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Transverse effects in shear flow of certain isotropic liquids
Journal of Non -Newtonian Fluid Mechanics, 14 ( 1984) 16 1- 112 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands
TRANSVERSE LIQUIDS *
161
EFFECTS IN SHEAR FLOW OF CERTAIN ISOTROPIC
F.M. LESLIE Department of Mathematics,
University of Strathclyde,
Glasgow (Great Britain)
(Received May 31, 1983)
Summary This paper discusses shear flow for a theory of isotropic liquids which differs from that for a Newtonian liquid solely by allowing angular velocity to be an independent variable. With the particular constitutive theory employed, one finds that transverse secondary flows occur despite the symmetry of the problem. These effects should prove more readily detectable in practice than existing predictions for such theory of variations in viscosity at small gapwidths.
1. Introduction In anisotropic liquids such as liquid crystals, transverse flow effects can occur rather naturally in shear flow due to asymmetric alignment of the anisotropic axis, stemming from the presence of a transverse magnetic field or particular surface conditions. Pieranski and Guyon [I] describe one experiment demonstrating such an effect, and Leslie [2] discusses other possibilities in his review of flow phenomena in liquid crystals. In this paper, however, we present an example in which transverse flow effects occur somewhat unexpectedly in a theory of isotropic liquids called polar fluid theory. Given that this theory is not far removed from that for a Newtonian liquid, the questions raised are not without some practical interest. Polar fluid theory differs from that of classical fluid mechanics in that it considers the angular velocity of a material element to be an independent
* Dedicated
to the memory
0377-0257/84/$03.00
of Professor
J.G. Oldroyd.
Q 1984 Elsevier Science Publishers
B.V.
162 kinematic variable, rather than regard it as equal to half the local vorticity. This leads to a more general treatment of conservation of angular momentum encompassing couple stress and body couples. A detailed account of this theory and some of its predictions is available in the review by Cowin [3]. For viscometric flows, the novel feature of such theory is that the effective viscosity can vary with the dimensions of the viscometer. When the dimensions are large compared with a characteristic material length, the flow is essentially Newtonian, but non-Newtonian behaviour occurs when the dimensions are comparable with this length, due to the boundaries basically inhibiting rotation of material elements. As Cowin discusses, detection of such effects in practice requires viscosity measurements with rather narrow gaps, and at present no conclusive experimental evidence seems to be available of such size effects. All viscometric studies’based on polar fluid theory employ constitutive relations that are linear isotropic functions of certain kinematic variables. However, as Leslie [4] points out, experience with liquid crystal theory shows that such constitutive assumptions can be too restrictive for an adequate description of certain materials, and therefore it appears reasonable to extend the constitutive theory to linear hemitropic functions of these variables. In this paper, therefore, we investigate the predictions of this more general theory for shear flow. The outcome is a little surprising in that we find that transverse flows occur naturally although the problem is outwardly symmetric about the plane of imposed shear. While the possibility of such additional complications in an isotropic theory is in itself of interest, there is the further point that these transverse flows may be more readily observable in experiments than existing predictions. 2. Polar fluid theory Cowin’s review [3] provides a full account of polar fluid theory, and therefore it suffices here to give only a summary of the relevant equations. In doing so, it is convenient to use Cartesian tensor notation; consequently repeated indices are subject to the summation convention, and a comma preceding a subscript implies partial differentiation with respect to the corresponding spatial coordinate. Also, a superposed dot denotes a material time derivative. Polar fluid theory employs two independent vector fields to describe flow, the velocity 0 and the angular velocity w of the fluid element, the latter an axial vector. With the assumption of incompressibility, the former is of course subject to the constraint 2);,i = o.,
(2.1)
163 q,and mass conservation reduces to a statement that density p is constant. Conservation of linear momentum remains unchanged by the inclusion of the second kinematic variable, and thus po; = p& + tij,j, (2.2) where F denotes body force per unit mass, and i the stress tensor. Here, however, the latter is no longer symmetric, since conservation of angular momentum now takes the more general form pk2h, = pG; + gi + sij,j,
gi=f?.
