, CA 94720, USA Received 4 April 1996
Abstract
The classical Newtonian theory of incompressible viscous fluid flow is extended so as to contain explicitly the vorticity vector and spin of vorticity vector. Examples of the theory are given, some of which relate to the problem of turbulent flow, and reference is made to some experimental results. Keywords: Viscous fluid flow; Vorticity
I. Introduction
In the context of a thermodynamical theory, the classical formulation of viscous fluids can be based on the equations of mass conservation, linear m o m e n t u m balance, as well as on entropy balance [1,2], together with appropriate constitutive equations which include that for a linear (Newtonian) viscous fluid. The equations for mass conservation, m o m e n t u m and entropy balance are , 6 + p div v = 0 piJ = pb + div T PO = p ( s + ~ ) - d i v p
(1) t = Tn k=p
(2) •n
(3)
In the above equations, p is the mass density of the fluid, v is the velocity vector, b is the external body force per unit mass, T is the Cauchy stress tensor which is related to the stress vector t by the second part o f Eq. (2), r/ is the entropy per unit mass, s is the external rate of • Corresponding author. t Decreased. 0377-0257/96/$15.00 Copyright © 1996 Elsevier Science B.V. All rights reserved. PH S0377-0257(96)01478-4
234
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supply of entropy per unit mass, ~ is the internal rate of generation of entropy per unit mass and p is the entropy flux vector which is related to the entropy flux k by the second part of Eq. (3). Also in Eqs. (1)-(3) a superposed dot designates material time differentiation and "div" denotes the divergence operator with respect to the current position x in the fluid. Further, constitutive equations for a Newtonian viscous fluid in the presence of thermal effects can be stated as T = p l + 221 tr D + 21tD
p = p 2 O__~ ap
80
p = x ~---~
'I -
~~0k
(4)
where the kinematical variable D=½{~v/Ox+(Ov/Ox)r}, 0 ( > 0 ) is the absolute temperature, ~, = ~,(p, 0) is the specific Helmholtz free energy, ! is the identity tensor, the scalar x in the third part of Eq. (4) is the conductivity coefficient and 2 and/L are the viscosity coefficients. The system of Eqs. (1)-(4) has been used successfully in the study of a wide range of problems in fluid dynamics which can also be verified experimentally. It has not, however, been possible to explain many features of the flow on the basis of Eqs. (1)-(4) alone. For example, problems concerned with turbulent flow are not covered by these equations and most of the theoretical developments in this direction involve additional equations which are rather ad hoc. It is clear from extensive available experimental results and also to some extent from existing numerical simulations that not only the vorticity (or spin) but also the rate of change of vorticity (or "spin of the spin") affect the structure of turbulent flows. Given this premise it is natural to see if a satisfactory theory can be constructed with the vorticity vector w defined by Eq. (9) and the spin of vorticity vector u defined in Eq. (10) explicitly displayed in the theory, instead of just being evaluated at the end of the solution of problems from the velocity vector v. Moreover, such a theory if properly based on continuum thermodynamic equations would have interest in its own right, apart from its possible application to turbulent flow. A general theory involving vorticity and spin of vorticity explicitly, together with two temperatures, the second of which is motivated by turbulent flow as described by Marshall and Naghdi [3,4] is presented in Sections 2 and 3. In order to have a theory with as few extra constitutive coefficients as possible, a restricted theory for an incompressible fluid is deduced in Section 4. Application to particular problems is given in Section 5-8 with comparison with experimental results where possible as far as turbulent flow is concerned.
2. Kinematic and thermal variables
Consider a body ~ with particles (material points) X embedded in a three-dimensional Euclidean space g3.Choosing a fixed origin in e3, we identify each material point of ~ by the position vector X in a fixed reference configuration Xo of 8 . In the current configuration K of at time t, the material point which was at X in ro takes the place x. The region of space occupied by M in Ko will be denoted by Ro bounded by a smooth closed surface 0Ro, while the corresponding region occupied by ~' in its current configuration x at time t will be denoted by R bounded by a smooth closed surface OR. Also, an arbitrary subset of ~ in K at time t will be denoted by S, bounded by 0S,, and occupies a region P ( ~ R ) bounded by a smooth closed surface 0P with unit outward normal n.
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235
A motion o f ~ , defined in terms of a sufficiently smooth vector function •, assigns the place x to each material point in the current configuration r of ~ ' at time t such that x = x(X, t)
(5)
and the particle velocity at x is v = .~ = v(X, t)
(6)
where a superposed dot denotes material time differentiation holding X fixed. The velocity gradient L is given by Ov L -
(7)
Ox
We recall the unique decomposition of the velocity gradient in terms o f a rate of deformation tensor D and the vorticity (spin) tensor W: L = O + W
O = ½(t. + L T)
W = l(L -- L T)
(8)
where L v denotes the transpose of L. The vorticity tensor W is related to the vorticity vector (or spin vector) w by w = curl v
2 Wci = w × 4i
(9)
for every vector ~i. We define the "spin o f the spin" vector or spin of the vorticity vector u by u = curl w = curl curl v = grad div v - V2v
(10)
where " g r a d " and "div" denote the gradient and the divergence operators both with respect to x. Further, we define the gradients o f the vorticity vector and of the spin o f the vorticity vector by
~w N=-Ox
~u P=-Ox
(11)
We note that the skew part of N, namely l ( N - - N T) is related to the axial vector u by (N - N V ) a = u × a
(12)
for every vector ~i, which is analogous to Eq. (9). Motivated by the notion o f turbulent temperature given by Marshall and Naghdi [3,4] and the more recent ideas on temperature and the basic thermal variables of Green and Naghdi [2,5] we introduce two thermal displacement magnitudes ~H, ~r and two temperature fields On, Or such that I ~ , = ~H(X, t) > 0
OH = ~n > 0
i In line with the developments in (Section 7) the scalars ~tt¢, ctr (on the macroscopic scale) are regarded as representing (on the molecular scale) some "mean" thermal displacement magnitudes which for brevity may be called thermal displacements. Additional background information is given in the paragraph containing (7.3) of Ref. [5].
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236
o~r=OtT(X,t)>O
OT=~T)O
(13)
The scalar On will be identified with ordinary temperature in classical thermomechanics (here the classical theory of linear viscous fluid) and the second temperature will be identified as an additional independent thermal variable (such as the turbulent temperature) along with an additional kinematic variable (such as the spin of the vorticity vector). For later use, we record here the effect of a superposed rigid body motion (SRBM) on the foregoing kinematic and thermal variables. Thus, under a SRBM in which x in the current configuration x moves to the place x + in the configuration x + at time t + = t + a, a being a constant, we have (14)
x + = Qx + a
where Q is a proper orthogonal second order tensor function of time which satisfies det Q = I
QQT = QTQ = I
(15)
a is a vector function of time and ! is the second order identity tensor. F r o m Eqs. (14) and (15) it follows that 0 = fiQ
v + = £c+ = fi + Q v + f i Q x
fi + fl T = 0
(16)
where fi is a skew tensor function of time t which represents rigid body angular velocity and satisfies fi~i = to x ti
for every vector ~i
(17)
to being the rigid body angular velocity vector. F r o m Eq. (16) and Eqs. (8-12) follow the results O + =QDQ
T
w + =QWQr+fi
w+
= QW+2to
(18)
and u + = Qu
p+ = QpQT
N + = QNQ T
(19)
Under SRBM, the thermal variables in Eq. (13) transform according to a + = o~i~
o~+ = O~y
O~ = On
O~ = Or
(20)
3. Basic equations It is clear that vortices, represented by the vorticity tensor W in Eq. (8) or the vorticity vector w in Eq. (9) have a significant effect on the behaviour of turbulent flow. Usually the vorticity vector w is calculated from solutions of equations of the type (1)-(4) so it does not have a direct effect on the type of flow. Here we add a spin or vorticity vector w explicitly to the theory which in turn leads to new response functions, external forces and equations of motion. We also add further structure explicitly to the theory by including a spin of spin or spin of vorticity vector u defined in Eq. (10). This also requires new response functions and equations of motion. The basic kinematic variables are therfore taken to be
A.E. Green, P.M. Naghdi / J. Non-Newtonian Fluid Mech. 66 (1996) 233--255
r, w, u
237
(21)
Also, following Refs. [3,4] we use two temperature fields
On, Or
(22)
representing the usual temperature 0/~ and a turbulent temperature 0T. We adopt the procedure of Green and Naghdi [6] and derive the basic mechanical and thermal equations from the First Law of Thermodynamics which we state in two parts as follows: (a) For any subset S, of the body ~ in the current configuration K, the total rate of change of kinetic energy K(S,) and heat energy H(S,) is balanced by the sum of the external rate of supply work R(S,) to S,, the external rate of supply of heat Q(S,) to S, and the internal rate of supply of energy W(S,) to S,. Thus d dt [K(S,) + H(S,)] + R(S,) + Q(S,) + W(S,) = 0
(23)
(b) The total rate of external work and heat, i.e. R(S,) + Q(S,), supplied to or extracted from S, in a cycle is zero. Based on the variables (21) and (22) we assume that the contributions in (a) to the First Law have the forms
K(S, ) = fp ½p (r • ~ + 2ylor " w + 2y2ov • u + yll w " w + 2yl2w " u + Y22U • u) dv
(24)
H(S, ) = .te P(tlz-IOH+ tlTOr) dv
(25)
de
da P
W(St) = tP {pw + p f " r +½mr
+ P # I " w --F P # 2 " u --b p ~ H O H + p ~ r O r + mYIHOH "-b mr'/TO T 1
• I~-k-m l o v • w + m20 v " u + 2 m l l W • w + m l z w • u + ½m22u " u } dv
(28)
In Eqs. (24)-(28) Yl0, Y2o, Y~1, Yl2, Y2z, m, m~0, m2o, mll, m12, m22, tlH, tlT, SH, ST, k H , kr are scalar-valued functions, while b, c~, c2, t, ml, m2 are vector-valued functions of (X, t); and t, ml, m 2 , k H , kr, in addition to depending on (X, t) depend on the outward unit normal n to the boundary surface 8P enclosing the arbitrary volume P of the body in the current configuration x. Moreover, except for the scalar w in Eq. (28) the integrands in Eq. (25)-(27) are linear in the independent variables (22) and (22); and the integrand in Eq. (24) is quadratic in the variables (21) while the integrand in (28) is quadratic in (21). We further assume that the scalar w is linear in the temperature rates 0t4, Or, the temperature gradients 00it/8x, 80T/OX and the gradients of the variables (21), namely L, N, P.
