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Saddle connections near degenerate critical points in Stokes flow within cavities
Applied Mathematics and Computation 172 (2006) 1133–1144
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Saddle connections near degenerate critical points in Stokes flow within cavities F. Gu¨rcan *, A. Deliceog˘lu Department of Mathematics, Erciyes University, Kayseri 38039, Turkey
Abstract Streamline patterns and their bifurcations in two-dimensional incompressible flow near a simple degenerate critical point away from boundaries are investigated by using the normal form theory for the streamfunction obtained by [M. Brøns, J.N. Hartnack, Streamline topologies near simple degenerate critical points in two-dimensional flow away from boundaries, Phys. Fluids 11 (1999) 314–324]. For the normal form of order six, a bifurcation diagram is constructed with two bifurcation parameters. The theory is applied to the patterns and bifurcations found numerically in the studies of Stokes flow in a double-lid-driven rectangular cavity with two control parameters (the cavity aspect ratio A (height to width), and the speed ratio S). Bifurcations in the cavity are obtained using an analytic solution for the streamfunction developed for any value of S and A. Using this solution for special values (S, A) a global bifurcation is identified with a single heteroclinic connection which connects three saddle points in a triangle and does not appear in the unfolding of the simple linear degenerate critical points lying in a line y = constant. 2005 Elsevier Inc. All rights reserved. Keywords: Bifurcations; Normal forms; Eigenfunction solution; Biorthogonal series; Stokes flow
*
Corresponding author. E-mail address: [email protected] (F. Gu¨rcan).
0096-3003/$ - see front matter 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2005.03.012
1134
F. G€ urcan, A. Deliceog˘lu / Appl. Math. Comput. 172 (2006) 1133–1144
1. Introduction The main objective of this article is to investigate streamline patterns and their bifurcations of Stokes flow in a cavity with two moving lids and two rigid walls both using a method from non-linear dynamical systems (as in [1]) and an analytic solution. The idea of the qualitative study of differential equations described streamline patterns has been extensively used, for example, see [2–6] and the bifurcation in the fluid flow where the topological structure of the flow field may change qualitatively as a parameter passes through a critical value, has been recently investigated in a number of articles, see [1,7–9]. Brøns et al. [10,11] used an idea of a combination of bifurcation theory for two-dimensional dynamical systems and numerical simulations. They considered the streamlines as trajectories for the velocity field in a cylindrical container with rotating end-covers and applied methods from bifurcation theory to describe the qualitative changes that are possible. Gu¨rcan [12,13] considered the cavity problem to investigate flow patterns for varying A (the cavity aspect ratio (height to width)). He used both an analytic solution for the streamfunction and methods from non-linear dynamics for the velocity field and showed that bifurcations occur at stagnation points away from boundaries and near the boundary. In the present study, using the streamfunction for two-dimensional incompressible flow obtained from the normal form theory [1,7] we construct a bifurcation diagram with the normal form of order up to six for the steady incompressible flow which is symmetric about the x axis. As an application of this theoretical framework, a similar bifurcation diagram with two control parameters is produced for Stokes flow in a double-lid-driven rectangular cavity by using an analytical solution for the streamfunction developed for any value of S and A where S is the speed ratio of the lids. To determine the existence of simple degenerate critical points in the cavity we use Taylor expansion of a velocity vector field near the singular point of the system of differential equations for the streamlines. The cavity flow is studied both analytically and numerically by a number of authors (see [14–17] and the review paper by Shankar and Deshpande [18]). It is the first formulated and solved analytically by Joseph and Sturges [19] for a single lid moving. They showed the presence of four eddies with corner eddies for the cavity aspect ratio A = 5. Subsequently, Sturges [20] considered the symmetric flow with the lids moving in opposite directions (i.e., S = U1/ U2 = 1 where U1 and U2 are the upper and lower lid velocities, respectively). For four values of aspect ratio he showed four distinct flow structures which revealed the presence of side eddies attached to the stationary side walls. For any value of S an analytic solution for the streamfunction expressed as an infinite series of Papkovich–Faddle eigenfunctions is developed and used to con-
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struct a (S, A) control space diagram which is similar to that obtained by using the normal form theory. In addition, using the analytic solution more complex flow patterns are obtained and these are not captured by the normal form coordinates used by [1]. 2. Bifurcations near simple degenerate critical points An incompressible two-dimensional flow far from any boundaries is considered. A streamfunction, w, exists such that the streamlines are found from ow ow x_ ¼ u ¼ and y_ ¼ v ¼ . oy ox Following Brøns and Hartnack [1] we will determine local flow structures under the assumption of the reflection symmetry about x = 0 and the streamfunction is then an even function of x. The expansion of w about a given critical point which is taken to be the origin, becomes 1 X w¼ c2i;j x2i y j . ð1Þ iþj¼1