‘P4
t 4P’
(2.3)
in which k is an inertial constant, presumably small, G external body couple per unit mass, g intrinsic body couple per unit volume arising from the asymmetric stress, and s couple stress tensor. Lastly, the point form of the energy balance is pi = pr - qi,; + t,jO,,j + sijw;,j - g,oi,
(2.4)
where E is internal energy per unit mass, q heat flux vector, and Yheat supply per unit mass per unit time. The linear isotropic constitutive theory for polar fluids derived by Cowin [3] adds to that for viscous fluids a dependence upon angular velocity and its gradient, and also a corresponding prescription for the couple stress. As Leslie [4] discusses, their hemitropic counterparts for an incompressible liquid are tij = -pS,, + 2pD,, + 2reijkHk + (6 + v)ui., + (t - v)w,,~ + 2Se,jkfl.k, (2.5) ‘ij =
-ev)6ij+(P+Y)o,,_/+(P-Y)wj.i+aei~jke.k
(aWk,k
+ 2+Djj + 21eijk Hk, q, =
-
K8,i
+
6deijkak,j
+
(24 TH,,
(2.7)
where the rate of strain tensor D and the axial vector H are given by 2D,, = ui,, + o,,~,
Hi = wi - $eijkokqj,
(2.8)
and p denotes the arbitrary pressure arising from the constraint (2.1) and 8 temperature. The coefficients are at most functions of temperature, and to avoid ambiguities in sign it is preferable to restrict one’s choice of Cartesian axes to right-handed systems. Leslie attempts to restrict the possible values of the various coefficients in the above by thermodynamic considerations, and finds that an entropy production argument requires that II 2 0,
r > 0,
K > 0,
3a+2p+2exao, (2.9)
164 where 48=
a=a+w/&-Bdl/dB,
46 - m/9 + Bdv/d6,
(2.10)
x, L and v being coefficients that appear in his entropy flux vector. At solid surfaces it is natural to retain the customary no-slip hypothesis as a boundary condition for the velocity. While the choice of an appropriate condition for angular velocity is less clear, one possibility is simply to assume that the angular velocity at the surface coincides with that of the solid boundary. This is the condition most commonly used, although Cowin [3] does mention alternatives. Since our choice does not really affect our main conclusions, we adopt the more popular option and assume that both the velocity and angular velocity at a solid surface coincide with the respective values for the solid. 3. Shear flow solutions In this section, we consider shear flow between two parallel plates, when the body force is simply gravity and external couples and heat sources are absent. Initially, it is helpful to maintain some generality before turning to specific cases, although the complexity of the equations does necessitate some particular assumptions. In common with most viscometric studies, our analysis ignores the thermal dependence of the various coefficients. Also, it appears reasonable on occasion to place restrictions on the relative magnitudes of the material parameters, partly motivated by the inequalities (2.9). With a choice of right-handed Cartesian axes such that the z-axis is normal to the plates and the origin midway between them, it is convenient to consider solutions of the foregoing equations in which the components of velocity and angular velocity and temperature depend solely upon the spatial coordinate z, so that 0, = u(z),
0-Y= b(Z),
V, = 0,
w* =f(&
UjJ=g(z),
a, = h(z),
(3.1)
e = e(z>, this clearly consistent with the constraint (2.1). One’s first inclination is somewhat naturally to discuss solutions similar to those described by Cowin [3], in which the components of velocity n and of angular velocity f and h are all zero, and thermal effects are ignored. This leads, however, to an overdeterminacy of solutions in all except very special cases, and consequently one is quickly compelled to include the generality embraced above. With this choice, the equations for conservation of linear momentum reduce to F’,X= t,,.: 9
P..” = t.vz.z’
P., = L,:
Y
(3.2)
165 where p=p++,
i,, = t,, + p,
(3 -3)
and J/ denotes the gravitational potential. tion of angular momentum become
Also, the equations
and lastly the energy balance
can be expressed
4 2.2 = tXZU,Z+ fyre,, + W+
s,,g + %Z+
as (3.5)
this following after use of eqns. (3.4). Straightforwardly a separation of variables yields t xz =a+cz,
for conserva-
argument
applied to eqns. (3.2)
tyr=b+dz,
(3.6)
p=po++cx+dy+iZ,,
where a, b, c, d and p. are arbitrary determines the pressure, and the former constitutive relation (2.5) lead to
constants. The latter ultimately pair of equations along with the
(I_L+T)u’-227g+(<+v)f’=a+cz, (3.7)
(/~+++27f+(5+v)g’=b+dz,
the prime denoting differentiation with respect to z. Moreover, the first two of eqns. (3.4) with the constitutive assumptions (2.5) and (2.6) provide a further pair of equations (P+v)f”+
b+l)u”-
2([ + v)g’ - 2ru’-4rf
(~+y)g”+(C#J++“+2(~+Y)f’+2rU’-4Tg=o,
= 0,
(3.8)
which again involve soleI:: tne functions u, z), f and g as in (3.7). Hence (3.7) and (3.8) provide four equations to determine the flow and the first two components of angular velocity. Rather clearly an inconsistency arises if one sets any of these components equal to zero, except perhaps in very special cases. Once these quantities are determined, the last of eqns. (3.4) and the energy balance (3.5) form two equations for the third component of angular velocity h and the temperature 8. Leslie [4] discusses a problem involving coupling between temperature and angular velocity, the liquid otherwise at rest, but here our concern is with viscometry, and therefore we concentrate upon solutions of equations (3.7) and (3.8). Before proceeding to particular cases it is of interest to ask if solutions of eqns. (3.7) and (3.8) are uniquely determined by specification of the stress components (3.6). To this end, assume that two solutions exist for given
166 values of a, b, c and d, and consider 5 = #)
3 = u(1) - @),
_ rJ2) 7
it follows from the linearity
their difference.
Thus if
f=f’U_fW,
of the equations
g = g”’ - gC2’, (3.9)
that
(/A + 7)U’ - 27g’+ ([ + V)F = 0, (/.A+ 7)8
+ 27f-t
(< + v)g’ = 0,
(p -t y)f”
+ (+ + {)U” - 2(3 + V)g’ - 275’ - 47f’
0,
(p+
+ (++
0.
y)g”
Elimination
{)v”+
2(3+
of the velocity
kf” - mg’ - nf= 0,
I+‘+
27u’-47g=
components
(3.10)
leads to
Q” -I- mJ’ - ng = 0,
(3.11)
where k = (P + Yh m = 2(p(l+
+ 4
- b#J+ O(5
v) - T(+
+ 4,
(3.12)
n = 4/lr.