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238
The basic equations may now be derived from Eqs. (23)-(28) using the methods indicated by Green and Naghdi [6]. Here we quote the final results. The equations of mass and inertia balances are dt
p dv =
m dv
PYll dv = dt
PYI2 dv =
f,
-dr
PY~odv =
mlo dv
mll dv
dt
PY2odv =
m12 dv
dt
d;,
PY22dv =
m2o dv
f,
(29)
rn22 dv
The mechanical equations are dt
P(r + Ylow + Y2ou) d v =
dr,
dt dt
f,
P('F~°vWyJlwWyI2u) d v =
P(.V2oV+ ylzw + yzzu) dv =
{mv + mlow + m2ou + p(b + f ) } d v +
f,
t da
(30)
P
{rnloe+mllw+rn12u+p(el +Pl)} dv+
ml" da (31) P
{m2ov + m12w -l- m22u + p(c2 + It2)} d r +
mz da
(32)
P
The entropy balance equations are p t l . dv =
{m~H + p ( s . + ~:H)} dv -
k . da
(33)
P
-~
P~lr dv =
{m~lr + p(Sr + •r)} dv -
(34)
k r da P
With the usual smoothness assumptions and boundedness properties, the application of Eqs. (29)-(34) to an arbitrary tetrahedron results in t = Tn
ml =M. ln
m2 = M 2 n
k.=p.,
n
kr=pr" n
(35)
where T, Mi, M 2 are second order tensor functions and Phr, P r are vector functions of X, t. Next, from Eqs. (29)-(34), after using (35), we obtain /~ + p dive = m
PPlo + (/~ + p div v)ylo = ml0
PPll + (jb + p div v)yll =roll
P P 2 o + ( f ~ + p divv)y2o=m2o
/9))22 -st- (/9 -+- p div V)Yl2 = m12
PY22 + (,6 + p div v)Y22 = m22
(36)
P(6 + Ylofi' + Y2oti) = p(b + f ) + div T P(Ylo~ + y~lff + Yl2ti) = p(c~ + t~) + div M~ /00;2013 -/- Y12 ff -t- Y22 li) = p ( C 2 "4- ~/2) -4- div M 2
Pflu = p(sn + (,~) -- d i v p u
(37)
A.E. Green, P.M. Naghdi / J. Non-Newtonian Fluid Mech. 66 (1996) 233 255
239
POT = p ( S T + ~T) -- divpT
(38)
With the help of Eqs. (24)-(28) and Eqs. (35)-(38), the energy balance Eq. (23) yields the equation " OOH Ox
pw + T " L + MI " N + Mz "P-pH
P T " OOV OX
(39)
PqtIOH -- P?ITOT = 0
where N, P are defined in Eq. (11). Part (b) of the First Law now yields (for details, See section 3 of Ref. [2]) W(S,) -
dt
(40)
p~, dv
where ~, is a scalar function of X, t. F r o m Eq. (28) it follows that I
p w + p e t + m ~ + p ( f " v + # l " w + P2 " u ) + ½ m e • v + m l o v • w + m20v " u + s m l l w
+mlzw
• u + l m 2 2 u • u + P(~HOH -}- ~TOT) "}- m ( q t t O H + q TO T) = 0
' w
(41)
and when this is combined with Eq. (39) to eliminate w we then obtain OOH Ox
-p(~+qnt)H+qrOT)-m¢+T'L+M~'N+M2"P-pt4"
-
p ( f " v + ttl " w + la2 " u ) - - ½ m v
• v--mint
" w-
|
m2ov • u - - ~ m l j w
-- ½m22u • u -- P(~.I-IOH + ~ r O r ) -- m O l n O n + q r O r )
OOT P T " OX
• w-
ml2w • u
= 0
(42)
which is the residual energy equation. U n d e r a SRBM Eq. (14), in addition to Eqs. (16)-(20), we assume that T + = QTQ T
M~( = Q M , Q T
l ~- = Q f
la~i = Q p ,
~
p+ = p
= ~-r
m~-e = rn12
M~ = QM2Q x
# 7 = Qp2
m+ = m
P~t = QPH
m{o = mm
rn~-l = mll
m2~o= rn2o
m+2 = m22
(43)
We also assume that the residual energy equation (42) remains invariant under any SRBM Eq. (14), so that using Eqs. (14)-(20) and Eq. (43) we may deduce that m- 0
f:0
mlo = 0
m2o = 0
rnll = 0
1
(44)
pp~ + m12u = 2ei × Tei
where e~ is an orthonormal basis. With Eq. (44) the residual energy equation (42) reduces to ~0H -p(~)+rlnOr+rlTOr)+
T'D+M~'N+M2"P-pp2"u-pI4"
- ½m22u • u -- p ( ~ I 4 O H + ~ r O r )
= 0
~x
pT'~Or~x
(45)
A.E. Green, P.M. Naghdi / J. Non-Newtonian Fluid Mech. 66 (1996) 233-255
240
To complete the theory, constitutive equations must be specified for T, M l , M2, 1u2,PI~, Pr, ~l~I, ~IT, ~H, ~r
(46)
which appear in Eq. (45). The residual energy equation (45) is then interpreted as an identity for all motions and temperature distributions, which places restrictions on constitutive equations In view of Eq. (44) Eqs. (36) reduce to + p div v = 0 PJ~12 =
Ylo :
ml2
P.922 =
Y2o = y2o(X)
0
Y20 = 0
m22
Yll = 0
Yl0 = Y l 0 ( X )
Yll = y~l(X)
(47)
The theory, in its general form, leads to a set of differential equations with a large number of new coefficients, which are rather unmanageable from the point of view of understanding turbulent or other flows. Since we are seeking a theory which contains that of a Newtonian viscous fluid in the presence of thermal effects, in the rest of the paper we restrict out attention to one which will reduce the mechanical differential equations to one equation. Moreover, we only consider an incompressible fluid although this extra restriction may be dispensed with without undue complication. It does, however, at this stage exclude any comparison with theories for rarified gas dynamics.
4. Restricted theory for incompressible Newtonian fluids with vorticity and spin of vorticity When the fluid is incompressible div v = 0
(48)
and since the divergence of a curl operation on a vector is zero, from Eq. (10) we also have div w = 0
div u = 0
u = -V2v
(49)
The natural extension of the Newtonian viscous theory to include explicitly vorticity and spin of vorticity is to suppose that T, Mr, ME depend on N, P in addition to D. Bearing in mind the last paragraph of Section 3 we make the restricted assumption that T = - p l + 2/~D + 2t~lP Ml = - - p l l + ill S 312 = - p 2 1 + 2/~1g + 2(/~ 2t//u)P
(50)
where p, Pl, P2 are arbitrary scalar functions of (X, t), P is the usual viscous coefficient and ~/l is a second coefficient. F r o m Eqs. (44) and (50) we then have flfll + m l 2 H = fll curl u = --,/./l V 2 w
(51)
Next we take Yl0 ---~0
Yll = 0
Yl2 = 0
m22 = 0
/*2 = 0
(52)
A.E. Green, P.M. Naghdi /J. Non-Newtonian Fluid Mech. 66 (1996) 233-255
241
so that ml2 = 0
Y22 = Y22(X)
(53)
Y20 = y z o ( X )
Also. in the light of the first and third parts of Eq. (37) and of Eq. (50) y2o =/~,/~
y_,~ = ~2/~2
(54)
The equations of m o t i o n (35) now reduce to p {b + (/q//t)ti } = p b - g r a d p +/~/V2v - 2]./IV41~ 0 = pc~ - grad pt p ( l ~ / I t ) { p b + (/t,//t)ti} = pcz - grad pz + (/tl//t)(/~VZv - 2/t,V4v)
(55)
Finally we choose c~ = ( / t l / / t ) b
P2 = (lq/It)P
Cl = O
pl = constant
(56)
Then Eq. (55) reduce to the single equation p {13+ (/zl//t)ri} = p b - g r a d p +/~VZv --2/llV4V
(57)
which contains just one coefficient /q in addition to/~. At this point we use one aspect of the Second Law of T h e r m o d y n a m i c s (Ref. [7]) in the form (58)
p(~HOn + ~TOT) • 0
for all motions. T h e n from Eqs. (42), (50) and (52) we have P(~_HOH + ~TOr) = --p((~ + ti~OH+ q r O r ) - - p .
~0 H O0 T ".~---~-- - P T "--~X + 21tO " D + 41~,O" P
+ 2(/t~//t)P " P >~ 0
(59)
Guided by Eq. (59) we now assume further constitutive equations = c . { O H - - OH In 0/_/} + c r { O r - - O r l n 0T}
tiH= CH In 0 ,
tlr = Cr In 0 r
K/-t ~0H PH=-
tOT0 0 r
0 o OX
PT=
(60)
0 o OX
where cn, cr, KH, •r are constants and 0o is a constant standard temperature. With the help of Eqs. (59) and (60) we choose constitutive equations for ~n, ~T. The scalar ~ must contain the terms which correspond to those for a Newtonian viscous fluid so we assume that KH S0H SOn p ~ t O H = -~o SX P~TOT
S----X-+ 2ttD " D + (9
XT SOT. SOT
Ox
S--x-+ 4 / q D • P + 2(tt21/it)P . P -
c~
(61)
A.E. Green, P.M. Naghdi / J. Non-Newtonian Fluid Mech. 66 (1996) 233-255
242
where ¢ is a constant. (For some purposes ¢ could depend on temperatures.) The first two terms on the right-hand side the first part of Eq. (61) are those corresponding to a Newtonian viscous fluid while ¢ is an interaction term between ~H, ~r. Substituting the constitutive equations (60) and (61) into the entropy balance equations (38) gives KH
pcHO. = pSHOn + ~ o V20~ + 2/tD • D + ¢
pcTG = psTOT +
KT
V20
+ 4
(62)
,0" P +
• e -- ¢
(63)
It is instructive to express the single mechanical equations (57) and the thermal equations (62) and (63) in nondimensional form as an aid to further developments. We denote standard reference values of length, speed, density and temperature by the constant lo, Vo, Po, 0o respectively and introduce nondimensional coefficients by Re - polovo
Rl -- p°13v~°
/t
Pel - poloV -- o XnOo
f12 = Rl 2 Re
/Zl
Pe2 = poloV XTOo
(64)
The coefficient Re is a Reynolds number and Pel, Pe2 are P6clet numbers. Using a circumflex to denote nondimensional quantities, we set x =/02
v = vo~
t = __l°/,
w=
Vo
U0 ^
To
w
I)0 ,,
II = l"~0II U2OH
CH= Oo
p = pOtO
OH = 0o0 H
V~T
CT= O0
P=
0 T = 0o0 T
2^ povop Re
¢
2povo3Re q5
(65)
We express Eqs. (57), (62) and (63) in nondimensional form with the help of Eqs. (64) and (65). Thus, omitting the circumflex on all nondimensional quantities, we have P Re ~
L
= pb - grad p + V2v - l/~ V%
(66)
where b is nondimensional. Also, after setting external supplies of entropy equal to zero, p Pel cn(tu = 31~-72'02 --H+2(Pel/Re) ( D'D+
Re2 R~ ¢']]
p P % c r O r = l w52- a ~,T+ 2 4(Pe2/RI) { D " P +-2-~ Re P " P - ~~I Re ¢ }
(67) (68)
We observe that equations of the form of Eq. (66), without the extra inertia term involving u, have been proposed before in a somewhat ad hoc manner. Here the equation is based on the
A.E. Green, P.M. Naghdi / J. Non-Newtonian Fluid Mech. 66 (1996) 233-255
243
full equations both mechanical and thermal, of continuum mechanics after special choice of constitutive equations, applying only to incompressible fluids. It remains to be shown that the theory has any relevance to turbulent flow in viscous fluids, although the theory has some interest apart from this.