The linear approximation of the equations for the streamlines is 0 2c0;2 x_ c0;1 x þ ¼ . 2c2;0 0 0 y_ y
ð2Þ
If c0,1 = 0, the origin is a critical point and the local flow pattern is determined using the standard theory for Hamiltonian systems. If jJj = 4c0,2c2,0 = 0 (considering jJj is not the zero matrix) the critical point is degenerate. This case refers as a simple linear degeneracy. Thus higher-order terms become decisive for the streamline pattern. For a significant simplification of higher-order terms in the streamfunction and an easy determination of simple degenerate patterns and their bifurcations, Brøns and Hartnack [1] used normal form transformations found via a generating function. Their normal form procedure gives rise to the following theorem. Theorem. Let c0,1, c2,0 and c4;0 , c6;0 ; . . . ; c2½N=22;0 be small parameters. Assuming the non-degeneracy conditions c0,2 5 0, c2½N =2;0 6¼ 0 a normal form of order N for the streamfunction (1) is r wN ¼ y 2 þ f ðxÞ; ð3Þ 2 where f ðxÞ ¼
½N =2 X i¼1
c2i x2i ;
c2½N =2 ¼
1 ; 2½N =2
F. G€ urcan, A. Deliceog˘lu / Appl. Math. Comput. 172 (2006) 1133–1144
1136 c
r ¼ c2½N0;2=2;0 , and c2i,i = 1, 2, . . . , [N/2] are transformed small parameters, and c2k;0 is c2k,0 plus non-linear combinations of c2i,j where 2i + j < 2k, see [1]. Using the normal form for w, (3), the corresponding dynamical systems x_ ¼ ry;
y_ ¼ f 0 ðxÞ
ð4Þ
can be analysed. The origin is always a critical point and the total number of critical points is odd. All the critical points lie on the x axis. In addition to the local bifurcation occurring when f 0 (x) = 0 and f00 (x) = 0, a global bifurcation takes place when the streamfunction attains the same value at two saddle points. We will focus on the normal form of order six (N = 6) to obtain a bifurcation diagram with two control parameters. It is, however, presented flow patterns for the normal form of order four. For N = 4 the streamfunction is r 1 w 4 ¼ y 2 þ c2 x2 þ x4 . 2 4 The condition for bifurcations is f 0 ðxÞ ¼ 2c2 x þ x3 ¼ 0;
ð5Þ
f 00 ðxÞ ¼ 0;
ð6Þ
it is clear that for c2 P 0 there is a single critical point which is the origin. When r > 0 it is a centre and a saddle point pffiffiffiffiffiffiffiffiffiffi ffi when r < 0. For c2 < 0 there are three critical points (x1 ¼ 0; x2;3 ¼ 2c2 Þ. Their types depend on r. A simple calculation shows that f 0 (x2) = f 0 (x3) and f(x2) = f(x3) so a heteroclinic trajectory occurs for r < 0, and for r > 0 the possible flow pattern is a separatrix. 2.1. Normal form order 6 For N = 6 the streamfunction is r 1 w 6 ¼ y 2 þ c2 x2 þ c4 x4 þ x6 . 2 6
ð7Þ
The condition for bifurcations is ð8Þ f 0 ðxÞ ¼ 2c2 x þ 4c4 x3 þ x5 ¼ 0; f 00 ðxÞ ¼ 2c2 þ 12c4 x2 þ 5x4 ¼ 0. pffiffiffiffiffiffiffiffiffiffiffi By eliminating x = 0 and 2c4 we find that in the (c2, c4) parameter plane, bifurcations occur on c4 axis and on the curve c2 2c24 ¼ 0 for c4 6 0. The (c2, c4) parameter plane can be divided into the following subregion: (1) When 4c24 2c2 P 0, the system (4) has five critical points, by computing the Jacobian of every critical points, we have, C 1 ð0; 0Þ;
C2
! rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2c4 þ
4c24 2c2 ; 0
and
! rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
C 3 2c4 þ
4c24 2c2 ; 0
F. G€ urcan, A. Deliceog˘lu / Appl. Math. Comput. 172 (2006) 1133–1144
1137
are centers, while other singular points such as ! ! rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi S4 2c4 4c24 2c2 ; 0 and S 5 2c4 4c24 2c2 ; 0 are saddle points. Since f(S4) and f(S5) have the same value at the critical points, there is a heteroclinic bifurcation at this region. (2) When 4c24 2c2 ¼ 0, the system (4) has three critical points, these are pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi C 1 ð0; 0Þ; C 2 ¼ S 4 ¼ A1 2c4 ; 0 and C 3 ¼ S 5 ¼ A2 2c4 ; 0 . Since f 00 (A1) = f 00 (A2) = 0, A1 and A2 are the degenerate critical points. Also this gives us that 4c24 2c2 ¼ 0 is a bifurcation curve. To determine the topological type at the degenerate critical points, we expand the streamfunction at that point, we have f 000 5 0, showing that they are cusps. (3) When 4c24 2c2 < 0, the system has one critical points C1(0, 0) and it is saddle point. (4) When c2 =p 0,ffiffiffiffiffiffiffiffiffiffi there are three pffiffiffiffiffiffiffiffiffiffiffi critical points, which are ffi C 1 ð0; 0Þ; D2 ð 4c4 ; 0Þ and D3 ð 4c4 ; 0Þ. The critical points D2 and D3 are center, but C1 is a degenerate critical points. As in the case 2) we expand streamfunction at the point C1, we get f (4) = 24c4. If c4 < 0, the degenerate critical point is a center. If c4 < 0, the degenerate critical point is saddle. The complete bifurcation diagram is shown in Fig. 1a.
(a)
(b)
Fig. 1. Bifurcation diagrams obtained using: (a) methods from non-linear dynamics and (b) an analytic solution.
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3. Boundary value problem A rectangular cavity is considered, of depth 2H and width 2L with stationary side walls lying along X = ±L and lids lying along Y = ±H and moving in both directions with speeds U1 and U2. Speeds are non-dimensionalised with respect to U2 and lengths with respect to L such that the flow domain is given by jxj 6 1; jyj 6 A. For the two-dimensional Stokes flow of an incompressible fluid the velocity components are expressed in terms of streamfunction, w, which is the solution of the biharmonic equation r4 w ¼ 0.