+ <)),
From the inequalities (2.9), the parameter n is clearly positive. Further the first term in k is likewise positive, and it seems reasonable to assume that k itself is positive, particularly when the latter of (2.9) essentially restrict the magnitude of the remaining coefficients. Naturally we assume that both k and n are non-zero. Should the coefficient m be zero, the equations (3.11) are readily solved. Otherwise further elimination shows that f and g both satisfy
k2fro + ( m2 - 2nk)f” and for uniqueness values. If one sets f=
+ n2f= 0,
(3.13)
both f and g must vanish when subject to zero boundary
eihr,
(3.14)
where X is an unknown
constant,
it follows that
k2A4 + (2nk - m2)A2 + n2 = 0,
(3.15)
and thus that X = ( $-m + ( m2 - 4nk)“2)/2k. Three cases therefore
(3.16)
emerge:
(i)
m2 c 4nk,
X = + (n/k)“2e*‘“,
(ii)
m2 = 4nk 3
X= +
(iii)
m2 > 4nk,
X = + (n/k)“2e*“,
cos2 w = m2/4nk,
(n/k)“2,
(3.17) (3.18)
cash’ w = m2/4nk.
(3.19)
167 The corresponding (i)
solutions
for f and S are as follows:
f=AcosX,zcoshX,z+BsinX,zsinhX,z+Ccosh,zsinhX,z +D sin h,z cash X,z, g=.Qcosx, -A
z cash X,z - C sin h,z sinh X,z sin X,z cash X,z + B cos X,z sinh A,z),
A, = (n/k)“2 (ii)
A, = (n/~?)“~
cos W,
sin w,
(3.20) s = sgn m.
f= A cos Xz + Bz sin AZ + C sin AZ + Dz cos AZ, g = s(C cos AZ - Dz sin AZ - A sin AZ + Bz cos AZ), A = ( n/k)“2,
(iii)
(3.21)
s=sgnm.
f= A cos X,z + B cos X,z + C sin X,z + D sin h2z, g= s(C cos X,z + D cos A,z -A A, = (n/k)1’2
ew,
A, = (n/k)“2
sin X,z - B sin X,z), ePw,
(3.22)
s=sgnm.
The solution for the particular case that m vanishes follows from (i) above by setting w equal to r/2. Straightforwardly one finds that solutions are uniquely determined in cases (i) and (ii), but that non-uniqueness occurs in case (iii) whenever (A, - A,)h = rl7r,
(3.23)
h denoting half the gapwidth and n an integer. This would seem to indicate instability at certain gapwidths, whatever the flow. Since such a phenomenon appears not to have been observed in practice, it seems that one should disregard values of the material parameters that give rise to this case, at least for the present. In a recent paper, Fraser [5] also finds that stability considerations can lead to some restriction upon the choice of material parameters. Below we therefore concentrate our discussion upon solutions corresponding to case (i), this seeming the most likely to be relevant in any event. Results for the special cases m zero and (ii) are readily obtainable either directly or from the former by taking limits. 4. Simple shear flow Here we consider the simplest form of shear flow, that due to the relative motion of two plates in a given direction parallel to their own planes, there being no imposed pressure gradient parallel to the plates. Given the symmetry of the problem, one might initially anticipate flow solely in the plane of shear, but the discussion of the previous section clearly indicates otherwise.
168
We therefore examine solutions of the eqns. (3.7) and (3.8) with the constants c and d zero, and an appropriate choice of x- and y-axes leads to the boundary conditions u(h) = -24(-h)
= I/,
u(h)=u(-h)=O (4.1)
f(h)=f(-h)=g(h)=g(-h)=O,
V being a constant and h denoting half the gapwidth. Given these conditions, an inspection of the equations suggests that one seek solutions in which u and f are both odd functions of z and o and g are even, this at once requiring that b be zero. In this event, the eqns. (3.7) and (3.8) lead to kf” - mg’ - nf = 0,
kg”+mf’-ng=
where k, m and n are as defined
-27~2,
in (3.12), and the substitutions
g=g+a/2y
f=.A
(4.2)
(4.3)
reduce these to kj” - mg’ - ny= 0,
kg” + rnf’ - ng = 0.
(4.4)
For the reasons given at the end of the previous section, we restrict our attention to the case (i) of that section. Hence using the results (3.20) the relevant solution for the angular velocity components is f = “( A4 cos X,z sinh X,z - N sin X,z cash X,z), 2P
(4.5)
g = a(l 2P
(4.6)
- N cos A,z cash h,z - M sin h,z sinh X,z),
where it4 = sin h,h/sinh
X,h,
N = cos X,h,‘cosh h,h,
and the A,, A, and s are as defined in (3.20). The corresponding components follow from integration of eqns. (3.7), and are
u=az_
(4.7) velocity
cos h,z sinh X,z - N sin X,z cash X,z)
P
-
4P2;+
a((+v) O= - 2&L+7) -
T) (p sin A, z cash X,z - Q cos h,z sinh X,z), (1 _N
(4.8)
cos X,z cash X,z - A4 sin X,z sinh X2z) z cash h,z - cos X,h cash A,h)
+ Q(sin X,z sinh A,z - sin X,h sinh X,h)),
(4.9)
169 with Q = X,M - X,N.
P = X,N + A&f,
(4.10)
In the former the constant of integration is zero since u is an odd function. Setting z equal to h in eqn. (4.8) gives a relationship between the parameters defining the problem F’, h and a in the form V=ah/p--
akA,(cosh2X2h
- cos2 A,h)
(4.11)
4p2(p + T) sinh A,h cash X,h ’
or equivalently V=ah/p(l
(4.12)
-+rH(h)/(p++))
where H(h) = sin’ w(tanh X,h/X,h)(
1 + sin2 X,h/sinh2
h,h),
this using the definitions of X2 and the coefficient respectively). It is relatively easy to show that lim H = 1,
h-0
lim H = 0,
(4.13) n ((3.20)
and (3.12), (4.14)
h-m
and that this factor takes values between these limits for intermediate values. Since /_tand T are both positive, it follows that V and a always have the same sign. If one defines an apparent viscosity n in the usual way by 7) = ah/V,
(4.15)
the result (4.12) yields n = p/(1-
TH(h)/(p
(4.16)
+ T)).