5. Plane Poiseuille flow We use a rectangular system of nondimensional coordinates xi = (x, y, =) associated with an orthonormal basis ei and choose the x~ = x axis along the centre line of a channel of uniform thickness 21o whose sides are bounded by the planes y = + 1. We seek a state of flow along the ej direction with the velocity field defined by 5v v = v(y, t)el
a2v
w = - - ~ y e~
u -
ay 2 el
(169)
where v is a function of y, t. Then with body forces b = 0, Eq. (66) gives
Re(au \at
@ ay
o
1 ~3/.) ~ 2fl 2 at 5 y 2 J =
@ az
ap
a2V
a x ~- ~5 y
1 a% f14 5y4
o
(70)
if we take the nondimensional value of p to be one. We seek solutions of Eq. (70) which are even in the variable y and which include the classical steady-state Poiseuille flow when t = 0 and fl is large. Thus v = L + 5 (A 1 _y2)+
Be;" cosh(~fly)
p=
-Ax
(71)
where Re2(1 -- ~g i 2) ~---0~2fl2(1 -- 0~2)
(72)
and :¢, L, A, B are constants. To determine the constants L, B we assume that the velocity at the walls 3'-- + 1 is V so that V = L + B cosh(~fl)
(73)
Also, we assume that the component of the couple m~ at the walls in the direction e3 normal to the channel is zero when t = 0. Hence, from Eqs. (35) and (50) aw
m l • e3 = ( M l e 2 ) " e3 =/IN32 =/z ~ y = 0
(74)
a2v ay 2 = 0
(75)
i.e. ( y = +1)
A.E. Green, P.M. Naghdi / J. Non-Newtonian Fluid Mech. 66 (1996) 233-255
244 or
--A "-t-o~2f128 cosh(~fl) = 0
(76)
v = V + A{2(1-y2)+e~'[c°sh(°~flY)~c°sh(°~fl)~ 0~2~2 cosh(~fl) J
(77)
Then
When fl is large this solution reduces to the classical Poiseuille flow v = V+½A(1 - y 2 )
(78)
~2>2
(79)
If or
~2<1
then, from Eq. (72), A > 0 and the solution Eq. (77) becomes large with increasing time. However, if 1 < ~2 < 2
(80)
then A < 0 the solution Eq. (77) reduces to Eq. (78) when t is large. When =1
then
2=0
(81)
and Eq. (77) becomes a steady-state solution v = v+ A
{~
(1 _ y 2 ) ~
cosh(fly) - cosh fl ~ ~-~s~/~
J
(82)
The total rate of flow across any section of the channel, per unit width, in the el direction is tanh fl]~ The maximum value Vm of v occurs at the centre of the channel where y = 0 so that Um = V + m
If
-cosh,} fl
'~ ]~ 2 c o s h
,84,
(V/Vm) = k when y --- _+ 1 then
A {1
-~ ~ + ~2 cosh/~ )
~-
(85)
and cosh(fly) - cosh fl v (1 - k ) { 1(1 - y 2 ) + fl 2 cos-s-~ ) - - = k -~ /)m 1 1 - cosh fl 2 f12 cosh fl When fl ~
(85)
A.E. Green, P.M. Naghdi / J. Non-Newtonian Fluid Mech. 66 (1996) 233-255 U
- - = k + (1 - k)(1 - y2) = 1 - (1 - kZ)y 2
245
(86)
I) m
which is the classical result. However, when fl is small v 1-k -- = k + ~
(1 - y 2 ) ( 5 - y 2 )
(87)
/)m
We observe that neither the classical result Eq. (86) nor the result Eq. (87), which is a second steady state, depend on viscous coefficients Re or RI. Marshall and Naghdi [4] display a velocity profile for v/vm in their Fig. 3, obtained from their theory and c o m p a r e it with the experimental data of Laufer [8,12]. The graph of Marshall and Naghdi compares quite well with that of Laufer, except perhaps near the walls y = + 1 and y = 0.5. The m u c h simpler Eq. (87) depends on only one constant k and this constant essentially determines the value of v at the walls which would normally be chosen by experiment. Here we select a value k = 0.7. The graph from Eq. (87) then compares reasonably well with that of Laufer. T u r n i n g to the temperature 0 r we carry out the calculation for the case when fl is small or Re large. Assuming Or= 0r(Y) and that ~b is constant, and retaining only terms in Eq. (68) corresponding to large values of Re, we have d202r
2 Pe2
@ 2 - ] ~ - ~ i {~b -
(v,,)2}
(88)
where, from Eq. (82)
v = v + -fl- ~2A - (y2 _ 1)(y: - 5)
v'" = fl:Ay
(89)
We replace 0~- and ~b by 2PezA2f12 RI
and
(90)
A2f14a2
respectively where a is a constant, so that Eq. (88) becomes d20 2 @2 - a 2 - y 2
(91)
If 0T = 0- when y = _+ 1 and 0 r is an even function of y, then 0 ~ , - 62 = -~z(1 - y Z ) ( b - y 2 ) In order 6, b before the general and Laufer
b = 6a 2 - 1
(92)
to give a full interpretation to Eq. (92) we need to find values for the constants a clear comparison m a y be m a d e with the measurements of Laufer [8]. However, character of result from Eq. (92) follows that given by Marshall and Naghdi [4] [8].
A.E. Green, P.M. Naghdi / J. Non-Newtonian Fluid Mech. 66 (1996) 233-255
246
6. Poiseuiile flow in a circular tube We use the same system o f axes as in Section 5 where n o w the z axis is along the centre line o f a uniform circular tube o f radius 1o. We consider a steady flow along the e3 direction defined by r = (x 2 + y2)1/2
v = v(r)e3
(93)
where v is a function o f r. Then from Eqs. (9) and (10) it follows that w
=
~
el
-
r
- -
e2
u =
-
v"
+
(94)
e3
r
where a prime denotes differentiation with respect t o r. If b is zero Eq. (66) gives
r dr)k, ~P=o
~P=0
ax
ay
+
=o
(95)
An appropriate solution o f Eq. (95) is v = L
Ar 2 -4- + B I o ( f l r )
(96)
where, A, B, L are constants and Io is a modified Bessel function o f the first kind. We suppose that the value o f v at the wall r = 1 is V so that A V = L - ~- + BIo(]?)
(97)
v = V + ¼A(1 - r 2) + B { I o ( f l r ) - Io(fl)}
(98)
and
Also, we suppose that the eo c o m p o n e n t o f the couple m, which acts across the surface r = l, is zero so that v" = O(r = 1)
(99)
or
-½A + flZBI~(fl) = 0
(100)
Hence v = V+¼A(1 - r 2) 4 A { I o ( f l r ) - Io(fl)} 2fl2I~(fl)
(101)
When fl --* o% the value o f v tends to v = V + l A ( 1 -- r 2)
(102)
A.E. Green, P.M. Naghdi / J. Non-Newtonian Fluid Mech. 66 (1996) 233-255
247
the classical solution. H o w e v e r , w h e n fl is small Aft 2
v = V+~(I
- r Z ) ( 5 - r 2)
(1(13)
The m a x i m u m value Vm o f V Occurs at the centre o f the channel where r = 0, so that l A {1 - I o ( f l ) } Vm = V + ~A +
(104)
If V/Vm = k when r = 1, then l/k = 1 + v
(105)
-4
and v
(1 - k )
(1 - r 2)-4
2flzig(fl )
) (106)
- k + v,.
1
+ 1 - Io(fl) 4 2f12I~;(fl)
W h e n fl -~ oo u
- k + (1 - k)(1 - r 2) = 1 - (1 - k ) r 2
(107)
1)m
which is the classical result. W h e n fl is small --= v
(1 - k ) k + - -
Um
7. C i r c u l a r
(1 - r2)(5 - r 2)
(108)
5
jet
H e r e we consider the flow o f fluid f r o m a circular orifice. It w o u l d be convenient to use a direct theory o f flow in pipes (or jets) as d e v e l o p e d by G r e e n and N a g h d i [9] but in order to avoid detailed reference to this p a p e r we derive equivalent e q u a t i o n s f r o m Eq. (66). W e take the axis o f the jet to be the z axis in the e3 direction o f an o r t h o n o r m a l basis ei with x and y axes along el and e2 respectively. W e assume that the jet is in a state o f steady m o t i o n b o u n d e d by the surface (109)
z = z(x, y)
over which there is a c o n s t a n t pressure. A t the surface 8z - v 3 + V, x +
Oz
= 0
where the velocity vector v* o f the fluid has c o m p o n e n t s
(110)
248
A.E. Green, P.M. Naghdi / J. Non-Newtonian Fluid Mech. 66 (1996) 233-255
(111)
v* = vjel + v2e2 + v3e3
Here we assume that (112)
v* = v + xwl + ywz
where v, wl, w2 are functions of z of the forms wl = wl el + w2e2,
v = re3
w2 =
--W2el
+ wle2
(1 13)
SO that Vl = XWI - - y w 2
v2 = --xw2 + y w l
(114)
v3 = v
Since the fluid is incompressible 2WI + V'= 0
(115)
where a prime denotes differentiation with respect to z. We limit attention to jets which are symmetric with respect to the x and y axes and define
R=ffdxdy Rll=ffxdxdy=ffy2dxdy
(116)
where integrations are over sections of the jet normal to the e3 axis. The incompressibility condition Eq. (115) is, alternatively, ~V 1
802
~V3
~x + ~ y +--~-z = 0
(117)
We integrate Eq. (117) over a section and use the surface condition Eq. (110) to obtain (Rv) = 0
(118)
Further, we multiply Eq. (117) by x 2, and integrate over a section using the surface condition Eq. (ll0) to get ~z (Rll v 2) = 0
(119)
Similarly, we multiply equations Eqs. (66) in turn by l, x, y and integrate over a section. This yields equations of motion for steady states: 0 pReRi,
{
, v,,,]} 1
vw~ + w 2 - w ~ - ~ [ v w r '
°{[
=~-'-~ R,i
'1}[
2w'l-~w]"
-R
+wlwf-w2w~] --p+2wl--~zw'[
}
]
(121)
,,I.E. Green,P.M. Naghdi/ J. Non-NewtonianFluidMech.66 (1996)233-255
249
p ReRii{vw'2W2wiw2-+[vw'~'+wiw'~+wzw'(]} = a~ a
{[ Rli w'~- ']}" +~w2 ~-5 w~'
"
(122)
From Eqs. (118) and (119)
Rv=k
Rllv2=kl
(123)
where k, kn are constants. Then Eq. (120) may be integrated to give
1]
[
kv-~v"
p Re
+A=R - p + 2 v ' - - ~ v "
(124)
where A is a constant. Eliminating p between Eqs. (121) and (124) and using Eq. (115) yields P Re ( k v - ~__k v" ~
-
-
v
V r __
,,
) + a + ----2-pRek.{ - ~ v, ,, + (~')2-w~-/S.L-2V~ 1 F 1 h,+41v,v,,,_w2w~l}
2-~v )
20z
7
v"-
,]} -~V Iv
(125)
Also Eq. (122) becomes
p Rek' 2
t.