ð9Þ
The boundary value problem is shown in Fig. 2 where the boundary conditions include: (i) w = constant = 0 on all boundaries wð1; yÞ ¼ wðx; AÞ ¼ 0;
ð10Þ
(ii) no-slip on the upper and lower lids and on the side-walls ow ðx; AÞ ¼ S; oy
ow ðx; AÞ ¼ 1 and oy
ow ow ð1; yÞ ¼ ð1; yÞ ¼ 0. ox ox
The streamfunction for any value of S can be written as 1 X An esn ðyAÞ þ Bn esn ðyþAÞ n wðx; yÞ ¼ /1 ðx; sn Þ . s2n n¼1
ð11Þ
ð12Þ
The functions /n1 ðx; sn Þ are even Papkovich–Faddle eigenfunctions (see [19,20]) /n1 ¼ sn sin sn cos sn x sn x cos sn sin sn x
ð13Þ
and sn are complex eigenvalues determined by the side wall conditions w = ow/ ox = 0 at x = ±1, which yield the eigenvalue equation sin 2sn ¼ 2sn . The sn are easily determined via a Newton iteration procedure as described by Robbins and Smith [21] using an initial estimate of the form
Fig. 2. Dimensionless boundary value problem and attendant boundary conditions.
F. G€ urcan, A. Deliceog˘lu / Appl. Math. Comput. 172 (2006) 1133–1144
2sn ð2n 1.5Þp þ i logð4n 1Þp.
1139
ð14Þ
From sin 2sn ¼ 2sn it follows that sn ¼ sn where the overbar denotes complex n conjugate and this, in turn, yields /n 1 ðxÞ ¼ /1 ðxÞ. Since the streamfunction is n ; Bn ¼ B n . To determine the coefficients (An real, it follows that An ¼ A and Bn) we employs SmithÕs [22] biorthogonality relation technique. This technique is the first used by Joseph and Sturges [19] to obtain a solution for a cavity with a single moving lid. However, it is simple matter to modify the technique in order to account for the movement of the other lid. Considering boundary conditions and employing SmithÕs relation (see [23] for more detail) yields 1 X ð1 sn Þ ð1 þ sn Þ ðAl þ Bl e2sl A ÞK l þ An Bn e2sn A M l;n ¼ 4; sn sn n¼1 ð15Þ ðAl e
2sl A
þ Bl ÞK l þ
1 X
An e
2sn A
n¼1
ð1 sn Þ ð1 þ sn Þ Bn M l;n ¼ 4S; sn sn ð16Þ
where Kl ¼
Z
þ1 1
½wl2 /n1 þ ð2wl2 wl1 Þ/n2 dx
and M l;n ¼
Z
þ1
1
wl2 /n1 dx.
Consequently, the right hand side of Eq. (16) becomes equal to 4S, that is, this corresponds the second (lower) lid speed. Thus this change results in a solution giving by Eq. (12) which is valid for any value of S when the coefficients are determined. Eqs. (15) and (16) form an infinite set to be solved for the unknown coefficients An and Bn n = ±1, ±2, . . . which, in practice, can be determined only by truncating after N terms. When the coefficients have been determined, the streamfunction at any interior point in the liquid is obtained by simply summing a finite number of terms in the series (12). In the present study, we take N = 100. Before analysing flow patterns, we investigate whether the solution (12) imposes any restrictions on steady streamline patterns and their bifurcations in the general symmetrical case which is based on only the existence of the streamfunction (1). For this purpose, we consider a Taylor expansion of (12) about a critical point which is given by (1) with each coefficient c2i,j expressed as an
F. G€ urcan, A. Deliceog˘lu / Appl. Math. Comput. 172 (2006) 1133–1144
1140
infinite series depending on An, Bn, sn. Although the c2i,j can easily be calculated for a given critical point, it is not analytically possible to see any constraints on the c2i,j (since An, Bn cannot be solved analytically). More generally, to determine any restrictions for the c2i,j, the expression (1) is inserted into the biharmonic Eq. (9) and collecting terms yields a series of algebraic equations for the c2i,j. The equations of order zero and two are given by x0 y 0 :
c2;2 þ 3c4;0 þ 3c0;4 ¼ 0;
x0 y 2 :
c4;2 þ 2c2;4 þ 15c0;6 ¼ 0;
2 0
c2;4 þ 2c4;2 þ 15c6;0 ¼ 0.
xy :
ð17Þ
The non-zero equations of order k constitute an undetermined homogeneous linear system and there is no information about coefficients of order less than four. Since in our study we only consider the streamfunction of the normal form of order N = 6, coefficient of order more then four cannot be appear, see (7). We conclude that all topologies and bifurcations described in Section 2 can occur in steady Stokes flow. 3.1. Flow patterns One aim is to track the flow bifurcations arising in the cavity as A is gradually increased for speed ratio S = 8 · 104. Fig. 3a (where A = 1.75) shows a separatrix with two sub-eddies, an outer circulation and the side eddies attached to the stationary side walls near the upper lid. As A is increased to A = 1.767 for S = 8 · 104, a simple calculation shows that the coefficient, X sn sin sn þ cos sn
sn ðy 0 AÞ sn ðy 0 þAÞ c2;0 ¼ Bn e An e ; 2sn obtained by the expansion of (12) at a critical point, (x, y) = (0, y0 ’ 1.628), determined by the bisection method when u = v = 0, is vanished. Thus (0, y0 ’ 1.628) is a simple linear degenerate critical point at which (from Section 2.1) o3 w ð0; y 0 Þ 6¼ 0; ox3 and so two saddle-node (cusp) bifurcations arise within the upper sub-eddy, see Fig. 3b where A = 1.78. In Fig. 3b, there is a separatrix with two saddle points and three centers lying on the line almost y = 1.63. As A continues to increase, the two saddle points (corresponding to saddlenode bifurcations) approach each other and coalesce to the centre on x = 0 to produce a saddle point at A = 1.785 (see Fig. 3c where A = 1.79). Streamline patterns about the critical points in Fig. 3a–c correspond to the patterns in regions I, II and III of Fig. 1a, respectively. Hence a similar diagram to Fig. 1a is
F. G€ urcan, A. Deliceog˘lu / Appl. Math. Comput. 172 (2006) 1133–1144
1141
(b)
(a)
(c)
(d) 4
Fig. 3. Streamlines for S = 8 · 10 (a) A = 1.75, (b) A = 1.78, (c) A = 1.79, and for S = 4 · 104 (d) A = 1.70. In (b), (c) and (d) the upper part of the cavity is illustrated.