Also the limits (4.14) at once give Ifimon=p+r,
Jimm7)=p
in agreement
with the conclusions
(4.17) of Cowin [3].
5. Plane Poiseuille flow Here we consider the flow produced by a pressure gradient in a given direction parallel to two parallel stationary plates. Partly this is to complete our account of shear flow, but also to show a novel aspect of the solution obtained. Our previous results of course lead one to expect transverse flow, but the fact that a transverse component of shear stress is also induced is perhaps somewhat surprising. With the x-axis parallel to the imposed pressure gradient we therefore
170 examine solutions of eqns. (3.7) and (3.8) when c is non-zero The boundary conditions for this flow are
but d is zero.
U(h)=+h)=v(h)=u(-h)=O, (5.1)
f(h)=f(-h)=g(h)=g(-h)=O,
and these conditions along with an inspection of the equations suggest in this case that one seek solutions with u and f even functions but o and g odd. This entails setting the constant a equal to zero, but since b need not vanish we retain it for the present. From eqns. (3.7) and (3.8) one now obtains kf” - mg’ - nf = 27b - (C#B + [)c, kg”+mf’-ng=
(5.2)
-2wz,
with k, m and n as above. The change of variables f =j+
CA - b/2p,
g=g+cz/2p,
(5.3)
with A=(2&+S)--))/2pn,
(54
yields once more kp
- mg’ - nf= 0,
Again confining quickly give f=(cA-b/2p)(I + @(R 2P g = ch/2p(
-s(cA
kg” + mf’ - ng = 0.
attention
to the case (i) of Section
(5.5)
3, the results
(3.20)
-A’ cos h,z cash A,z - M sin X,z sinh A,z) cos h,z cash h2z - S sin X,z sinh X,z),
(54
- S cos X,z sinh X,z - R sin h,z cash X,z)
z/h
- b/2p)(M
cos X,z sinh X,z - N sin X,z cash X,z),
(5.7)
where M= R=
sin h,h sinh X,h
-,
sin A,h cash X,h
7
Also the velocity given by
N= S=
cos X,h
cash A,h
3
(5 4
cos X,h
sinh X,h *
components
follow from integration
of eqns. (3.7) and are
(~+7)U=(~/2)(z~-h~)-(~+~)f+2rj-~~gdz, (p++=bz-(E+v)g--2$fdz,
(5 -9)
171 the constant of integration being zero in the latter because o is odd. However, the boundary conditions are satisfied provided that one chooses the constant b such that k&(cosh2
X,h - cos2 h,h)
4p(p + T)h sinh X,h cash X,h
X,h - cos2 X,h) 4prh sinh h,h cash X,h
kX,(cosh2
(5.10) Since the coefficient of b is a non-zero multiple of the right-hand side of eqn. (4.1 l), it does not vanish and thus the above condition determines the constant. In plane Poiseuille flow therefore the induced transverse flow gives rise in general to a non-zero transverse component of shear stress. It is of interest to note that this component tends to zero as the gapwidth shrinks to zero, but has a non-zero limit as the gapwidth becomes very large. 6. Concluding remarks As mentioned earlier, the principal prediction of the theory of polar liquids described by Cowin [3] is an increase in effective viscosity near solid surfaces, which would become apparent in viscosity measurements with narrow gaps. In practice attempts to confirm this prediction meet problems in that such measurements are sensitive to surface irregularities and also any slight mis-alignment of the plates. However, the present prediction of transverse flow effects is not so readily masked by these practical difficulties. Clearly, if one reverses the direction of primary flow, the direction of the transverse secondary flow is also changed, this apparently allowing these secondary flows to be distinguished from effects due to the complications cited. Also, and more importantly, these transverse effects do not appear to be restricted to flows with narrow gaps, although they may be small due to the relevant material parameters being small compared with others. In practice, of course, the plates are finite and it is not clear how the channel boundaries affect these predictions. Naturally, one anticipates that their presence simply gives rise to some modification of the flow in their immediate vicinity. ‘Given that the secondary transverse flow is probably small, such edge effects are presumably insignificant. Finally, the question arises as to which liquids are likely to display such effects. Intuitively, one thinks of the rotation of a fluid element being more important when the constituent molecular structure is relatively large, and therefore liquids composed of relatively large molecules appear to be the
172 more probable candidates. However, the weaker material symmetry discussed in this paper presumably eliminates many liquids, and this consideration quickly brings to mind liquid crystals having this weaker symmetry. Consequently, if the theory described in this paper has relevance to real materials, one imagines that it must predict behaviour of cholesteric liquid crystals in their “isotropic” state. Certainly, it would be of interest to investigate shear flow of cholesteric liquid crystals at temperatures above the liquid crystal-isotropic liquid transition. However, the predictions obtained above are not necessarily confined to liquids that exhibit liquid crystalline properties. References 1 2 3 4 5
P. Pieranski F.M. Leslie, S.C. Cowin, F.M. Leslie, C. Fraser, J.
and E. Guyon, Phys. Lett., 49A (1974)237. Adv. Liq. Cryst., 4 (1979) 1. Adv. Appl. Mech., 14 (1974) 279. Arch. Ration. Mech. Anal., 70 (1979) 189. Appl. Math. Phys. (ZAMP), 32 (1981) 695.
TRANSVERSE LIQUIDS *
161
EFFECTS IN SHEAR FLOW OF CERTAIN ISOTROPIC
F.M. LESLIE Department of Mathematics,
University of Strathclyde,
Glasgow (Great Britain)
(Received May 31, 1983)
Summary This paper discusses shear flow for a theory of isotropic liquids which differs from that for a Newtonian liquid solely by allowing angular velocity to be an independent variable. With the particular constitutive theory employed, one finds that transverse secondary flows occur despite the symmetry of the problem. These effects should prove more readily detectable in practice than existing predictions for such theory of variations in viscosity at small gapwidths.
1. Introduction In anisotropic liquids such as liquid crystals, transverse flow effects can occur rather naturally in shear flow due to asymmetric alignment of the anisotropic axis, stemming from the presence of a transverse magnetic field or particular surface conditions. Pieranski and Guyon [I] describe one experiment demonstrating such an effect, and Leslie [2] discusses other possibilities in his review of flow phenomena in liquid crystals. In this paper, however, we present an example in which transverse flow effects occur somewhat unexpectedly in a theory of isotropic liquids called polar fluid theory. Given that this theory is not far removed from that for a Newtonian liquid, the questions raised are not without some practical interest. Polar fluid theory differs from that of classical fluid mechanics in that it considers the angular velocity of a material element to be an independent
* Dedicated
to the memory
0377-0257/84/$03.00
of Professor
J.G. Oldroyd.
Q 1984 Elsevier Science Publishers
B.V.
162 kinematic variable, rather than regard it as equal to half the local vorticity. This leads to a more general treatment of conservation of angular momentum encompassing couple stress and body couples. A detailed account of this theory and some of its predictions is available in the review by Cowin [3]. For viscometric flows, the novel feature of such theory is that the effective viscosity can vary with the dimensions of the viscometer. When the dimensions are large compared with a characteristic material length, the flow is essentially Newtonian, but non-Newtonian behaviour occurs when the dimensions are comparable with this length, due to the boundaries basically inhibiting rotation of material elements. As Cowin discusses, detection of such effects in practice requires viscosity measurements with rather narrow gaps, and at present no conclusive experimental evidence seems to be available of such size effects. All viscometric studies’based on polar fluid theory employ constitutive relations that are linear isotropic functions of certain kinematic variables. However, as Leslie [4] points out, experience with liquid crystal theory shows that such constitutive assumptions can be too restrictive for an adequate description of certain materials, and therefore it appears reasonable to extend the constitutive theory to linear hemitropic functions of these variables. In this paper, therefore, we investigate the predictions of this more general theory for shear flow. The outcome is a little surprising in that we find that transverse flows occur naturally although the problem is outwardly symmetric about the plane of imposed shear. While the possibility of such additional complications in an isotropic theory is in itself of interest, there is the further point that these transverse flows may be more readily observable in experiments than existing predictions. 2. Polar fluid theory Cowin’s review [3] provides a full account of polar fluid theory, and therefore it suffices here to give only a summary of the relevant equations. In doing so, it is convenient to use Cartesian tensor notation; consequently repeated indices are subject to the summation convention, and a comma preceding a subscript implies partial differentiation with respect to the corresponding spatial coordinate. Also, a superposed dot denotes a material time derivative. Polar fluid theory employs two independent vector fields to describe flow, the velocity 0 and the angular velocity w of the fluid element, the latter an axial vector. With the assumption of incompressibility, the former is of course subject to the constraint 2);,i = o.,
(2.1)
163 q,and mass conservation reduces to a statement that density p is constant. Conservation of linear momentum remains unchanged by the inclusion of the second kinematic variable, and thus po; = p& + tij,j, (2.2) where F denotes body force per unit mass, and i the stress tensor. Here, however, the latter is no longer symmetric, since conservation of angular momentum now takes the more general form pk2h, = pG; + gi + sij,j,
gi=f?.
‘P4
t 4P’
(2.3)
in which k is an inertial constant, presumably small, G external body couple per unit mass, g intrinsic body couple per unit volume arising from the asymmetric stress, and s couple stress tensor. Lastly, the point form of the energy balance is pi = pr - qi,; + t,jO,,j + sijw;,j - g,oi,
(2.4)
where E is internal energy per unit mass, q heat flux vector, and Yheat supply per unit mass per unit time. The linear isotropic constitutive theory for polar fluids derived by Cowin [3] adds to that for viscous fluids a dependence upon angular velocity and its gradient, and also a corresponding prescription for the couple stress. As Leslie [4] discusses, their hemitropic counterparts for an incompressible liquid are tij = -pS,, + 2pD,, + 2reijkHk + (6 + v)ui., + (t - v)w,,~ + 2Se,jkfl.k, (2.5) ‘ij =
-ev)6ij+(P+Y)o,,_/+(P-Y)wj.i+aei~jke.k
(aWk,k
+ 2+Djj + 21eijk Hk, q, =
-
K8,i
+
6deijkak,j
+
(24 TH,,
(2.7)
where the rate of strain tensor D and the axial vector H are given by 2D,, = ui,, + o,,~,
Hi = wi - $eijkokqj,
(2.8)
and p denotes the arbitrary pressure arising from the constraint (2.1) and 8 temperature. The coefficients are at most functions of temperature, and to avoid ambiguities in sign it is preferable to restrict one’s choice of Cartesian axes to right-handed systems. Leslie attempts to restrict the possible values of the various coefficients in the above by thermodynamic considerations, and finds that an entropy production argument requires that II 2 0,
r > 0,
K > 0,
3a+2p+2exao, (2.9)
164 where 48=
a=a+w/&-Bdl/dB,
46 - m/9 + Bdv/d6,
(2.10)
x, L and v being coefficients that appear in his entropy flux vector. At solid surfaces it is natural to retain the customary no-slip hypothesis as a boundary condition for the velocity. While the choice of an appropriate condition for angular velocity is less clear, one possibility is simply to assume that the angular velocity at the surface coincides with that of the solid boundary. This is the condition most commonly used, although Cowin [3] does mention alternatives. Since our choice does not really affect our main conclusions, we adopt the more popular option and assume that both the velocity and angular velocity at a solid surface coincide with the respective values for the solid. 3. Shear flow solutions In this section, we consider shear flow between two parallel plates, when the body force is simply gravity and external couples and heat sources are absent. Initially, it is helpful to maintain some generality before turning to specific cases, although the complexity of the equations does necessitate some particular assumptions. In common with most viscometric studies, our analysis ignores the thermal dependence of the various coefficients. Also, it appears reasonable on occasion to place restrictions on the relative magnitudes of the material parameters, partly motivated by the inequalities (2.9). With a choice of right-handed Cartesian axes such that the z-axis is normal to the plates and the origin midway between them, it is convenient to consider solutions of the foregoing equations in which the components of velocity and angular velocity and temperature depend solely upon the spatial coordinate z, so that 0, = u(z),
0-Y= b(Z),
V, = 0,
w* =f(&
UjJ=g(z),
a, = h(z),
(3.1)
e = e(z>, this clearly consistent with the constraint (2.1). One’s first inclination is somewhat naturally to discuss solutions similar to those described by Cowin [3], in which the components of velocity n and of angular velocity f and h are all zero, and thermal effects are ignored. This leads, however, to an overdeterminacy of solutions in all except very special cases, and consequently one is quickly compelled to include the generality embraced above. With this choice, the equations for conservation of linear momentum reduce to F’,X= t,,.: 9
P..” = t.vz.z’
P., = L,:
Y
(3.2)
165 where p=p++,
i,, = t,, + p,
(3 -3)
and J/ denotes the gravitational potential. tion of angular momentum become
Also, the equations
and lastly the energy balance
can be expressed
4 2.2 = tXZU,Z+ fyre,, + W+
s,,g + %Z+
as (3.5)
this following after use of eqns. (3.4). Straightforwardly a separation of variables yields t xz =a+cz,
for conserva-
argument
applied to eqns. (3.2)
tyr=b+dz,
(3.6)
p=po++cx+dy+iZ,,
where a, b, c, d and p. are arbitrary determines the pressure, and the former constitutive relation (2.5) lead to
constants. The latter ultimately pair of equations along with the
(I_L+T)u’-227g+(<+v)f’=a+cz, (3.7)
(/~+++27f+(5+v)g’=b+dz,
the prime denoting differentiation with respect to z. Moreover, the first two of eqns. (3.4) with the constitutive assumptions (2.5) and (2.6) provide a further pair of equations (P+v)f”+
b+l)u”-
2([ + v)g’ - 2ru’-4rf
(~+y)g”+(C#J++“+2(~+Y)f’+2rU’-4Tg=o,
= 0,
(3.8)
which again involve soleI:: tne functions u, z), f and g as in (3.7). Hence (3.7) and (3.8) provide four equations to determine the flow and the first two components of angular velocity. Rather clearly an inconsistency arises if one sets any of these components equal to zero, except perhaps in very special cases. Once these quantities are determined, the last of eqns. (3.4) and the energy balance (3.5) form two equations for the third component of angular velocity h and the temperature 8. Leslie [4] discusses a problem involving coupling between temperature and angular velocity, the liquid otherwise at rest, but here our concern is with viscometry, and therefore we concentrate upon solutions of equations (3.7) and (3.8). Before proceeding to particular cases it is of interest to ask if solutions of eqns. (3.7) and (3.8) are uniquely determined by specification of the stress components (3.6). To this end, assume that two solutions exist for given
166 values of a, b, c and d, and consider 5 = #)
3 = u(1) - @),
_ rJ2) 7
it follows from the linearity
their difference.
Thus if
f=f’U_fW,
of the equations
g = g”’ - gC2’, (3.9)
that
(/A + 7)U’ - 27g’+ ([ + V)F = 0, (/.A+ 7)8
+ 27f-t
(< + v)g’ = 0,
(p -t y)f”
+ (+ + {)U” - 2(3 + V)g’ - 275’ - 47f’
0,
(p+
+ (++
0.
y)g”
Elimination
{)v”+
2(3+
of the velocity
kf” - mg’ - nf= 0,
I+‘+
27u’-47g=
components
(3.10)
leads to
Q” -I- mJ’ - ng = 0,
(3.11)
where k = (P + Yh m = 2(p(l+
+ 4
- b#J+ O(5
v) - T(+
+ 4,
(3.12)
n = 4/lr.
+ <)),
From the inequalities (2.9), the parameter n is clearly positive. Further the first term in k is likewise positive, and it seems reasonable to assume that k itself is positive, particularly when the latter of (2.9) essentially restrict the magnitude of the remaining coefficients. Naturally we assume that both k and n are non-zero. Should the coefficient m be zero, the equations (3.11) are readily solved. Otherwise further elimination shows that f and g both satisfy
k2fro + ( m2 - 2nk)f” and for uniqueness values. If one sets f=
+ n2f= 0,
(3.13)
both f and g must vanish when subject to zero boundary
eihr,
(3.14)
where X is an unknown
constant,
it follows that
k2A4 + (2nk - m2)A2 + n2 = 0,
(3.15)
and thus that X = ( $-m + ( m2 - 4nk)“2)/2k. Three cases therefore
(3.16)
emerge:
(i)
m2 c 4nk,
X = + (n/k)“2e*‘“,
(ii)
m2 = 4nk 3
X= +
(iii)
m2 > 4nk,
X = + (n/k)“2e*“,
cos2 w = m2/4nk,
(n/k)“2,
(3.17) (3.18)
cash’ w = m2/4nk.
(3.19)
167 The corresponding (i)
solutions
for f and S are as follows:
f=AcosX,zcoshX,z+BsinX,zsinhX,z+Ccosh,zsinhX,z +D sin h,z cash X,z, g=.Qcosx, -A
z cash X,z - C sin h,z sinh X,z sin X,z cash X,z + B cos X,z sinh A,z),
A, = (n/k)“2 (ii)
A, = (n/~?)“~
cos W,
sin w,
(3.20) s = sgn m.
f= A cos Xz + Bz sin AZ + C sin AZ + Dz cos AZ, g = s(C cos AZ - Dz sin AZ - A sin AZ + Bz cos AZ), A = ( n/k)“2,
(iii)
(3.21)
s=sgnm.
f= A cos X,z + B cos X,z + C sin X,z + D sin h2z, g= s(C cos X,z + D cos A,z -A A, = (n/k)1’2
ew,
A, = (n/k)“2
sin X,z - B sin X,z), ePw,
(3.22)
s=sgnm.
The solution for the particular case that m vanishes follows from (i) above by setting w equal to r/2. Straightforwardly one finds that solutions are uniquely determined in cases (i) and (ii), but that non-uniqueness occurs in case (iii) whenever (A, - A,)h = rl7r,
(3.23)
h denoting half the gapwidth and n an integer. This would seem to indicate instability at certain gapwidths, whatever the flow. Since such a phenomenon appears not to have been observed in practice, it seems that one should disregard values of the material parameters that give rise to this case, at least for the present. In a recent paper, Fraser [5] also finds that stability considerations can lead to some restriction upon the choice of material parameters. Below we therefore concentrate our discussion upon solutions corresponding to case (i), this seeming the most likely to be relevant in any event. Results for the special cases m zero and (ii) are readily obtainable either directly or from the former by taking limits. 4. Simple shear flow Here we consider the simplest form of shear flow, that due to the relative motion of two plates in a given direction parallel to their own planes, there being no imposed pressure gradient parallel to the plates. Given the symmetry of the problem, one might initially anticipate flow solely in the plane of shear, but the discussion of the previous section clearly indicates otherwise.
168
We therefore examine solutions of the eqns. (3.7) and (3.8) with the constants c and d zero, and an appropriate choice of x- and y-axes leads to the boundary conditions u(h) = -24(-h)
= I/,
u(h)=u(-h)=O (4.1)
f(h)=f(-h)=g(h)=g(-h)=O,
V being a constant and h denoting half the gapwidth. Given these conditions, an inspection of the equations suggests that one seek solutions in which u and f are both odd functions of z and o and g are even, this at once requiring that b be zero. In this event, the eqns. (3.7) and (3.8) lead to kf” - mg’ - nf = 0,
kg”+mf’-ng=
where k, m and n are as defined
-27~2,
in (3.12), and the substitutions
g=g+a/2y
f=.A
(4.2)
(4.3)
reduce these to kj” - mg’ - ny= 0,
kg” + rnf’ - ng = 0.
(4.4)
For the reasons given at the end of the previous section, we restrict our attention to the case (i) of that section. Hence using the results (3.20) the relevant solution for the angular velocity components is f = “( A4 cos X,z sinh X,z - N sin X,z cash X,z), 2P
(4.5)
g = a(l 2P
(4.6)
- N cos A,z cash h,z - M sin h,z sinh X,z),
where it4 = sin h,h/sinh
X,h,
N = cos X,h,‘cosh h,h,
and the A,, A, and s are as defined in (3.20). The corresponding components follow from integration of eqns. (3.7), and are
u=az_
(4.7) velocity
cos h,z sinh X,z - N sin X,z cash X,z)
P
-
4P2;+
a((+v) O= - 2&L+7) -
T) (p sin A, z cash X,z - Q cos h,z sinh X,z), (1 _N
(4.8)
cos X,z cash X,z - A4 sin X,z sinh X2z) z cash h,z - cos X,h cash A,h)
+ Q(sin X,z sinh A,z - sin X,h sinh X,h)),
(4.9)
169 with Q = X,M - X,N.
P = X,N + A&f,
(4.10)
In the former the constant of integration is zero since u is an odd function. Setting z equal to h in eqn. (4.8) gives a relationship between the parameters defining the problem F’, h and a in the form V=ah/p--
akA,(cosh2X2h
- cos2 A,h)
(4.11)
4p2(p + T) sinh A,h cash X,h ’
or equivalently V=ah/p(l
(4.12)
-+rH(h)/(p++))
where H(h) = sin’ w(tanh X,h/X,h)(
1 + sin2 X,h/sinh2
h,h),
this using the definitions of X2 and the coefficient respectively). It is relatively easy to show that lim H = 1,
h-0
lim H = 0,
(4.13) n ((3.20)
and (3.12), (4.14)
h-m
and that this factor takes values between these limits for intermediate values. Since /_tand T are both positive, it follows that V and a always have the same sign. If one defines an apparent viscosity n in the usual way by 7) = ah/V,
(4.15)
the result (4.12) yields n = p/(1-
TH(h)/(p
(4.16)
+ T)).
Also the limits (4.14) at once give Ifimon=p+r,
Jimm7)=p
in agreement
with the conclusions
(4.17) of Cowin [3].
5. Plane Poiseuille flow Here we consider the flow produced by a pressure gradient in a given direction parallel to two parallel stationary plates. Partly this is to complete our account of shear flow, but also to show a novel aspect of the solution obtained. Our previous results of course lead one to expect transverse flow, but the fact that a transverse component of shear stress is also induced is perhaps somewhat surprising. With the x-axis parallel to the imposed pressure gradient we therefore
170 examine solutions of eqns. (3.7) and (3.8) when c is non-zero The boundary conditions for this flow are
but d is zero.
U(h)=+h)=v(h)=u(-h)=O, (5.1)
f(h)=f(-h)=g(h)=g(-h)=O,
and these conditions along with an inspection of the equations suggest in this case that one seek solutions with u and f even functions but o and g odd. This entails setting the constant a equal to zero, but since b need not vanish we retain it for the present. From eqns. (3.7) and (3.8) one now obtains kf” - mg’ - nf = 27b - (C#B + [)c, kg”+mf’-ng=
(5.2)
-2wz,
with k, m and n as above. The change of variables f =j+
CA - b/2p,
g=g+cz/2p,
(5.3)
with A=(2&+S)--))/2pn,
(54
yields once more kp
- mg’ - nf= 0,
Again confining quickly give f=(cA-b/2p)(I + @(R 2P g = ch/2p(
-s(cA
kg” + mf’ - ng = 0.
attention
to the case (i) of Section
(5.5)
3, the results
(3.20)
-A’ cos h,z cash A,z - M sin X,z sinh A,z) cos h,z cash h2z - S sin X,z sinh X,z),
(54
- S cos X,z sinh X,z - R sin h,z cash X,z)
z/h
- b/2p)(M
cos X,z sinh X,z - N sin X,z cash X,z),
(5.7)
where M= R=
sin h,h sinh X,h
-,
sin A,h cash X,h
7
Also the velocity given by
N= S=
cos X,h
cash A,h
3
(5 4
cos X,h
sinh X,h *
components
follow from integration
of eqns. (3.7) and are
(~+7)U=(~/2)(z~-h~)-(~+~)f+2rj-~~gdz, (p++=bz-(E+v)g--2$fdz,
(5 -9)
171 the constant of integration being zero in the latter because o is odd. However, the boundary conditions are satisfied provided that one chooses the constant b such that k&(cosh2
X,h - cos2 h,h)
4p(p + T)h sinh X,h cash X,h
X,h - cos2 X,h) 4prh sinh h,h cash X,h
kX,(cosh2
(5.10) Since the coefficient of b is a non-zero multiple of the right-hand side of eqn. (4.1 l), it does not vanish and thus the above condition determines the constant. In plane Poiseuille flow therefore the induced transverse flow gives rise in general to a non-zero transverse component of shear stress. It is of interest to note that this component tends to zero as the gapwidth shrinks to zero, but has a non-zero limit as the gapwidth becomes very large. 6. Concluding remarks As mentioned earlier, the principal prediction of the theory of polar liquids described by Cowin [3] is an increase in effective viscosity near solid surfaces, which would become apparent in viscosity measurements with narrow gaps. In practice attempts to confirm this prediction meet problems in that such measurements are sensitive to surface irregularities and also any slight mis-alignment of the plates. However, the present prediction of transverse flow effects is not so readily masked by these practical difficulties. Clearly, if one reverses the direction of primary flow, the direction of the transverse secondary flow is also changed, this apparently allowing these secondary flows to be distinguished from effects due to the complications cited. Also, and more importantly, these transverse effects do not appear to be restricted to flows with narrow gaps, although they may be small due to the relevant material parameters being small compared with others. In practice, of course, the plates are finite and it is not clear how the channel boundaries affect these predictions. Naturally, one anticipates that their presence simply gives rise to some modification of the flow in their immediate vicinity. ‘Given that the secondary transverse flow is probably small, such edge effects are presumably insignificant. Finally, the question arises as to which liquids are likely to display such effects. Intuitively, one thinks of the rotation of a fluid element being more important when the constituent molecular structure is relatively large, and therefore liquids composed of relatively large molecules appear to be the
172 more probable candidates. However, the weaker material symmetry discussed in this paper presumably eliminates many liquids, and this consideration quickly brings to mind liquid crystals having this weaker symmetry. Consequently, if the theory described in this paper has relevance to real materials, one imagines that it must predict behaviour of cholesteric liquid crystals in their “isotropic” state. Certainly, it would be of interest to investigate shear flow of cholesteric liquid crystals at temperatures above the liquid crystal-isotropic liquid transition. However, the predictions obtained above are not necessarily confined to liquids that exhibit liquid crystalline properties. References 1 2 3 4 5
P. Pieranski F.M. Leslie, S.C. Cowin, F.M. Leslie, C. Fraser, J.
and E. Guyon, Phys. Lett., 49A (1974)237. Adv. Liq. Cryst., 4 (1979) 1. Adv. Appl. Mech., 14 (1974) 279. Arch. Ration. Mech. Anal., 70 (1979) 189. Appl. Math. Phys. (ZAMP), 32 (1981) 695.