.t~
L
--21v'w~-21"v w2J~])
Alternatively, Eqs. (125) and (126) may be expressed in the forms pRe
kv 4 - - ~ v3v" + Av 3 +p Rekiv - vv"+
= 3kV2kV ,(
+w2w~
(v')2-w2+~- 5
1~5v"')+kiv'[v"--~zv'"l-k'V[v"--~Sv"
p Rekiv vw'2-v'w2-~-~LVW2 -2v'w~-2v"w2
{
,
(127)
7}
1 w~,}+kiv{w~__~ 5w{,,}+kv2w ~ = - 2kl v' {w'2 _ ~5
(128)
We seek solutions of Eqs. (127) and (128) in the light of experimental work reported by Liepmann and Gharib [10] who considered a jet issuing from a nozzle at a Reynolds number Re = 5500. The diagrams in their paper suggest that the cross-sections of the jet have, approximately, the forms
r = a(z) + b(z) sin2{2n(z)O}
(129)
A.E. Green, P.M. Naghdi / J. Non-Newtonian Fluid Mech. 66 (1996) 233-255
250
where the z axis is along the central axis o f the jet and (r, 0) are polar coordinates in each section o f the jet. Also a, b, n are functions of z while n is an integer ___1 and the exit to the nozzle is at z = 0. F r o m Eq. (116) we then have =-=zc
a2 + 2ab +
/)
kl rc I 9a262 5ab3 3564) R , , - v2 - 4 , a 4 + 2a3b + - - - - 4 - - +----4-- + - i - 2 - ~ ;
(130)
The Reynolds n u m b e r of the issuing jet is large, which suggests that fl is small, so that in the n e i g h b o u r h o o d of the nozzle we seek solutions of Eqs. (127) and (128) of the f o r m v = Me-"'- + • • •
w2 = M e -m:
(131)
where M , )17I, ft, m are constants. The d o m i n a n t terms in the equations are satisfied if -1
=0
m
-1
or
m=2ti
(m-2ri)=0
(132)
Then ri=fl
and
m=fl
(133)
We choose ri = / ?
m = 2Ji = 2/?
(134)
so that the d o m i n a n t terms in Eqs. (129) become w2 = / O e -2/~z
v = M e -a~
(135)
If we choose the nondimensional velocity so that v = 1 when z = O, then w2 = Jl~re-2/~
v = e -pz
(136)
The velocity c o m p o n e n t s in any section of the jet are 1 ,=½r/?e-/Sz radial-velocity rwl = - srv
transverse velocity rw2 = r~Ie-2/t._ longitudinal velocity v = e -l~"
(137)
with vorticity c o m p o n e n t s radial rw'2 = 2 f l ~ I r e -2/J-" transverse rw'l
=
--~rv 1 . = --½r/?2e - p z
longitudinal 2w2 = 2 ~ t e -2/~-"
(138)
F r o m Eq. (124) pk = -(A
+ k/?)e - a :
(139)
A.E. Green, P.M. Naghdi / J. Non-Newtonian Fluid Mech. 66 (1996) 233-255
251
so that k/¢ < - A
A< 0
(140)
which is in line with the conditions that fl is small. Eq. (129) and (135) yield a = A l e ~1/2)/~:
b = Ble ~l/z)l~:
(141)
where k A = zc A~ + 2 A , B , +---~-- ]
~{ 9A 2lBi2 5AIB~ 35B 4] k l B = - ~ Aa + 2 A 3 B , + ----4-- + - - - 4 - - - + - T - ~
(142)
If the (nondimensional) value o f a is such that a = 1 when z = 0, then A~ = 1 and a = e (l/2)/j:= 1 + ½flz
b = Ble(I/2)/J: = B l ( l + ½flz)
(143)
since fl is small and provided z is not too large. At z = 0 the integer n is very large so that the section is approximately circular. As z increases in some range 0 < z < z~ the value of the integer n decreases until n = 2 when the boundary is star-shaped with eight maxima, i.e. r = a + b sin 2 40
(144)
Again, guided by Leipmann and Gharib [10], we assume that there is some structural breakdown in some range z~ < z < z2. Then in the region o f z = z2 the effective value o f the Reynolds number is much smaller, so that /~ is larger, and we look for a different type of solution of Eq. (127) and (128). We assume that v = N z -~
w2 = IVz -~'
(z2 < z)
(145)
where N, ~7, n, m are constants and h is not the same as ti in Eq. (129). When fl is large the dominant terms in Eqs. (127) and (128) are satisfied if h= 2
rh = 2h -- 1 = 3
(146)
Hence V = N2 -2
W2 =/Vz -3
(147)
with velocity components radial rw~ = - ~ 1r v p = r N z -3 transverse rw 2 = r l V z - 3 longitudinal v = N z - 2 F r o m Eqs. (123) and (145)
(148)
252
A.E. Green, P.M. Naghdi / J. Non-Newtonian Fluid Mech. 66 (1996) 233-255
k Z2 R = ~[
kl z4 Rll =~-~
(149)
with RI, RII still being given by Eq. (130). Hence
a = Az
b = Bz
(150)
where N -- ~
kl
+ 2~/B +
re(
9i/2B 2
5t/B 3
35B4~
--~ __~ ~4 + 2 2 3 ~ + ---T-- + - - T - + 1 - ~ - / U
(151)
The sections of the jet are given by
r = {A + B sinZ(2nO)}z
(152)
where we could maintain n = 2 as in Eq. (144) or vary it with z as necessary. Liepmann and Gharib [10] note that after the region of breakdown the jet grows linearly with distance downstream in agreement with Eq. (150). The foregoing solution indicates general agreement with what Liepmann and Gharib find experimentally.
8. Some time-dependent problems Choosing the non-dimensional value of p to be 1, Eq. (66) becomes Re
{,} iJ+~su
=pb-gradp+V2v-~V%
u=-V2v
(153)
where fl is given by Eq. (71). Suppose a liquid of depth lo is at rest and at the instant t = 0 a uniform tangential stress f begins to act on the surface in the x direction. With the origin at the bottom of the liquid and the surface at y = 1 the conditions to be satisfied are v~ = 0 ~Vl
(t = 0)
v~ = 0
(y = 0)
1 ~3Vl
Oy flz 0y2 - f ( Y = 1)
(154)
where we have used the first part of Eq. (50). With zero body force we have from Eq. (66) Re(.~_Vl
\ 8t
1
03v, "~ 02v,
2fl2 OtOyE] = 0y2
1 O4v, f12 0y4
p = const
(155)
The solution of Eq. (155) under conditions (154) is
8 (e_kZtsin(2)_~_ie_9kEtsin(3__~) + " ' ' + } ] where
(156)
A.E. Green, P.M. Naghdi / J. Non-Newtonian Fluid Mech. 66 (1996) 233-255
253
n 2
1
Rek 2-
+fl-5 n~
2s+ 1 n-~rc
s=0,1,2,...
(157)
1-~ 2/~2 As pointed out by Lamb [11], calculations of the above type have sometimes been used (in the case when f l ~ ~ ) to illustrate the action of wind in producing ocean currents; however, inserting numerical values for Re the final state would, according to the formula (156), be approached with extraordinary slowness. A more practical interpretation is obtained by using an ad hoc "coefficient of turbulence". However, if in Eq. (157) we suppose that fl is small then Re k 2 is replaced by two (instead of one when fl is large). This would halve the time, which would go some way towards a practical interpretation. As a second time-dependent example consider motion in circles around the axis of z, the velocity vo = rf being a function of the distance r from this axis and also a function of time. Then, from Eq. (66) I~ t Re
1 ~h] 1 ~p 1 /'~2h 3 ~ h ) 2fl 2 ~ = - r - I ~ - - ~ + h - ~ 5 ~ r E + r ~ r
1
~p
(158) (159)
where O2f . 3 ~f h = ~ r 2 + r ar
(160)
Then p = - A O + H(r, t) where A is a constant and ~H = R e r 2 [ f 2 - ~ - ~1 f h ] ~---~
(161)
Re
(162)
= -~ 4- h - --~ ~ ~r 2 + r --~rJ
The vorticity ( of the fluid about the z axis is
af
(=r~r+2
f
(163)
and h = - -r ~r
(164)
We consider a solution for an infinite fluid when A = 0 and we have an isolated vortex at z = 0. Then when fl in Eq. (162) is very large we have the classical solution [11]:
A.E. Green; P.M. Naghdi / J. Non-Newtonian Fluid Mech. 66 (1996) 233-255
254 K
f = 2--~-5r2[1 (=
-- e -
Re
r2/(4t)]
(165)
Re x e - Rer2/(at) 4m
where k is the value of the circulation when t ~ ~ . As t increases from 0 to oo the velocity diminishes from k/(2nr) to 0. The vorticity increases from zero to a maximum and then falls to zero.
In contrast, when fl is very small (so that the Reynolds number Re is large) Eq. (162) reduces to Re0h 2 0t
02h 3Sh - + 0r r ~r
(166)
From our present vortex problem Eq. (166) with Eq. (160) reduces to
Re Of_ O2f 3 Of 2 &
~r
r 0r
(167)
which is the same as Eq. (162) when fl is large but with Re replaced by Re/2. The solution corresponding to Eq. (165) is then h"
f = 2-~r2 [1 - - e -
Re
K e_Rer2/(8t )
r2/(St)] (168)
The rate of diffusion of the vortex into the surrounding fluid is therefore halved at high Reynolds number, compared with that of Eq. (165) for low Reynolds numbers. Other time-dependent problems show similar effects of the present theory compared with the classical Newtonian viscous fluid theory, especially for large Reynolds numbers, and give support to the suggestion that the theory can be of value in relation to turbulence as indicated in Sections 5-7.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
A.E. Green and P.M. Naghdi, Proc. R. Soc. London, 357 (1977) 253-270. A.E. Green and P.M. Naghdi, J. Appl. Math. Phys. 42 (1991)159-168. J.S. Marshall and P.M. Naghdi, Philos. Trans. R. Soc. London, A327 (1989) 415-449. J.S. Marshall and P.M. Naghdi, Philos. Trans. R. Soc. London, A327 (1989) 449-475. A.E. Green and P.M. Naghdi, Proc. R. Soc. London, 432 (1991) 171-]L94; A438 (1992) 605. A.E. Green and P.M. Naghdi, Philos. Trans. R. Soc. London, A448 (1995) 335-356; A448 (1995) 357-377; A448 (1995) 379-388. J.S. Marshall and P.M. Naghdi, Contin. Thermodyn., 3 (1991) 65-77. J. Laufer, Natl. Advis. Comm. Aeronaut. Rep., 1053 (1951) 1247-1265. A.E. Green and P.M. Naghdi, Philos. Trans. R. Soc. London, A432 (1993) 525-542. D. Liepmann and M. Gharib, J. Fluid Mech., 245 (1992) 643-668.
A.E. Green, P.M. Naghdi / J. Non-Newtonian Fluid Mech. 66 (1996) 233 255
[11] H. Lamb, Hydrodynamics, 6th edn., 1932, University Press, Cambridge, pp. 592, 593. [12] J. Laufer, Natl. Advis. Comm. Aeronaut. Rep., 1174 (1954) 417-434.
255
ELSEVIER
J. Non-Newtonian Fluid Mech., 66 (1996) 233-255
An extended theory for incompressible viscous fluid flow A . E . G r e e n ~'*, P . M . N a g h d i b't ~Mathematical Institute, 24-29 St. Giles, Oxford, OXI 3LB, UK bDepartment of Mechanical Engineering, University of California, Berkeley, CA 94720, USA Received 4 April 1996
Abstract
The classical Newtonian theory of incompressible viscous fluid flow is extended so as to contain explicitly the vorticity vector and spin of vorticity vector. Examples of the theory are given, some of which relate to the problem of turbulent flow, and reference is made to some experimental results. Keywords: Viscous fluid flow; Vorticity
I. Introduction
In the context of a thermodynamical theory, the classical formulation of viscous fluids can be based on the equations of mass conservation, linear m o m e n t u m balance, as well as on entropy balance [1,2], together with appropriate constitutive equations which include that for a linear (Newtonian) viscous fluid. The equations for mass conservation, m o m e n t u m and entropy balance are , 6 + p div v = 0 piJ = pb + div T PO = p ( s + ~ ) - d i v p
(1) t = Tn k=p
(2) •n
(3)
In the above equations, p is the mass density of the fluid, v is the velocity vector, b is the external body force per unit mass, T is the Cauchy stress tensor which is related to the stress vector t by the second part o f Eq. (2), r/ is the entropy per unit mass, s is the external rate of • Corresponding author. t Decreased. 0377-0257/96/$15.00 Copyright © 1996 Elsevier Science B.V. All rights reserved. PH S0377-0257(96)01478-4
234
A.E. Green, P.M. Naghdi / J. Non-Newtonian Fluid Mech. 66 (1996) 233-255
supply of entropy per unit mass, ~ is the internal rate of generation of entropy per unit mass and p is the entropy flux vector which is related to the entropy flux k by the second part of Eq. (3). Also in Eqs. (1)-(3) a superposed dot designates material time differentiation and "div" denotes the divergence operator with respect to the current position x in the fluid. Further, constitutive equations for a Newtonian viscous fluid in the presence of thermal effects can be stated as T = p l + 221 tr D + 21tD
p = p 2 O__~ ap
80
p = x ~---~
'I -
~~0k
(4)
where the kinematical variable D=½{~v/Ox+(Ov/Ox)r}, 0 ( > 0 ) is the absolute temperature, ~, = ~,(p, 0) is the specific Helmholtz free energy, ! is the identity tensor, the scalar x in the third part of Eq. (4) is the conductivity coefficient and 2 and/L are the viscosity coefficients. The system of Eqs. (1)-(4) has been used successfully in the study of a wide range of problems in fluid dynamics which can also be verified experimentally. It has not, however, been possible to explain many features of the flow on the basis of Eqs. (1)-(4) alone. For example, problems concerned with turbulent flow are not covered by these equations and most of the theoretical developments in this direction involve additional equations which are rather ad hoc. It is clear from extensive available experimental results and also to some extent from existing numerical simulations that not only the vorticity (or spin) but also the rate of change of vorticity (or "spin of the spin") affect the structure of turbulent flows. Given this premise it is natural to see if a satisfactory theory can be constructed with the vorticity vector w defined by Eq. (9) and the spin of vorticity vector u defined in Eq. (10) explicitly displayed in the theory, instead of just being evaluated at the end of the solution of problems from the velocity vector v. Moreover, such a theory if properly based on continuum thermodynamic equations would have interest in its own right, apart from its possible application to turbulent flow. A general theory involving vorticity and spin of vorticity explicitly, together with two temperatures, the second of which is motivated by turbulent flow as described by Marshall and Naghdi [3,4] is presented in Sections 2 and 3. In order to have a theory with as few extra constitutive coefficients as possible, a restricted theory for an incompressible fluid is deduced in Section 4. Application to particular problems is given in Section 5-8 with comparison with experimental results where possible as far as turbulent flow is concerned.
2. Kinematic and thermal variables
Consider a body ~ with particles (material points) X embedded in a three-dimensional Euclidean space g3.Choosing a fixed origin in e3, we identify each material point of ~ by the position vector X in a fixed reference configuration Xo of 8 . In the current configuration K of at time t, the material point which was at X in ro takes the place x. The region of space occupied by M in Ko will be denoted by Ro bounded by a smooth closed surface 0Ro, while the corresponding region occupied by ~' in its current configuration x at time t will be denoted by R bounded by a smooth closed surface OR. Also, an arbitrary subset of ~ in K at time t will be denoted by S, bounded by 0S,, and occupies a region P ( ~ R ) bounded by a smooth closed surface 0P with unit outward normal n.
A.E. Green, P.M. Naghdi / J. Non-Newtonian Fluid Mech. 66 (1996) 233-255
235
A motion o f ~ , defined in terms of a sufficiently smooth vector function •, assigns the place x to each material point in the current configuration r of ~ ' at time t such that x = x(X, t)
(5)
and the particle velocity at x is v = .~ = v(X, t)
(6)
where a superposed dot denotes material time differentiation holding X fixed. The velocity gradient L is given by Ov L -
(7)
Ox
We recall the unique decomposition of the velocity gradient in terms o f a rate of deformation tensor D and the vorticity (spin) tensor W: L = O + W
O = ½(t. + L T)
W = l(L -- L T)
(8)
where L v denotes the transpose of L. The vorticity tensor W is related to the vorticity vector (or spin vector) w by w = curl v
2 Wci = w × 4i
(9)
for every vector ~i. We define the "spin o f the spin" vector or spin of the vorticity vector u by u = curl w = curl curl v = grad div v - V2v
(10)
where " g r a d " and "div" denote the gradient and the divergence operators both with respect to x. Further, we define the gradients o f the vorticity vector and of the spin o f the vorticity vector by
~w N=-Ox
~u P=-Ox
(11)
We note that the skew part of N, namely l ( N - - N T) is related to the axial vector u by (N - N V ) a = u × a
(12)
for every vector ~i, which is analogous to Eq. (9). Motivated by the notion o f turbulent temperature given by Marshall and Naghdi [3,4] and the more recent ideas on temperature and the basic thermal variables of Green and Naghdi [2,5] we introduce two thermal displacement magnitudes ~H, ~r and two temperature fields On, Or such that I ~ , = ~H(X, t) > 0
OH = ~n > 0
i In line with the developments in (Section 7) the scalars ~tt¢, ctr (on the macroscopic scale) are regarded as representing (on the molecular scale) some "mean" thermal displacement magnitudes which for brevity may be called thermal displacements. Additional background information is given in the paragraph containing (7.3) of Ref. [5].
A.E. Green, P.M. Naghdi / J. Non-Newtonian Fluid Mech. 66 (1996) 233-255
236
o~r=OtT(X,t)>O
OT=~T)O
(13)
The scalar On will be identified with ordinary temperature in classical thermomechanics (here the classical theory of linear viscous fluid) and the second temperature will be identified as an additional independent thermal variable (such as the turbulent temperature) along with an additional kinematic variable (such as the spin of the vorticity vector). For later use, we record here the effect of a superposed rigid body motion (SRBM) on the foregoing kinematic and thermal variables. Thus, under a SRBM in which x in the current configuration x moves to the place x + in the configuration x + at time t + = t + a, a being a constant, we have (14)
x + = Qx + a
where Q is a proper orthogonal second order tensor function of time which satisfies det Q = I
QQT = QTQ = I
(15)
a is a vector function of time and ! is the second order identity tensor. F r o m Eqs. (14) and (15) it follows that 0 = fiQ
v + = £c+ = fi + Q v + f i Q x
fi + fl T = 0
(16)
where fi is a skew tensor function of time t which represents rigid body angular velocity and satisfies fi~i = to x ti
for every vector ~i
(17)
to being the rigid body angular velocity vector. F r o m Eq. (16) and Eqs. (8-12) follow the results O + =QDQ
T
w + =QWQr+fi
w+
= QW+2to
(18)
and u + = Qu
p+ = QpQT
N + = QNQ T
(19)
Under SRBM, the thermal variables in Eq. (13) transform according to a + = o~i~
o~+ = O~y
O~ = On
O~ = Or
(20)
3. Basic equations It is clear that vortices, represented by the vorticity tensor W in Eq. (8) or the vorticity vector w in Eq. (9) have a significant effect on the behaviour of turbulent flow. Usually the vorticity vector w is calculated from solutions of equations of the type (1)-(4) so it does not have a direct effect on the type of flow. Here we add a spin or vorticity vector w explicitly to the theory which in turn leads to new response functions, external forces and equations of motion. We also add further structure explicitly to the theory by including a spin of spin or spin of vorticity vector u defined in Eq. (10). This also requires new response functions and equations of motion. The basic kinematic variables are therfore taken to be
A.E. Green, P.M. Naghdi / J. Non-Newtonian Fluid Mech. 66 (1996) 233--255
r, w, u
237
(21)
Also, following Refs. [3,4] we use two temperature fields
On, Or
(22)
representing the usual temperature 0/~ and a turbulent temperature 0T. We adopt the procedure of Green and Naghdi [6] and derive the basic mechanical and thermal equations from the First Law of Thermodynamics which we state in two parts as follows: (a) For any subset S, of the body ~ in the current configuration K, the total rate of change of kinetic energy K(S,) and heat energy H(S,) is balanced by the sum of the external rate of supply work R(S,) to S,, the external rate of supply of heat Q(S,) to S, and the internal rate of supply of energy W(S,) to S,. Thus d dt [K(S,) + H(S,)] + R(S,) + Q(S,) + W(S,) = 0
(23)
(b) The total rate of external work and heat, i.e. R(S,) + Q(S,), supplied to or extracted from S, in a cycle is zero. Based on the variables (21) and (22) we assume that the contributions in (a) to the First Law have the forms
K(S, ) = fp ½p (r • ~ + 2ylor " w + 2y2ov • u + yll w " w + 2yl2w " u + Y22U • u) dv
(24)
H(S, ) = .te P(tlz-IOH+ tlTOr) dv
(25)
de
da P
W(St) = tP {pw + p f " r +½mr
+ P # I " w --F P # 2 " u --b p ~ H O H + p ~ r O r + mYIHOH "-b mr'/TO T 1
• I~-k-m l o v • w + m20 v " u + 2 m l l W • w + m l z w • u + ½m22u " u } dv
(28)
In Eqs. (24)-(28) Yl0, Y2o, Y~1, Yl2, Y2z, m, m~0, m2o, mll, m12, m22, tlH, tlT, SH, ST, k H , kr are scalar-valued functions, while b, c~, c2, t, ml, m2 are vector-valued functions of (X, t); and t, ml, m 2 , k H , kr, in addition to depending on (X, t) depend on the outward unit normal n to the boundary surface 8P enclosing the arbitrary volume P of the body in the current configuration x. Moreover, except for the scalar w in Eq. (28) the integrands in Eq. (25)-(27) are linear in the independent variables (22) and (22); and the integrand in Eq. (24) is quadratic in the variables (21) while the integrand in (28) is quadratic in (21). We further assume that the scalar w is linear in the temperature rates 0t4, Or, the temperature gradients 00it/8x, 80T/OX and the gradients of the variables (21), namely L, N, P.
A.E. Green, P.M. Naghdi / J. Non-Newtonian Fluid Mech. 66 (1996) 233-255
238
The basic equations may now be derived from Eqs. (23)-(28) using the methods indicated by Green and Naghdi [6]. Here we quote the final results. The equations of mass and inertia balances are dt
p dv =
m dv
PYll dv = dt
PYI2 dv =
f,
-dr
PY~odv =
mlo dv
mll dv
dt
PY2odv =
m12 dv
dt
d;,
PY22dv =
m2o dv
f,
(29)
rn22 dv
The mechanical equations are dt
P(r + Ylow + Y2ou) d v =
dr,
dt dt
f,
P('F~°vWyJlwWyI2u) d v =
P(.V2oV+ ylzw + yzzu) dv =
{mv + mlow + m2ou + p(b + f ) } d v +
f,
t da
(30)
P
{rnloe+mllw+rn12u+p(el +Pl)} dv+
ml" da (31) P
{m2ov + m12w -l- m22u + p(c2 + It2)} d r +
mz da
(32)
P
The entropy balance equations are p t l . dv =
{m~H + p ( s . + ~:H)} dv -
k . da
(33)
P
-~
P~lr dv =
{m~lr + p(Sr + •r)} dv -
(34)
k r da P
With the usual smoothness assumptions and boundedness properties, the application of Eqs. (29)-(34) to an arbitrary tetrahedron results in t = Tn
ml =M. ln
m2 = M 2 n
k.=p.,
n
kr=pr" n
(35)
where T, Mi, M 2 are second order tensor functions and Phr, P r are vector functions of X, t. Next, from Eqs. (29)-(34), after using (35), we obtain /~ + p dive = m
PPlo + (/~ + p div v)ylo = ml0
PPll + (jb + p div v)yll =roll
P P 2 o + ( f ~ + p divv)y2o=m2o
/9))22 -st- (/9 -+- p div V)Yl2 = m12
PY22 + (,6 + p div v)Y22 = m22
(36)
P(6 + Ylofi' + Y2oti) = p(b + f ) + div T P(Ylo~ + y~lff + Yl2ti) = p(c~ + t~) + div M~ /00;2013 -/- Y12 ff -t- Y22 li) = p ( C 2 "4- ~/2) -4- div M 2
Pflu = p(sn + (,~) -- d i v p u
(37)
A.E. Green, P.M. Naghdi / J. Non-Newtonian Fluid Mech. 66 (1996) 233 255
239
POT = p ( S T + ~T) -- divpT
(38)
With the help of Eqs. (24)-(28) and Eqs. (35)-(38), the energy balance Eq. (23) yields the equation " OOH Ox
pw + T " L + MI " N + Mz "P-pH
P T " OOV OX
(39)
PqtIOH -- P?ITOT = 0
where N, P are defined in Eq. (11). Part (b) of the First Law now yields (for details, See section 3 of Ref. [2]) W(S,) -
dt
(40)
p~, dv
where ~, is a scalar function of X, t. F r o m Eq. (28) it follows that I
p w + p e t + m ~ + p ( f " v + # l " w + P2 " u ) + ½ m e • v + m l o v • w + m20v " u + s m l l w
+mlzw
• u + l m 2 2 u • u + P(~HOH -}- ~TOT) "}- m ( q t t O H + q TO T) = 0
' w
(41)
and when this is combined with Eq. (39) to eliminate w we then obtain OOH Ox
-p(~+qnt)H+qrOT)-m¢+T'L+M~'N+M2"P-pt4"
-
p ( f " v + ttl " w + la2 " u ) - - ½ m v
• v--mint
" w-
|
m2ov • u - - ~ m l j w
-- ½m22u • u -- P(~.I-IOH + ~ r O r ) -- m O l n O n + q r O r )
OOT P T " OX
• w-
ml2w • u
= 0
(42)
which is the residual energy equation. U n d e r a SRBM Eq. (14), in addition to Eqs. (16)-(20), we assume that T + = QTQ T
M~( = Q M , Q T
l ~- = Q f
la~i = Q p ,
~
p+ = p
= ~-r
m~-e = rn12
M~ = QM2Q x
# 7 = Qp2
m+ = m
P~t = QPH
m{o = mm
rn~-l = mll
m2~o= rn2o
m+2 = m22
(43)
We also assume that the residual energy equation (42) remains invariant under any SRBM Eq. (14), so that using Eqs. (14)-(20) and Eq. (43) we may deduce that m- 0
f:0
mlo = 0
m2o = 0
rnll = 0
1
(44)
pp~ + m12u = 2ei × Tei
where e~ is an orthonormal basis. With Eq. (44) the residual energy equation (42) reduces to ~0H -p(~)+rlnOr+rlTOr)+
T'D+M~'N+M2"P-pp2"u-pI4"
- ½m22u • u -- p ( ~ I 4 O H + ~ r O r )
= 0
~x
pT'~Or~x
(45)
A.E. Green, P.M. Naghdi / J. Non-Newtonian Fluid Mech. 66 (1996) 233-255
240
To complete the theory, constitutive equations must be specified for T, M l , M2, 1u2,PI~, Pr, ~l~I, ~IT, ~H, ~r
(46)
which appear in Eq. (45). The residual energy equation (45) is then interpreted as an identity for all motions and temperature distributions, which places restrictions on constitutive equations In view of Eq. (44) Eqs. (36) reduce to + p div v = 0 PJ~12 =
Ylo :
ml2
P.922 =
Y2o = y2o(X)
0
Y20 = 0
m22
Yll = 0
Yl0 = Y l 0 ( X )
Yll = y~l(X)
(47)
The theory, in its general form, leads to a set of differential equations with a large number of new coefficients, which are rather unmanageable from the point of view of understanding turbulent or other flows. Since we are seeking a theory which contains that of a Newtonian viscous fluid in the presence of thermal effects, in the rest of the paper we restrict out attention to one which will reduce the mechanical differential equations to one equation. Moreover, we only consider an incompressible fluid although this extra restriction may be dispensed with without undue complication. It does, however, at this stage exclude any comparison with theories for rarified gas dynamics.
4. Restricted theory for incompressible Newtonian fluids with vorticity and spin of vorticity When the fluid is incompressible div v = 0
(48)
and since the divergence of a curl operation on a vector is zero, from Eq. (10) we also have div w = 0
div u = 0
u = -V2v
(49)
The natural extension of the Newtonian viscous theory to include explicitly vorticity and spin of vorticity is to suppose that T, Mr, ME depend on N, P in addition to D. Bearing in mind the last paragraph of Section 3 we make the restricted assumption that T = - p l + 2/~D + 2t~lP Ml = - - p l l + ill S 312 = - p 2 1 + 2/~1g + 2(/~ 2t//u)P
(50)
where p, Pl, P2 are arbitrary scalar functions of (X, t), P is the usual viscous coefficient and ~/l is a second coefficient. F r o m Eqs. (44) and (50) we then have flfll + m l 2 H = fll curl u = --,/./l V 2 w
(51)
Next we take Yl0 ---~0
Yll = 0
Yl2 = 0
m22 = 0
/*2 = 0
(52)
A.E. Green, P.M. Naghdi /J. Non-Newtonian Fluid Mech. 66 (1996) 233-255
241
so that ml2 = 0
Y22 = Y22(X)
(53)
Y20 = y z o ( X )
Also. in the light of the first and third parts of Eq. (37) and of Eq. (50) y2o =/~,/~
y_,~ = ~2/~2
(54)
The equations of m o t i o n (35) now reduce to p {b + (/q//t)ti } = p b - g r a d p +/~/V2v - 2]./IV41~ 0 = pc~ - grad pt p ( l ~ / I t ) { p b + (/t,//t)ti} = pcz - grad pz + (/tl//t)(/~VZv - 2/t,V4v)
(55)
Finally we choose c~ = ( / t l / / t ) b
P2 = (lq/It)P
Cl = O
pl = constant
(56)
Then Eq. (55) reduce to the single equation p {13+ (/zl//t)ri} = p b - g r a d p +/~VZv --2/llV4V
(57)
which contains just one coefficient /q in addition to/~. At this point we use one aspect of the Second Law of T h e r m o d y n a m i c s (Ref. [7]) in the form (58)
p(~HOn + ~TOT) • 0
for all motions. T h e n from Eqs. (42), (50) and (52) we have P(~_HOH + ~TOr) = --p((~ + ti~OH+ q r O r ) - - p .
~0 H O0 T ".~---~-- - P T "--~X + 21tO " D + 41~,O" P
+ 2(/t~//t)P " P >~ 0
(59)
Guided by Eq. (59) we now assume further constitutive equations = c . { O H - - OH In 0/_/} + c r { O r - - O r l n 0T}
tiH= CH In 0 ,
tlr = Cr In 0 r
K/-t ~0H PH=-
tOT0 0 r
0 o OX
PT=
(60)
0 o OX
where cn, cr, KH, •r are constants and 0o is a constant standard temperature. With the help of Eqs. (59) and (60) we choose constitutive equations for ~n, ~T. The scalar ~ must contain the terms which correspond to those for a Newtonian viscous fluid so we assume that KH S0H SOn p ~ t O H = -~o SX P~TOT
S----X-+ 2ttD " D + (9
XT SOT. SOT
Ox
S--x-+ 4 / q D • P + 2(tt21/it)P . P -
c~
(61)
A.E. Green, P.M. Naghdi / J. Non-Newtonian Fluid Mech. 66 (1996) 233-255
242
where ¢ is a constant. (For some purposes ¢ could depend on temperatures.) The first two terms on the right-hand side the first part of Eq. (61) are those corresponding to a Newtonian viscous fluid while ¢ is an interaction term between ~H, ~r. Substituting the constitutive equations (60) and (61) into the entropy balance equations (38) gives KH
pcHO. = pSHOn + ~ o V20~ + 2/tD • D + ¢
pcTG = psTOT +
KT
V20
+ 4
(62)
,0" P +
• e -- ¢
(63)
It is instructive to express the single mechanical equations (57) and the thermal equations (62) and (63) in nondimensional form as an aid to further developments. We denote standard reference values of length, speed, density and temperature by the constant lo, Vo, Po, 0o respectively and introduce nondimensional coefficients by Re - polovo
Rl -- p°13v~°
/t
Pel - poloV -- o XnOo
f12 = Rl 2 Re
/Zl
Pe2 = poloV XTOo
(64)
The coefficient Re is a Reynolds number and Pel, Pe2 are P6clet numbers. Using a circumflex to denote nondimensional quantities, we set x =/02
v = vo~
t = __l°/,
w=
Vo
U0 ^
To
w
I)0 ,,
II = l"~0II U2OH
CH= Oo
p = pOtO
OH = 0o0 H
V~T
CT= O0
P=
0 T = 0o0 T
2^ povop Re
¢
2povo3Re q5
(65)
We express Eqs. (57), (62) and (63) in nondimensional form with the help of Eqs. (64) and (65). Thus, omitting the circumflex on all nondimensional quantities, we have P Re ~
L
= pb - grad p + V2v - l/~ V%
(66)
where b is nondimensional. Also, after setting external supplies of entropy equal to zero, p Pel cn(tu = 31~-72'02 --H+2(Pel/Re) ( D'D+
Re2 R~ ¢']]
p P % c r O r = l w52- a ~,T+ 2 4(Pe2/RI) { D " P +-2-~ Re P " P - ~~I Re ¢ }
(67) (68)
We observe that equations of the form of Eq. (66), without the extra inertia term involving u, have been proposed before in a somewhat ad hoc manner. Here the equation is based on the
A.E. Green, P.M. Naghdi / J. Non-Newtonian Fluid Mech. 66 (1996) 233-255
243
full equations both mechanical and thermal, of continuum mechanics after special choice of constitutive equations, applying only to incompressible fluids. It remains to be shown that the theory has any relevance to turbulent flow in viscous fluids, although the theory has some interest apart from this.
5. Plane Poiseuille flow We use a rectangular system of nondimensional coordinates xi = (x, y, =) associated with an orthonormal basis ei and choose the x~ = x axis along the centre line of a channel of uniform thickness 21o whose sides are bounded by the planes y = + 1. We seek a state of flow along the ej direction with the velocity field defined by 5v v = v(y, t)el
a2v
w = - - ~ y e~
u -
ay 2 el
(169)
where v is a function of y, t. Then with body forces b = 0, Eq. (66) gives
Re(au \at
@ ay
o
1 ~3/.) ~ 2fl 2 at 5 y 2 J =
@ az
ap
a2V
a x ~- ~5 y
1 a% f14 5y4
o
(70)
if we take the nondimensional value of p to be one. We seek solutions of Eq. (70) which are even in the variable y and which include the classical steady-state Poiseuille flow when t = 0 and fl is large. Thus v = L + 5 (A 1 _y2)+
Be;" cosh(~fly)
p=
-Ax
(71)
where Re2(1 -- ~g i 2) ~---0~2fl2(1 -- 0~2)
(72)
and :¢, L, A, B are constants. To determine the constants L, B we assume that the velocity at the walls 3'-- + 1 is V so that V = L + B cosh(~fl)
(73)
Also, we assume that the component of the couple m~ at the walls in the direction e3 normal to the channel is zero when t = 0. Hence, from Eqs. (35) and (50) aw
m l • e3 = ( M l e 2 ) " e3 =/IN32 =/z ~ y = 0
(74)
a2v ay 2 = 0
(75)
i.e. ( y = +1)
A.E. Green, P.M. Naghdi / J. Non-Newtonian Fluid Mech. 66 (1996) 233-255
244 or
--A "-t-o~2f128 cosh(~fl) = 0
(76)
v = V + A{2(1-y2)+e~'[c°sh(°~flY)~c°sh(°~fl)~ 0~2~2 cosh(~fl) J
(77)
Then
When fl is large this solution reduces to the classical Poiseuille flow v = V+½A(1 - y 2 )
(78)
~2>2
(79)
If or
~2<1
then, from Eq. (72), A > 0 and the solution Eq. (77) becomes large with increasing time. However, if 1 < ~2 < 2
(80)
then A < 0 the solution Eq. (77) reduces to Eq. (78) when t is large. When =1
then
2=0
(81)
and Eq. (77) becomes a steady-state solution v = v+ A
{~
(1 _ y 2 ) ~
cosh(fly) - cosh fl ~ ~-~s~/~
J
(82)
The total rate of flow across any section of the channel, per unit width, in the el direction is tanh fl]~ The maximum value Vm of v occurs at the centre of the channel where y = 0 so that Um = V + m
If
-cosh,} fl
'~ ]~ 2 c o s h
,84,
(V/Vm) = k when y --- _+ 1 then
A {1
-~ ~ + ~2 cosh/~ )
~-
(85)
and cosh(fly) - cosh fl v (1 - k ) { 1(1 - y 2 ) + fl 2 cos-s-~ ) - - = k -~ /)m 1 1 - cosh fl 2 f12 cosh fl When fl ~
(85)
A.E. Green, P.M. Naghdi / J. Non-Newtonian Fluid Mech. 66 (1996) 233-255 U
- - = k + (1 - k)(1 - y2) = 1 - (1 - kZ)y 2
245
(86)
I) m
which is the classical result. However, when fl is small v 1-k -- = k + ~
(1 - y 2 ) ( 5 - y 2 )
(87)
/)m
We observe that neither the classical result Eq. (86) nor the result Eq. (87), which is a second steady state, depend on viscous coefficients Re or RI. Marshall and Naghdi [4] display a velocity profile for v/vm in their Fig. 3, obtained from their theory and c o m p a r e it with the experimental data of Laufer [8,12]. The graph of Marshall and Naghdi compares quite well with that of Laufer, except perhaps near the walls y = + 1 and y = 0.5. The m u c h simpler Eq. (87) depends on only one constant k and this constant essentially determines the value of v at the walls which would normally be chosen by experiment. Here we select a value k = 0.7. The graph from Eq. (87) then compares reasonably well with that of Laufer. T u r n i n g to the temperature 0 r we carry out the calculation for the case when fl is small or Re large. Assuming Or= 0r(Y) and that ~b is constant, and retaining only terms in Eq. (68) corresponding to large values of Re, we have d202r
2 Pe2
@ 2 - ] ~ - ~ i {~b -
(v,,)2}
(88)
where, from Eq. (82)
v = v + -fl- ~2A - (y2 _ 1)(y: - 5)
v'" = fl:Ay
(89)
We replace 0~- and ~b by 2PezA2f12 RI
and
(90)
A2f14a2
respectively where a is a constant, so that Eq. (88) becomes d20 2 @2 - a 2 - y 2
(91)
If 0T = 0- when y = _+ 1 and 0 r is an even function of y, then 0 ~ , - 62 = -~z(1 - y Z ) ( b - y 2 ) In order 6, b before the general and Laufer
b = 6a 2 - 1
(92)
to give a full interpretation to Eq. (92) we need to find values for the constants a clear comparison m a y be m a d e with the measurements of Laufer [8]. However, character of result from Eq. (92) follows that given by Marshall and Naghdi [4] [8].
A.E. Green, P.M. Naghdi / J. Non-Newtonian Fluid Mech. 66 (1996) 233-255
246
6. Poiseuiile flow in a circular tube We use the same system o f axes as in Section 5 where n o w the z axis is along the centre line o f a uniform circular tube o f radius 1o. We consider a steady flow along the e3 direction defined by r = (x 2 + y2)1/2
v = v(r)e3
(93)
where v is a function o f r. Then from Eqs. (9) and (10) it follows that w
=
~
el
-
r
- -
e2
u =
-
v"
+
(94)
e3
r
where a prime denotes differentiation with respect t o r. If b is zero Eq. (66) gives
r dr)k, ~P=o
~P=0
ax
ay
+
=o
(95)
An appropriate solution o f Eq. (95) is v = L
Ar 2 -4- + B I o ( f l r )
(96)
where, A, B, L are constants and Io is a modified Bessel function o f the first kind. We suppose that the value o f v at the wall r = 1 is V so that A V = L - ~- + BIo(]?)
(97)
v = V + ¼A(1 - r 2) + B { I o ( f l r ) - Io(fl)}
(98)
and
Also, we suppose that the eo c o m p o n e n t o f the couple m, which acts across the surface r = l, is zero so that v" = O(r = 1)
(99)
or
-½A + flZBI~(fl) = 0
(100)
Hence v = V+¼A(1 - r 2) 4 A { I o ( f l r ) - Io(fl)} 2fl2I~(fl)
(101)
When fl --* o% the value o f v tends to v = V + l A ( 1 -- r 2)
(102)
A.E. Green, P.M. Naghdi / J. Non-Newtonian Fluid Mech. 66 (1996) 233-255
247
the classical solution. H o w e v e r , w h e n fl is small Aft 2
v = V+~(I
- r Z ) ( 5 - r 2)
(1(13)
The m a x i m u m value Vm o f V Occurs at the centre o f the channel where r = 0, so that l A {1 - I o ( f l ) } Vm = V + ~A +
(104)
If V/Vm = k when r = 1, then l/k = 1 + v
(105)
-4
and v
(1 - k )
(1 - r 2)-4
2flzig(fl )
) (106)
- k + v,.
1
+ 1 - Io(fl) 4 2f12I~;(fl)
W h e n fl -~ oo u
- k + (1 - k)(1 - r 2) = 1 - (1 - k ) r 2
(107)
1)m
which is the classical result. W h e n fl is small --= v
(1 - k ) k + - -
Um
7. C i r c u l a r
(1 - r2)(5 - r 2)
(108)
5
jet
H e r e we consider the flow o f fluid f r o m a circular orifice. It w o u l d be convenient to use a direct theory o f flow in pipes (or jets) as d e v e l o p e d by G r e e n and N a g h d i [9] but in order to avoid detailed reference to this p a p e r we derive equivalent e q u a t i o n s f r o m Eq. (66). W e take the axis o f the jet to be the z axis in the e3 direction o f an o r t h o n o r m a l basis ei with x and y axes along el and e2 respectively. W e assume that the jet is in a state o f steady m o t i o n b o u n d e d by the surface (109)
z = z(x, y)
over which there is a c o n s t a n t pressure. A t the surface 8z - v 3 + V, x +
Oz
= 0
where the velocity vector v* o f the fluid has c o m p o n e n t s
(110)
248
A.E. Green, P.M. Naghdi / J. Non-Newtonian Fluid Mech. 66 (1996) 233-255
(111)
v* = vjel + v2e2 + v3e3
Here we assume that (112)
v* = v + xwl + ywz
where v, wl, w2 are functions of z of the forms wl = wl el + w2e2,
v = re3
w2 =
--W2el
+ wle2
(1 13)
SO that Vl = XWI - - y w 2
v2 = --xw2 + y w l
(114)
v3 = v
Since the fluid is incompressible 2WI + V'= 0
(115)
where a prime denotes differentiation with respect to z. We limit attention to jets which are symmetric with respect to the x and y axes and define
R=ffdxdy Rll=ffxdxdy=ffy2dxdy
(116)
where integrations are over sections of the jet normal to the e3 axis. The incompressibility condition Eq. (115) is, alternatively, ~V 1
802
~V3
~x + ~ y +--~-z = 0
(117)
We integrate Eq. (117) over a section and use the surface condition Eq. (110) to obtain (Rv) = 0
(118)
Further, we multiply Eq. (117) by x 2, and integrate over a section using the surface condition Eq. (ll0) to get ~z (Rll v 2) = 0
(119)
Similarly, we multiply equations Eqs. (66) in turn by l, x, y and integrate over a section. This yields equations of motion for steady states: 0 pReRi,
{
, v,,,]} 1
vw~ + w 2 - w ~ - ~ [ v w r '
°{[
=~-'-~ R,i
'1}[
2w'l-~w]"
-R
+wlwf-w2w~] --p+2wl--~zw'[
}
]
(121)
,,I.E. Green,P.M. Naghdi/ J. Non-NewtonianFluidMech.66 (1996)233-255
249
p ReRii{vw'2W2wiw2-+[vw'~'+wiw'~+wzw'(]} = a~ a
{[ Rli w'~- ']}" +~w2 ~-5 w~'
"
(122)
From Eqs. (118) and (119)
Rv=k
Rllv2=kl
(123)
where k, kn are constants. Then Eq. (120) may be integrated to give
1]
[
kv-~v"
p Re
+A=R - p + 2 v ' - - ~ v "
(124)
where A is a constant. Eliminating p between Eqs. (121) and (124) and using Eq. (115) yields P Re ( k v - ~__k v" ~
-
-
v
V r __
,,
) + a + ----2-pRek.{ - ~ v, ,, + (~')2-w~-/S.L-2V~ 1 F 1 h,+41v,v,,,_w2w~l}
2-~v )
20z
7
v"-
,]} -~V Iv
(125)
Also Eq. (122) becomes
p Rek' 2
t.
.t~
L
--21v'w~-21"v w2J~])
Alternatively, Eqs. (125) and (126) may be expressed in the forms pRe
kv 4 - - ~ v3v" + Av 3 +p Rekiv - vv"+
= 3kV2kV ,(
+w2w~
(v')2-w2+~- 5
1~5v"')+kiv'[v"--~zv'"l-k'V[v"--~Sv"
p Rekiv vw'2-v'w2-~-~LVW2 -2v'w~-2v"w2
{
,
(127)
7}
1 w~,}+kiv{w~__~ 5w{,,}+kv2w ~ = - 2kl v' {w'2 _ ~5
(128)
We seek solutions of Eqs. (127) and (128) in the light of experimental work reported by Liepmann and Gharib [10] who considered a jet issuing from a nozzle at a Reynolds number Re = 5500. The diagrams in their paper suggest that the cross-sections of the jet have, approximately, the forms
r = a(z) + b(z) sin2{2n(z)O}
(129)
A.E. Green, P.M. Naghdi / J. Non-Newtonian Fluid Mech. 66 (1996) 233-255
250
where the z axis is along the central axis o f the jet and (r, 0) are polar coordinates in each section o f the jet. Also a, b, n are functions of z while n is an integer ___1 and the exit to the nozzle is at z = 0. F r o m Eq. (116) we then have =-=zc
a2 + 2ab +
/)
kl rc I 9a262 5ab3 3564) R , , - v2 - 4 , a 4 + 2a3b + - - - - 4 - - +----4-- + - i - 2 - ~ ;
(130)
The Reynolds n u m b e r of the issuing jet is large, which suggests that fl is small, so that in the n e i g h b o u r h o o d of the nozzle we seek solutions of Eqs. (127) and (128) of the f o r m v = Me-"'- + • • •
w2 = M e -m:
(131)
where M , )17I, ft, m are constants. The d o m i n a n t terms in the equations are satisfied if -1
=0
m
-1
or
m=2ti
(m-2ri)=0
(132)
Then ri=fl
and
m=fl
(133)
We choose ri = / ?
m = 2Ji = 2/?
(134)
so that the d o m i n a n t terms in Eqs. (129) become w2 = / O e -2/~z
v = M e -a~
(135)
If we choose the nondimensional velocity so that v = 1 when z = O, then w2 = Jl~re-2/~
v = e -pz
(136)
The velocity c o m p o n e n t s in any section of the jet are 1 ,=½r/?e-/Sz radial-velocity rwl = - srv
transverse velocity rw2 = r~Ie-2/t._ longitudinal velocity v = e -l~"
(137)
with vorticity c o m p o n e n t s radial rw'2 = 2 f l ~ I r e -2/J-" transverse rw'l
=
--~rv 1 . = --½r/?2e - p z
longitudinal 2w2 = 2 ~ t e -2/~-"
(138)
F r o m Eq. (124) pk = -(A
+ k/?)e - a :
(139)
A.E. Green, P.M. Naghdi / J. Non-Newtonian Fluid Mech. 66 (1996) 233-255
251
so that k/¢ < - A
A< 0
(140)
which is in line with the conditions that fl is small. Eq. (129) and (135) yield a = A l e ~1/2)/~:
b = Ble ~l/z)l~:
(141)
where k A = zc A~ + 2 A , B , +---~-- ]
~{ 9A 2lBi2 5AIB~ 35B 4] k l B = - ~ Aa + 2 A 3 B , + ----4-- + - - - 4 - - - + - T - ~
(142)
If the (nondimensional) value o f a is such that a = 1 when z = 0, then A~ = 1 and a = e (l/2)/j:= 1 + ½flz
b = Ble(I/2)/J: = B l ( l + ½flz)
(143)
since fl is small and provided z is not too large. At z = 0 the integer n is very large so that the section is approximately circular. As z increases in some range 0 < z < z~ the value of the integer n decreases until n = 2 when the boundary is star-shaped with eight maxima, i.e. r = a + b sin 2 40
(144)
Again, guided by Leipmann and Gharib [10], we assume that there is some structural breakdown in some range z~ < z < z2. Then in the region o f z = z2 the effective value o f the Reynolds number is much smaller, so that /~ is larger, and we look for a different type of solution of Eq. (127) and (128). We assume that v = N z -~
w2 = IVz -~'
(z2 < z)
(145)
where N, ~7, n, m are constants and h is not the same as ti in Eq. (129). When fl is large the dominant terms in Eqs. (127) and (128) are satisfied if h= 2
rh = 2h -- 1 = 3
(146)
Hence V = N2 -2
W2 =/Vz -3
(147)
with velocity components radial rw~ = - ~ 1r v p = r N z -3 transverse rw 2 = r l V z - 3 longitudinal v = N z - 2 F r o m Eqs. (123) and (145)
(148)
252
A.E. Green, P.M. Naghdi / J. Non-Newtonian Fluid Mech. 66 (1996) 233-255
k Z2 R = ~[
kl z4 Rll =~-~
(149)
with RI, RII still being given by Eq. (130). Hence
a = Az
b = Bz
(150)
where N -- ~
kl
+ 2~/B +
re(
9i/2B 2
5t/B 3
35B4~
--~ __~ ~4 + 2 2 3 ~ + ---T-- + - - T - + 1 - ~ - / U
(151)
The sections of the jet are given by
r = {A + B sinZ(2nO)}z
(152)
where we could maintain n = 2 as in Eq. (144) or vary it with z as necessary. Liepmann and Gharib [10] note that after the region of breakdown the jet grows linearly with distance downstream in agreement with Eq. (150). The foregoing solution indicates general agreement with what Liepmann and Gharib find experimentally.
8. Some time-dependent problems Choosing the non-dimensional value of p to be 1, Eq. (66) becomes Re
{,} iJ+~su
=pb-gradp+V2v-~V%
u=-V2v
(153)
where fl is given by Eq. (71). Suppose a liquid of depth lo is at rest and at the instant t = 0 a uniform tangential stress f begins to act on the surface in the x direction. With the origin at the bottom of the liquid and the surface at y = 1 the conditions to be satisfied are v~ = 0 ~Vl
(t = 0)
v~ = 0
(y = 0)
1 ~3Vl
Oy flz 0y2 - f ( Y = 1)
(154)
where we have used the first part of Eq. (50). With zero body force we have from Eq. (66) Re(.~_Vl
\ 8t
1
03v, "~ 02v,
2fl2 OtOyE] = 0y2
1 O4v, f12 0y4
p = const
(155)
The solution of Eq. (155) under conditions (154) is
8 (e_kZtsin(2)_~_ie_9kEtsin(3__~) + " ' ' + } ] where
(156)
A.E. Green, P.M. Naghdi / J. Non-Newtonian Fluid Mech. 66 (1996) 233-255
253
n 2
1
Rek 2-
+fl-5 n~
2s+ 1 n-~rc
s=0,1,2,...
(157)
1-~ 2/~2 As pointed out by Lamb [11], calculations of the above type have sometimes been used (in the case when f l ~ ~ ) to illustrate the action of wind in producing ocean currents; however, inserting numerical values for Re the final state would, according to the formula (156), be approached with extraordinary slowness. A more practical interpretation is obtained by using an ad hoc "coefficient of turbulence". However, if in Eq. (157) we suppose that fl is small then Re k 2 is replaced by two (instead of one when fl is large). This would halve the time, which would go some way towards a practical interpretation. As a second time-dependent example consider motion in circles around the axis of z, the velocity vo = rf being a function of the distance r from this axis and also a function of time. Then, from Eq. (66) I~ t Re
1 ~h] 1 ~p 1 /'~2h 3 ~ h ) 2fl 2 ~ = - r - I ~ - - ~ + h - ~ 5 ~ r E + r ~ r
1
~p
(158) (159)
where O2f . 3 ~f h = ~ r 2 + r ar
(160)
Then p = - A O + H(r, t) where A is a constant and ~H = R e r 2 [ f 2 - ~ - ~1 f h ] ~---~
(161)
Re
(162)
= -~ 4- h - --~ ~ ~r 2 + r --~rJ
The vorticity ( of the fluid about the z axis is
af
(=r~r+2
f
(163)
and h = - -r ~r
(164)
We consider a solution for an infinite fluid when A = 0 and we have an isolated vortex at z = 0. Then when fl in Eq. (162) is very large we have the classical solution [11]:
A.E. Green; P.M. Naghdi / J. Non-Newtonian Fluid Mech. 66 (1996) 233-255
254 K
f = 2--~-5r2[1 (=
-- e -
Re
r2/(4t)]
(165)
Re x e - Rer2/(at) 4m
where k is the value of the circulation when t ~ ~ . As t increases from 0 to oo the velocity diminishes from k/(2nr) to 0. The vorticity increases from zero to a maximum and then falls to zero.
In contrast, when fl is very small (so that the Reynolds number Re is large) Eq. (162) reduces to Re0h 2 0t
02h 3Sh - + 0r r ~r
(166)
From our present vortex problem Eq. (166) with Eq. (160) reduces to
Re Of_ O2f 3 Of 2 &
~r
r 0r
(167)
which is the same as Eq. (162) when fl is large but with Re replaced by Re/2. The solution corresponding to Eq. (165) is then h"
f = 2-~r2 [1 - - e -
Re
K e_Rer2/(8t )
r2/(St)] (168)
The rate of diffusion of the vortex into the surrounding fluid is therefore halved at high Reynolds number, compared with that of Eq. (165) for low Reynolds numbers. Other time-dependent problems show similar effects of the present theory compared with the classical Newtonian viscous fluid theory, especially for large Reynolds numbers, and give support to the suggestion that the theory can be of value in relation to turbulence as indicated in Sections 5-7.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
A.E. Green and P.M. Naghdi, Proc. R. Soc. London, 357 (1977) 253-270. A.E. Green and P.M. Naghdi, J. Appl. Math. Phys. 42 (1991)159-168. J.S. Marshall and P.M. Naghdi, Philos. Trans. R. Soc. London, A327 (1989) 415-449. J.S. Marshall and P.M. Naghdi, Philos. Trans. R. Soc. London, A327 (1989) 449-475. A.E. Green and P.M. Naghdi, Proc. R. Soc. London, 432 (1991) 171-]L94; A438 (1992) 605. A.E. Green and P.M. Naghdi, Philos. Trans. R. Soc. London, A448 (1995) 335-356; A448 (1995) 357-377; A448 (1995) 379-388. J.S. Marshall and P.M. Naghdi, Contin. Thermodyn., 3 (1991) 65-77. J. Laufer, Natl. Advis. Comm. Aeronaut. Rep., 1053 (1951) 1247-1265. A.E. Green and P.M. Naghdi, Philos. Trans. R. Soc. London, A432 (1993) 525-542. D. Liepmann and M. Gharib, J. Fluid Mech., 245 (1992) 643-668.
A.E. Green, P.M. Naghdi / J. Non-Newtonian Fluid Mech. 66 (1996) 233 255
[11] H. Lamb, Hydrodynamics, 6th edn., 1932, University Press, Cambridge, pp. 592, 593. [12] J. Laufer, Natl. Advis. Comm. Aeronaut. Rep., 1174 (1954) 417-434.
255