expected to be present for the cavity flow with two control parameters. Indeed, for 1.75 < A < 1.80 and 1 · 102 < S < 1 · 104, we obtain a (S, A) control space diagram, Fig. 1b, exhibiting bifurcation curves. The type and form of the bifurcations in (S, A) space are exactly the same as those obtained via the theoretical framework, Fig. 1a. Furthermore, for a special value (A = 1.7,S = 4 · 104) a global bifurcation is identified with a heteroclinic connection which connects three saddle points, see Fig. 3d where the saddles are not located on a line. The configurations with critical points in a triangle are not captured by the streamfunction (3) which has the unfolding of a simple linear degenerate in which all critical points lie on a line in the normal form coordinates. Using the analytic solution (12), investigations show that once again a heteroclinic connection appear in the cavity when the cavity aspect ratio A becomes A = 4.623 for S = 0.0031, see Fig. 4e. Fig. 4 shows a sequence of flow patterns as A is increased from four and many bifurcations occur in the central section of the cavity. The first three bifurcations correspond to the appearance of side eddies, a saddle node bifurcation near each side wall and
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(d)
(b)
(a)
(c)
(e)
(f)
(g)
Fig. 4. Flow structure development in the central section of cavity as A increases from 4 for S = 0.0031; (a) A = 4, (b) A = 4.55, (c) A = 4.6, (d) A = 4.6225, (e) A = 4.6236, (f) A = 4.627 and (g) A = 4.678.
F. G€ urcan, A. Deliceog˘lu / Appl. Math. Comput. 172 (2006) 1133–1144
1143
thirdly a saddle node bifurcation on x = 0, see Fig. 4a–d. As the aspect ratio A is reached to A = 4.623, Fig. 4e the three saddles become connected by a single streamline. As A is increased gradually the heteroclinic connection transforms into two separatrices with separatrix Sp2 enclosed by Sp1, Fig. 4f. When A = 4.645 the two saddle points of Sp2 coalesce at the centre on x = 0 to produce a saddle, Fig. 4g for A = 4.678. These patterns show that flow structure development for any values of A and S would be more intricate. The dynamics near simple and non-simple degenerate critical points of Stokes flow in a cavity with the two control parameters are under an in-depth investigation.
References [1] M. Brøns, J.N. Hartnack, Streamline topologies near simple degenerate critical points in twodimensional flow away from boundaries, Phys. Fluids 11 (1999) 314–324. [2] J.C.R. Hunt, C.J. Abel, H. Woo, Kinematical studies of the flow around free or surfacemounted obstacles; applying topology of flow visualization, J. Fluid Mech. 86 (1978) 179–200. [3] M. Tobak, D.J. Peake, Topology of three-dimensional separated flows, Ann. Rev. Fluid Mech. 14 (1982) 61–85. [4] A.E. Perry, M.S. Chong, A description of eddying motions and flow patterns using criticalpoint concepts, Ann. Rev. Fluid Mech. 19 (1987) 125–137. [5] P.G. Bakker, Bifurcation in flow patterns, Thesis, Technical University of Delft, Netherlands, 1989. [6] U. Dallmann, Three-dimensional vortex structures and vorticity topology, Fluid Dyn. Res. 3 (1988) 183. [7] J.N. Hartnack, Streamlines topologies near a stationary wall using normal forms, Acta Mech. 136 (1999) 55–75. [8] T. Ma, S. Wang, Topology of 2D incompressible flows and applications to geophysical fluid dynamics, RACSAM, Rev. R. Acad. Cien. Serie A. Mater. 96 (3) (2002) 447–459. [9] M. Ghil, T. Ma, S. Wang, Structural bifurcation of 2D incompressible flows, Indiana Univ. Math. J. (Special issue) 50 (1) (2001) 159–180. [10] M. Brøns, L.K. Voigt, J.N. Sørensen, Streamline topology of steady axisymmetric vortex breakdown in a cylinder with co- and counter-rotating end-covers, J. Fluid Mech. 401 (1999) 275–292. [11] M. Brøns, L.K. Voigt, J.N. Sørensen, Topology of vortex breakdown bubbles in a cylinder with a rotating bottom and a free surface, J. Fluid Mech. 428 (2001) 133–148. [12] F. Gu¨rcan, Streamline topologies in Stokes flow within lid-driven cavities, Theoret. Comput. Fluid Dyn. 17 (1) (2003) 19–30. [13] F. Gu¨rcan, Streamline topologies near a stationary wall of Stokes flow in a cavity, Appl. Math. Comput. 165 (2) (2005) 329–345. [14] O. Burggraf, Analytical and numerical studies of the structure of steady separated flows, J. Fluid Mech. 24 (1966) 113–151. [15] T.B. Gatski, C.E. Grosch, M.E. Rose, A numerical study of the two dimensional Navier– Stokes equation in vorticity-velocity variables, J. Comput. Phys. 48 (1982) 1–22. [16] P.H. Gaskell, A.K.C. Lau, N.G. Wright, Comparison of two solution strategies for use with higher-order discretization schemes in fluid flow simulation, Int. J. Numer. Meth. Eng. 8 (1988) 1203–1215.
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[17] P.H. Gaskell, M.D. Savage, J.L. Summers, H.M. Thompson, Stokes flow in closed rectangular domains, Appl. Math. Modell. 22 (1998) 727–743. [18] P.N. Shankar, M.D. Deshpande, Fluid mechanics in the driven cavity, Annu. Rev. Fluid Mech. 32 (2000) 93–136. [19] D.D. Joseph, L. Sturges, The convergence of biorthogonal series for biharmonic and stokes flow edge problems: Part II, SIAM. J. Appl. Math. 34 (1978) 7–27. [20] L.D. Sturges, Stokes flow in a two-dimensional cavity with moving end walls, Phys. Fluids 29 (5) (1986) 1731–1734. [21] C.I. Robbins, R.C.T. Smith, A table of roots of Sin z = z, Phil. Mag. 39 (7) (1948) 1005. [22] R.C.T. Smith, The bending of a semi-infinite strip, Austral. J. Sci. Res. 5 (1952) 227–237. [23] F. Gu¨rcan, Flow bifurcations in rectangular, Lid-Driven, Cavity Flows, PhD Thesis, University of Leeds, 1997.
www.elsevier.com/locate/amc
Saddle connections near degenerate critical points in Stokes flow within cavities F. Gu¨rcan *, A. Deliceog˘lu Department of Mathematics, Erciyes University, Kayseri 38039, Turkey
Abstract Streamline patterns and their bifurcations in two-dimensional incompressible flow near a simple degenerate critical point away from boundaries are investigated by using the normal form theory for the streamfunction obtained by [M. Brøns, J.N. Hartnack, Streamline topologies near simple degenerate critical points in two-dimensional flow away from boundaries, Phys. Fluids 11 (1999) 314–324]. For the normal form of order six, a bifurcation diagram is constructed with two bifurcation parameters. The theory is applied to the patterns and bifurcations found numerically in the studies of Stokes flow in a double-lid-driven rectangular cavity with two control parameters (the cavity aspect ratio A (height to width), and the speed ratio S). Bifurcations in the cavity are obtained using an analytic solution for the streamfunction developed for any value of S and A. Using this solution for special values (S, A) a global bifurcation is identified with a single heteroclinic connection which connects three saddle points in a triangle and does not appear in the unfolding of the simple linear degenerate critical points lying in a line y = constant. 2005 Elsevier Inc. All rights reserved. Keywords: Bifurcations; Normal forms; Eigenfunction solution; Biorthogonal series; Stokes flow
*
Corresponding author. E-mail address: [email protected] (F. Gu¨rcan).
0096-3003/$ - see front matter 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2005.03.012
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F. G€ urcan, A. Deliceog˘lu / Appl. Math. Comput. 172 (2006) 1133–1144
1. Introduction The main objective of this article is to investigate streamline patterns and their bifurcations of Stokes flow in a cavity with two moving lids and two rigid walls both using a method from non-linear dynamical systems (as in [1]) and an analytic solution. The idea of the qualitative study of differential equations described streamline patterns has been extensively used, for example, see [2–6] and the bifurcation in the fluid flow where the topological structure of the flow field may change qualitatively as a parameter passes through a critical value, has been recently investigated in a number of articles, see [1,7–9]. Brøns et al. [10,11] used an idea of a combination of bifurcation theory for two-dimensional dynamical systems and numerical simulations. They considered the streamlines as trajectories for the velocity field in a cylindrical container with rotating end-covers and applied methods from bifurcation theory to describe the qualitative changes that are possible. Gu¨rcan [12,13] considered the cavity problem to investigate flow patterns for varying A (the cavity aspect ratio (height to width)). He used both an analytic solution for the streamfunction and methods from non-linear dynamics for the velocity field and showed that bifurcations occur at stagnation points away from boundaries and near the boundary. In the present study, using the streamfunction for two-dimensional incompressible flow obtained from the normal form theory [1,7] we construct a bifurcation diagram with the normal form of order up to six for the steady incompressible flow which is symmetric about the x axis. As an application of this theoretical framework, a similar bifurcation diagram with two control parameters is produced for Stokes flow in a double-lid-driven rectangular cavity by using an analytical solution for the streamfunction developed for any value of S and A where S is the speed ratio of the lids. To determine the existence of simple degenerate critical points in the cavity we use Taylor expansion of a velocity vector field near the singular point of the system of differential equations for the streamlines. The cavity flow is studied both analytically and numerically by a number of authors (see [14–17] and the review paper by Shankar and Deshpande [18]). It is the first formulated and solved analytically by Joseph and Sturges [19] for a single lid moving. They showed the presence of four eddies with corner eddies for the cavity aspect ratio A = 5. Subsequently, Sturges [20] considered the symmetric flow with the lids moving in opposite directions (i.e., S = U1/ U2 = 1 where U1 and U2 are the upper and lower lid velocities, respectively). For four values of aspect ratio he showed four distinct flow structures which revealed the presence of side eddies attached to the stationary side walls. For any value of S an analytic solution for the streamfunction expressed as an infinite series of Papkovich–Faddle eigenfunctions is developed and used to con-
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struct a (S, A) control space diagram which is similar to that obtained by using the normal form theory. In addition, using the analytic solution more complex flow patterns are obtained and these are not captured by the normal form coordinates used by [1]. 2. Bifurcations near simple degenerate critical points An incompressible two-dimensional flow far from any boundaries is considered. A streamfunction, w, exists such that the streamlines are found from ow ow x_ ¼ u ¼ and y_ ¼ v ¼ . oy ox Following Brøns and Hartnack [1] we will determine local flow structures under the assumption of the reflection symmetry about x = 0 and the streamfunction is then an even function of x. The expansion of w about a given critical point which is taken to be the origin, becomes 1 X w¼ c2i;j x2i y j . ð1Þ iþj¼1
The linear approximation of the equations for the streamlines is 0 2c0;2 x_ c0;1 x þ ¼ . 2c2;0 0 0 y_ y
ð2Þ
If c0,1 = 0, the origin is a critical point and the local flow pattern is determined using the standard theory for Hamiltonian systems. If jJj = 4c0,2c2,0 = 0 (considering jJj is not the zero matrix) the critical point is degenerate. This case refers as a simple linear degeneracy. Thus higher-order terms become decisive for the streamline pattern. For a significant simplification of higher-order terms in the streamfunction and an easy determination of simple degenerate patterns and their bifurcations, Brøns and Hartnack [1] used normal form transformations found via a generating function. Their normal form procedure gives rise to the following theorem. Theorem. Let c0,1, c2,0 and c4;0 , c6;0 ; . . . ; c2½N=22;0 be small parameters. Assuming the non-degeneracy conditions c0,2 5 0, c2½N =2;0 6¼ 0 a normal form of order N for the streamfunction (1) is r wN ¼ y 2 þ f ðxÞ; ð3Þ 2 where f ðxÞ ¼
½N =2 X i¼1
c2i x2i ;
c2½N =2 ¼
1 ; 2½N =2
F. G€ urcan, A. Deliceog˘lu / Appl. Math. Comput. 172 (2006) 1133–1144
1136 c
r ¼ c2½N0;2=2;0 , and c2i,i = 1, 2, . . . , [N/2] are transformed small parameters, and c2k;0 is c2k,0 plus non-linear combinations of c2i,j where 2i + j < 2k, see [1]. Using the normal form for w, (3), the corresponding dynamical systems x_ ¼ ry;
y_ ¼ f 0 ðxÞ
ð4Þ
can be analysed. The origin is always a critical point and the total number of critical points is odd. All the critical points lie on the x axis. In addition to the local bifurcation occurring when f 0 (x) = 0 and f00 (x) = 0, a global bifurcation takes place when the streamfunction attains the same value at two saddle points. We will focus on the normal form of order six (N = 6) to obtain a bifurcation diagram with two control parameters. It is, however, presented flow patterns for the normal form of order four. For N = 4 the streamfunction is r 1 w 4 ¼ y 2 þ c2 x2 þ x4 . 2 4 The condition for bifurcations is f 0 ðxÞ ¼ 2c2 x þ x3 ¼ 0;
ð5Þ
f 00 ðxÞ ¼ 0;
ð6Þ
it is clear that for c2 P 0 there is a single critical point which is the origin. When r > 0 it is a centre and a saddle point pffiffiffiffiffiffiffiffiffiffi ffi when r < 0. For c2 < 0 there are three critical points (x1 ¼ 0; x2;3 ¼ 2c2 Þ. Their types depend on r. A simple calculation shows that f 0 (x2) = f 0 (x3) and f(x2) = f(x3) so a heteroclinic trajectory occurs for r < 0, and for r > 0 the possible flow pattern is a separatrix. 2.1. Normal form order 6 For N = 6 the streamfunction is r 1 w 6 ¼ y 2 þ c2 x2 þ c4 x4 þ x6 . 2 6
ð7Þ
The condition for bifurcations is ð8Þ f 0 ðxÞ ¼ 2c2 x þ 4c4 x3 þ x5 ¼ 0; f 00 ðxÞ ¼ 2c2 þ 12c4 x2 þ 5x4 ¼ 0. pffiffiffiffiffiffiffiffiffiffiffi By eliminating x = 0 and 2c4 we find that in the (c2, c4) parameter plane, bifurcations occur on c4 axis and on the curve c2 2c24 ¼ 0 for c4 6 0. The (c2, c4) parameter plane can be divided into the following subregion: (1) When 4c24 2c2 P 0, the system (4) has five critical points, by computing the Jacobian of every critical points, we have, C 1 ð0; 0Þ;
C2
! rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2c4 þ
4c24 2c2 ; 0
and
! rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
C 3 2c4 þ
4c24 2c2 ; 0
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1137
are centers, while other singular points such as ! ! rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi S4 2c4 4c24 2c2 ; 0 and S 5 2c4 4c24 2c2 ; 0 are saddle points. Since f(S4) and f(S5) have the same value at the critical points, there is a heteroclinic bifurcation at this region. (2) When 4c24 2c2 ¼ 0, the system (4) has three critical points, these are pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi C 1 ð0; 0Þ; C 2 ¼ S 4 ¼ A1 2c4 ; 0 and C 3 ¼ S 5 ¼ A2 2c4 ; 0 . Since f 00 (A1) = f 00 (A2) = 0, A1 and A2 are the degenerate critical points. Also this gives us that 4c24 2c2 ¼ 0 is a bifurcation curve. To determine the topological type at the degenerate critical points, we expand the streamfunction at that point, we have f 000 5 0, showing that they are cusps. (3) When 4c24 2c2 < 0, the system has one critical points C1(0, 0) and it is saddle point. (4) When c2 =p 0,ffiffiffiffiffiffiffiffiffiffi there are three pffiffiffiffiffiffiffiffiffiffiffi critical points, which are ffi C 1 ð0; 0Þ; D2 ð 4c4 ; 0Þ and D3 ð 4c4 ; 0Þ. The critical points D2 and D3 are center, but C1 is a degenerate critical points. As in the case 2) we expand streamfunction at the point C1, we get f (4) = 24c4. If c4 < 0, the degenerate critical point is a center. If c4 < 0, the degenerate critical point is saddle. The complete bifurcation diagram is shown in Fig. 1a.
(a)
(b)
Fig. 1. Bifurcation diagrams obtained using: (a) methods from non-linear dynamics and (b) an analytic solution.
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3. Boundary value problem A rectangular cavity is considered, of depth 2H and width 2L with stationary side walls lying along X = ±L and lids lying along Y = ±H and moving in both directions with speeds U1 and U2. Speeds are non-dimensionalised with respect to U2 and lengths with respect to L such that the flow domain is given by jxj 6 1; jyj 6 A. For the two-dimensional Stokes flow of an incompressible fluid the velocity components are expressed in terms of streamfunction, w, which is the solution of the biharmonic equation r4 w ¼ 0.
ð9Þ
The boundary value problem is shown in Fig. 2 where the boundary conditions include: (i) w = constant = 0 on all boundaries wð1; yÞ ¼ wðx; AÞ ¼ 0;
ð10Þ
(ii) no-slip on the upper and lower lids and on the side-walls ow ðx; AÞ ¼ S; oy
ow ðx; AÞ ¼ 1 and oy
ow ow ð1; yÞ ¼ ð1; yÞ ¼ 0. ox ox
The streamfunction for any value of S can be written as 1 X An esn ðyAÞ þ Bn esn ðyþAÞ n wðx; yÞ ¼ /1 ðx; sn Þ . s2n n¼1
ð11Þ
ð12Þ
The functions /n1 ðx; sn Þ are even Papkovich–Faddle eigenfunctions (see [19,20]) /n1 ¼ sn sin sn cos sn x sn x cos sn sin sn x
ð13Þ
and sn are complex eigenvalues determined by the side wall conditions w = ow/ ox = 0 at x = ±1, which yield the eigenvalue equation sin 2sn ¼ 2sn . The sn are easily determined via a Newton iteration procedure as described by Robbins and Smith [21] using an initial estimate of the form
Fig. 2. Dimensionless boundary value problem and attendant boundary conditions.
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2sn ð2n 1.5Þp þ i logð4n 1Þp.
1139
ð14Þ
From sin 2sn ¼ 2sn it follows that sn ¼ sn where the overbar denotes complex n conjugate and this, in turn, yields /n 1 ðxÞ ¼ /1 ðxÞ. Since the streamfunction is n ; Bn ¼ B n . To determine the coefficients (An real, it follows that An ¼ A and Bn) we employs SmithÕs [22] biorthogonality relation technique. This technique is the first used by Joseph and Sturges [19] to obtain a solution for a cavity with a single moving lid. However, it is simple matter to modify the technique in order to account for the movement of the other lid. Considering boundary conditions and employing SmithÕs relation (see [23] for more detail) yields 1 X ð1 sn Þ ð1 þ sn Þ ðAl þ Bl e2sl A ÞK l þ An Bn e2sn A M l;n ¼ 4; sn sn n¼1 ð15Þ ðAl e
2sl A
þ Bl ÞK l þ
1 X
An e
2sn A
n¼1
ð1 sn Þ ð1 þ sn Þ Bn M l;n ¼ 4S; sn sn ð16Þ
where Kl ¼
Z
þ1 1
½wl2 /n1 þ ð2wl2 wl1 Þ/n2 dx
and M l;n ¼
Z
þ1
1
wl2 /n1 dx.
Consequently, the right hand side of Eq. (16) becomes equal to 4S, that is, this corresponds the second (lower) lid speed. Thus this change results in a solution giving by Eq. (12) which is valid for any value of S when the coefficients are determined. Eqs. (15) and (16) form an infinite set to be solved for the unknown coefficients An and Bn n = ±1, ±2, . . . which, in practice, can be determined only by truncating after N terms. When the coefficients have been determined, the streamfunction at any interior point in the liquid is obtained by simply summing a finite number of terms in the series (12). In the present study, we take N = 100. Before analysing flow patterns, we investigate whether the solution (12) imposes any restrictions on steady streamline patterns and their bifurcations in the general symmetrical case which is based on only the existence of the streamfunction (1). For this purpose, we consider a Taylor expansion of (12) about a critical point which is given by (1) with each coefficient c2i,j expressed as an
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1140
infinite series depending on An, Bn, sn. Although the c2i,j can easily be calculated for a given critical point, it is not analytically possible to see any constraints on the c2i,j (since An, Bn cannot be solved analytically). More generally, to determine any restrictions for the c2i,j, the expression (1) is inserted into the biharmonic Eq. (9) and collecting terms yields a series of algebraic equations for the c2i,j. The equations of order zero and two are given by x0 y 0 :
c2;2 þ 3c4;0 þ 3c0;4 ¼ 0;
x0 y 2 :
c4;2 þ 2c2;4 þ 15c0;6 ¼ 0;
2 0
c2;4 þ 2c4;2 þ 15c6;0 ¼ 0.
xy :
ð17Þ
The non-zero equations of order k constitute an undetermined homogeneous linear system and there is no information about coefficients of order less than four. Since in our study we only consider the streamfunction of the normal form of order N = 6, coefficient of order more then four cannot be appear, see (7). We conclude that all topologies and bifurcations described in Section 2 can occur in steady Stokes flow. 3.1. Flow patterns One aim is to track the flow bifurcations arising in the cavity as A is gradually increased for speed ratio S = 8 · 104. Fig. 3a (where A = 1.75) shows a separatrix with two sub-eddies, an outer circulation and the side eddies attached to the stationary side walls near the upper lid. As A is increased to A = 1.767 for S = 8 · 104, a simple calculation shows that the coefficient, X sn sin sn þ cos sn
sn ðy 0 AÞ sn ðy 0 þAÞ c2;0 ¼ Bn e An e ; 2sn obtained by the expansion of (12) at a critical point, (x, y) = (0, y0 ’ 1.628), determined by the bisection method when u = v = 0, is vanished. Thus (0, y0 ’ 1.628) is a simple linear degenerate critical point at which (from Section 2.1) o3 w ð0; y 0 Þ 6¼ 0; ox3 and so two saddle-node (cusp) bifurcations arise within the upper sub-eddy, see Fig. 3b where A = 1.78. In Fig. 3b, there is a separatrix with two saddle points and three centers lying on the line almost y = 1.63. As A continues to increase, the two saddle points (corresponding to saddlenode bifurcations) approach each other and coalesce to the centre on x = 0 to produce a saddle point at A = 1.785 (see Fig. 3c where A = 1.79). Streamline patterns about the critical points in Fig. 3a–c correspond to the patterns in regions I, II and III of Fig. 1a, respectively. Hence a similar diagram to Fig. 1a is
F. G€ urcan, A. Deliceog˘lu / Appl. Math. Comput. 172 (2006) 1133–1144
1141
(b)
(a)
(c)
(d) 4
Fig. 3. Streamlines for S = 8 · 10 (a) A = 1.75, (b) A = 1.78, (c) A = 1.79, and for S = 4 · 104 (d) A = 1.70. In (b), (c) and (d) the upper part of the cavity is illustrated.
expected to be present for the cavity flow with two control parameters. Indeed, for 1.75 < A < 1.80 and 1 · 102 < S < 1 · 104, we obtain a (S, A) control space diagram, Fig. 1b, exhibiting bifurcation curves. The type and form of the bifurcations in (S, A) space are exactly the same as those obtained via the theoretical framework, Fig. 1a. Furthermore, for a special value (A = 1.7,S = 4 · 104) a global bifurcation is identified with a heteroclinic connection which connects three saddle points, see Fig. 3d where the saddles are not located on a line. The configurations with critical points in a triangle are not captured by the streamfunction (3) which has the unfolding of a simple linear degenerate in which all critical points lie on a line in the normal form coordinates. Using the analytic solution (12), investigations show that once again a heteroclinic connection appear in the cavity when the cavity aspect ratio A becomes A = 4.623 for S = 0.0031, see Fig. 4e. Fig. 4 shows a sequence of flow patterns as A is increased from four and many bifurcations occur in the central section of the cavity. The first three bifurcations correspond to the appearance of side eddies, a saddle node bifurcation near each side wall and
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(d)
(b)
(a)
(c)
(e)
(f)
(g)
Fig. 4. Flow structure development in the central section of cavity as A increases from 4 for S = 0.0031; (a) A = 4, (b) A = 4.55, (c) A = 4.6, (d) A = 4.6225, (e) A = 4.6236, (f) A = 4.627 and (g) A = 4.678.
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1143
thirdly a saddle node bifurcation on x = 0, see Fig. 4a–d. As the aspect ratio A is reached to A = 4.623, Fig. 4e the three saddles become connected by a single streamline. As A is increased gradually the heteroclinic connection transforms into two separatrices with separatrix Sp2 enclosed by Sp1, Fig. 4f. When A = 4.645 the two saddle points of Sp2 coalesce at the centre on x = 0 to produce a saddle, Fig. 4g for A = 4.678. These patterns show that flow structure development for any values of A and S would be more intricate. The dynamics near simple and non-simple degenerate critical points of Stokes flow in a cavity with the two control parameters are under an in-depth investigation.
References [1] M. Brøns, J.N. Hartnack, Streamline topologies near simple degenerate critical points in twodimensional flow away from boundaries, Phys. Fluids 11 (1999) 314–324. [2] J.C.R. Hunt, C.J. Abel, H. Woo, Kinematical studies of the flow around free or surfacemounted obstacles; applying topology of flow visualization, J. Fluid Mech. 86 (1978) 179–200. [3] M. Tobak, D.J. Peake, Topology of three-dimensional separated flows, Ann. Rev. Fluid Mech. 14 (1982) 61–85. [4] A.E. Perry, M.S. Chong, A description of eddying motions and flow patterns using criticalpoint concepts, Ann. Rev. Fluid Mech. 19 (1987) 125–137. [5] P.G. Bakker, Bifurcation in flow patterns, Thesis, Technical University of Delft, Netherlands, 1989. [6] U. Dallmann, Three-dimensional vortex structures and vorticity topology, Fluid Dyn. Res. 3 (1988) 183. [7] J.N. Hartnack, Streamlines topologies near a stationary wall using normal forms, Acta Mech. 136 (1999) 55–75. [8] T. Ma, S. Wang, Topology of 2D incompressible flows and applications to geophysical fluid dynamics, RACSAM, Rev. R. Acad. Cien. Serie A. Mater. 96 (3) (2002) 447–459. [9] M. Ghil, T. Ma, S. Wang, Structural bifurcation of 2D incompressible flows, Indiana Univ. Math. J. (Special issue) 50 (1) (2001) 159–180. [10] M. Brøns, L.K. Voigt, J.N. Sørensen, Streamline topology of steady axisymmetric vortex breakdown in a cylinder with co- and counter-rotating end-covers, J. Fluid Mech. 401 (1999) 275–292. [11] M. Brøns, L.K. Voigt, J.N. Sørensen, Topology of vortex breakdown bubbles in a cylinder with a rotating bottom and a free surface, J. Fluid Mech. 428 (2001) 133–148. [12] F. Gu¨rcan, Streamline topologies in Stokes flow within lid-driven cavities, Theoret. Comput. Fluid Dyn. 17 (1) (2003) 19–30. [13] F. Gu¨rcan, Streamline topologies near a stationary wall of Stokes flow in a cavity, Appl. Math. Comput. 165 (2) (2005) 329–345. [14] O. Burggraf, Analytical and numerical studies of the structure of steady separated flows, J. Fluid Mech. 24 (1966) 113–151. [15] T.B. Gatski, C.E. Grosch, M.E. Rose, A numerical study of the two dimensional Navier– Stokes equation in vorticity-velocity variables, J. Comput. Phys. 48 (1982) 1–22. [16] P.H. Gaskell, A.K.C. Lau, N.G. Wright, Comparison of two solution strategies for use with higher-order discretization schemes in fluid flow simulation, Int. J. Numer. Meth. Eng. 8 (1988) 1203–1215.
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F. G€ urcan, A. Deliceog˘lu / Appl. Math. