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A note on secondary flow of a non-Newtonian fluid in a non-circular pipe
391
Journal ofNon-Ne~~on~n Fluid ikfeehanics, 1 (1976) @ Elsevier Scientific Publishing Company, Amsterdam
391-395 - Printed in The Netherlands
Short Communication A NOTE ON SECONDARY FLOW OF A NON-NEWTONIAN FLUID IN A NON-CIRCULAR PIPE
R.S. RIVLIN Center for the Applicut~on Pa. 18015 (USA) (Received
of mathematics,
Lehigh University,
Rethlehe~,
May 2, 1976)
1. Introduction If an incompressible Newtonian fluid of viscosity+crr flows in a straight pipe of uniform non-circular cross-section under a constant pressure gradient P, then, unless the velocity is so large that the flow becomes turbulent, each fluid particle moves down the tube in a rectilinear path with velocity u3 which depends on its position on the transverse cross-section and is given by the solution of the equation:
v2v, =$
)
(1)
where V2 is the two-dimensional Laplace operator in the transverse planes, with u3 = 0 at the wall of the pipe. It was shown by Ericksen [l] that if we attempt to obtain a solution to this problem for an incompressible non-Ne~on~n fluid and assume that the fluid particles move in rectilinear paths along the pipe, then the equations of motion are, in general, incompatible. Langlois and Rivhn [Z] showed that for second-order and third-order Rivlin-Ericksen fluids, solutions are possible for which the particle velocities are longitudinal. However, if the fourth-order constitutive equation is valid for the fluid, a steady flow in transverse planes is superposed on the longitudinal flow. In ref. 2 the stream-line pattern for this flow was determined in the case when the cross-section of the pipe is elliptical. It was found that the physical constants in the fourth-order constitutive equation enter into the expression for the stream-function through a single combination, l? say. Ballal and Rivlin [3,4] have analyzed the flow of a non-Newtonian fluid in the annular region between two infinitely long circular cylinders with parallel axes. In [3 J they considered the case when the flow is produced by a uniform pressure gradient and in [4] they considered the case when it is produced by
392
relative motion of the cylinders, with constant velocity parallel to their lengths. In both cases, it was found that, with the third-order constitutive equation, the velocity at each point of the fluid is parallel to the lengths of the cylinders. However, if the fourth-order constitutive equation is used, a steady flow in transverse planes is superposed on this longitudinal flow. In both cases, the physical constants in the fourth-order constitutive equation enter into the expressions for the stream-function through the same single combination F, as okcurs in the stream-function for the transverse flow in a pipe with elliptical cross-section. In this short communication, we consider an infinitely long cylinder of uniform cross-section, which may possibly contain one or more infinitely long cylinders, with uniform cross-sections and with their axes parallel to that of the outer cylinder. The fluid is contained between the outer cylinder and the inner cylinders and flow is produced in it by a uniform longitudinal pressure gradient, or by longitudinal motions with constant velocities of one or more of the cylinders, or by both. It is shown that the transverse secondary flow predicted by the fourthorder Rivlin-Ericksen constitutive equation involves only the single combination I’ of the physical constants for the fluid. 2. Basic equations We consider slow steady flows in an incompressible viscoelastic fluid for which the velocity EP has components euI in the rectangular Cartesian reference system x. The Rivlin-Ericksen tensors PA, (e = 1, 2, . ..) have components e”A?’11 in the system x given by A;? = $(ui j + u.I.1.) 9
LI;~+~’ = ukA,fja; + uk jA;;'
(2) +
u,,,A$
Let Uij be the components in the system x of the Cauchy stress 9. We assume that the velocity eg is sufficiently slow so that the fourth-order Rivlin-Ericksen constitutive equation is valid. We then have [ 21 (3) where
sl = %Al,
s,=a,A_2+~3&,
4 is the unit cartesian tensor and p is an arbitrary hydrostatic pressure.
393
3. Proof of the theorem We consider the mass of fluid to be of infinite extent in the direction of the x,-axis and to be bounded in the x,x,-planes by one or more infinitely long cylindrical surfaces with uniform cross-sections and their lengths parallel to the xs-axis. We suppose that the flow is produced in the fluid by a uniform pressure gradient parallel to the xs-axis, or by the motion of one or more of the cylinders with uniform velocities parallel to the xs-axis, or by a combination of these. The velocity field CZJ must satisfy no-slip boundary conditions on the cylindrical surfaces, so that on them Ui
(5)
=Om
Throughout the fluid
We now write: -v = -v(l)
_ + ev’“’ -4 &J(3) __ + &P),
p = &,
(7)
where e?(l) is the flow-field if the fluid is Newtonian and has viscosity.$, (i.e. if it satisfies the first-order constitu~ive equation _a= e& - p& ), e2zj2) is the correction to this obtained by assuming that the second-order constitutive equation 9 = es, + e2+S2-p& is satisfied, and so on. It follows from the analysis carried out in [2-41 that w="g) ua
= uk3'=
()
(0 = 1, a,
up =us”’ = w(x1,x2) YS3) = up =W(q,
)
x2
(8)
say, say,
where uL*)(CX= 1,2) is derivable from a stream-function ~!J(x~,x~). w satisfies eqn. (1) with the boundary condition (5). W satisfies the equation
+.Y,V% + (& + 03) g I + gJw,J i
+ 33w.2
)=
0,
where
I = (w.1j2+ (w,212,
(10)
and W = 0 on the cylindrical boundaries. u&*)is given in terms of the streamfunction $ by (4) = $/2 , Ul
d4' = -3L.1,
where # is given by the equation
(11)
394
;c.Qv*J/ + (2&J ++Y,)cp + Y@ = 0,
(12)
with cp = _?Y-
(W,,W,J - Q?Q?)
@= j&
~wQ2
axlax
-
+ $(T$
@,212
-
II + (2
2
2)
- 2)
kl~,2
+ w.2~.1),
UW,lW,2),
(13)
1
‘Y=5Y4+4Y5+4^13+2Yd, vi*’ = ~$4) = 0 on the bounding cylinders. From eqns. (l), (7) and (8) we have
and
v2w SE
c72wJ = v2w,2 = 0.
,
a1
(14)
Using eqns. (14) we obtain from the first of eqns. (13) @ = +$72iC2
- W,2V2izJ).
(15)
We now use eqns. (14) to substitute for G/e1 in eqn. (9) and obtain ol,v21i7 + (01 + &)A = 0,
(16)
where A = IV2W + 1.1 w.1 + I.2w.2.
(17)
From eqns. (15) and (16), we have @=
Pl +P3 -~(w,lA,2
(18)
--,2fyl).
Using the expression (10) for I and eqn. (14), we obtain, after considerable algebraic manipulation, cp = _(Pl
+ P3!, a1
(19)
’
where a =E
{[(w.1)2 - (7q2)21
+ w.20$2
-
w.12
--w,1y2(w,11
3$1)w,111 - w,,($l
-%?2))
- 3W22N.222-
+
(20)
395
Again, using eqns. (10) and (14) in the second of eqns. (13), we obtain, after lengthy algebraic manipulation, 0=2a.
(21)
Introducing eqns. (20) and (21) into eqn. (12), we have VQ = 2r52,
(22)
where (23) Since the boundary conditions envisaged are independent of the physical constants occurring in the constitutive equation, it is evident that the latter enter into the determination of the secondary flow in the transverse planes only through the single constant r. If G = 0, we find, with eqns. (14), that eqn. (20) yields s2=
w,2($2
-
3w~i)W,rii - w.1 (W?r - 3WT2M.222-
(24)
This result can also be readily obtained from eqn. (3.16) of [4]. Conclusion A non-Newtonian fluid contained in an infinitely long uniform pipe of noncircular crosssection, or in the multiply-connected region formed by a number of infinitely long pipes with parallel axes, and flowing under a uniform longitudinal pressure gradient, or as a result of the relative longitudinal motion with uniform velocity of the pipes, or as a result of a combination of these, has been considered. It has been shown that if the flow is such that the fourth-order Rivlin-Ericksen constitutive equation is valid, then the secondary flow in transverse planes involves only a single combination of the material constants occurring in this equation. Acknowledgement This work was carried out with the support of a grant from the National Science Foundation to Lehigh University. References 1 2 3 4
J.L. Ericksen, Q. Appl. Math., 14 (1956) 299. W.E. Langlois and R.S. Rivlin, Rend. Mat., 22 (1963) 169. B.Y. Ballal and R.S. Rivlin, pending publication. B.Y. Ballal and R.S. Rivlin, Rheol. Acta, 14 (1975) 861.
Journal ofNon-Ne~~on~n Fluid ikfeehanics, 1 (1976) @ Elsevier Scientific Publishing Company, Amsterdam
391-395 - Printed in The Netherlands
Short Communication A NOTE ON SECONDARY FLOW OF A NON-NEWTONIAN FLUID IN A NON-CIRCULAR PIPE
R.S. RIVLIN Center for the Applicut~on Pa. 18015 (USA) (Received
of mathematics,
Lehigh University,
Rethlehe~,
May 2, 1976)
1. Introduction If an incompressible Newtonian fluid of viscosity+crr flows in a straight pipe of uniform non-circular cross-section under a constant pressure gradient P, then, unless the velocity is so large that the flow becomes turbulent, each fluid particle moves down the tube in a rectilinear path with velocity u3 which depends on its position on the transverse cross-section and is given by the solution of the equation:
v2v, =$
)
(1)
where V2 is the two-dimensional Laplace operator in the transverse planes, with u3 = 0 at the wall of the pipe. It was shown by Ericksen [l] that if we attempt to obtain a solution to this problem for an incompressible non-Ne~on~n fluid and assume that the fluid particles move in rectilinear paths along the pipe, then the equations of motion are, in general, incompatible. Langlois and Rivhn [Z] showed that for second-order and third-order Rivlin-Ericksen fluids, solutions are possible for which the particle velocities are longitudinal. However, if the fourth-order constitutive equation is valid for the fluid, a steady flow in transverse planes is superposed on the longitudinal flow. In ref. 2 the stream-line pattern for this flow was determined in the case when the cross-section of the pipe is elliptical. It was found that the physical constants in the fourth-order constitutive equation enter into the expression for the stream-function through a single combination, l? say. Ballal and Rivlin [3,4] have analyzed the flow of a non-Newtonian fluid in the annular region between two infinitely long circular cylinders with parallel axes. In [3 J they considered the case when the flow is produced by a uniform pressure gradient and in [4] they considered the case when it is produced by
392
relative motion of the cylinders, with constant velocity parallel to their lengths. In both cases, it was found that, with the third-order constitutive equation, the velocity at each point of the fluid is parallel to the lengths of the cylinders. However, if the fourth-order constitutive equation is used, a steady flow in transverse planes is superposed on this longitudinal flow. In both cases, the physical constants in the fourth-order constitutive equation enter into the expressions for the stream-function through the same single combination F, as okcurs in the stream-function for the transverse flow in a pipe with elliptical cross-section. In this short communication, we consider an infinitely long cylinder of uniform cross-section, which may possibly contain one or more infinitely long cylinders, with uniform cross-sections and with their axes parallel to that of the outer cylinder. The fluid is contained between the outer cylinder and the inner cylinders and flow is produced in it by a uniform longitudinal pressure gradient, or by longitudinal motions with constant velocities of one or more of the cylinders, or by both. It is shown that the transverse secondary flow predicted by the fourthorder Rivlin-Ericksen constitutive equation involves only the single combination I’ of the physical constants for the fluid. 2. Basic equations We consider slow steady flows in an incompressible viscoelastic fluid for which the velocity EP has components euI in the rectangular Cartesian reference system x. The Rivlin-Ericksen tensors PA, (e = 1, 2, . ..) have components e”A?’11 in the system x given by A;? = $(ui j + u.I.1.) 9
LI;~+~’ = ukA,fja; + uk jA;;'
(2) +
u,,,A$
Let Uij be the components in the system x of the Cauchy stress 9. We assume that the velocity eg is sufficiently slow so that the fourth-order Rivlin-Ericksen constitutive equation is valid. We then have [ 21 (3) where
sl = %Al,
s,=a,A_2+~3&,
4 is the unit cartesian tensor and p is an arbitrary hydrostatic pressure.
393
3. Proof of the theorem We consider the mass of fluid to be of infinite extent in the direction of the x,-axis and to be bounded in the x,x,-planes by one or more infinitely long cylindrical surfaces with uniform cross-sections and their lengths parallel to the xs-axis. We suppose that the flow is produced in the fluid by a uniform pressure gradient parallel to the xs-axis, or by the motion of one or more of the cylinders with uniform velocities parallel to the xs-axis, or by a combination of these. The velocity field CZJ must satisfy no-slip boundary conditions on the cylindrical surfaces, so that on them Ui
(5)
=Om
Throughout the fluid
We now write: -v = -v(l)
_ + ev’“’ -4 &J(3) __ + &P),
p = &,
(7)
where e?(l) is the flow-field if the fluid is Newtonian and has viscosity.$, (i.e. if it satisfies the first-order constitu~ive equation _a= e& - p& ), e2zj2) is the correction to this obtained by assuming that the second-order constitutive equation 9 = es, + e2+S2-p& is satisfied, and so on. It follows from the analysis carried out in [2-41 that w="g) ua
= uk3'=
()
(0 = 1, a,
up =us”’ = w(x1,x2) YS3) = up =W(q,
)
x2
(8)
say, say,
where uL*)(CX= 1,2) is derivable from a stream-function ~!J(x~,x~). w satisfies eqn. (1) with the boundary condition (5). W satisfies the equation
+.Y,V% + (& + 03) g I + gJw,J i
+ 33w.2
)=
0,
where
I = (w.1j2+ (w,212,
(10)
and W = 0 on the cylindrical boundaries. u&*)is given in terms of the streamfunction $ by (4) = $/2 , Ul
d4' = -3L.1,
where # is given by the equation
(11)
394
;c.Qv*J/ + (2&J ++Y,)cp + Y@ = 0,
(12)
with cp = _?Y-
(W,,W,J - Q?Q?)
@= j&
~wQ2
axlax
-
+ $(T$
@,212
-
II + (2
2
2)
- 2)
kl~,2
+ w.2~.1),
UW,lW,2),
(13)
1
‘Y=5Y4+4Y5+4^13+2Yd, vi*’ = ~$4) = 0 on the bounding cylinders. From eqns. (l), (7) and (8) we have
and
v2w SE
c72wJ = v2w,2 = 0.
,
a1
(14)
Using eqns. (14) we obtain from the first of eqns. (13) @ = +$72iC2
- W,2V2izJ).
(15)
We now use eqns. (14) to substitute for G/e1 in eqn. (9) and obtain ol,v21i7 + (01 + &)A = 0,
(16)
where A = IV2W + 1.1 w.1 + I.2w.2.
(17)
From eqns. (15) and (16), we have @=
Pl +P3 -~(w,lA,2
(18)
--,2fyl).
Using the expression (10) for I and eqn. (14), we obtain, after considerable algebraic manipulation, cp = _(Pl
+ P3!, a1
(19)
’
where a =E
{[(w.1)2 - (7q2)21
+ w.20$2
-
w.12
--w,1y2(w,11
3$1)w,111 - w,,($l
-%?2))
- 3W22N.222-
+
(20)
395
Again, using eqns. (10) and (14) in the second of eqns. (13), we obtain, after lengthy algebraic manipulation, 0=2a.
(21)
Introducing eqns. (20) and (21) into eqn. (12), we have VQ = 2r52,
(22)
where (23) Since the boundary conditions envisaged are independent of the physical constants occurring in the constitutive equation, it is evident that the latter enter into the determination of the secondary flow in the transverse planes only through the single constant r. If G = 0, we find, with eqns. (14), that eqn. (20) yields s2=
w,2($2
-
3w~i)W,rii - w.1 (W?r - 3WT2M.222-
(24)
This result can also be readily obtained from eqn. (3.16) of [4]. Conclusion A non-Newtonian fluid contained in an infinitely long uniform pipe of noncircular crosssection, or in the multiply-connected region formed by a number of infinitely long pipes with parallel axes, and flowing under a uniform longitudinal pressure gradient, or as a result of the relative longitudinal motion with uniform velocity of the pipes, or as a result of a combination of these, has been considered. It has been shown that if the flow is such that the fourth-order Rivlin-Ericksen constitutive equation is valid, then the secondary flow in transverse planes involves only a single combination of the material constants occurring in this equation. Acknowledgement This work was carried out with the support of a grant from the National Science Foundation to Lehigh University. References 1 2 3 4
J.L. Ericksen, Q. Appl. Math., 14 (1956) 299. W.E. Langlois and R.S. Rivlin, Rend. Mat., 22 (1963) 169. B.Y. Ballal and R.S. Rivlin, pending publication. B.Y. Ballal and R.S. Rivlin, Rheol. Acta, 14 (1975) 861.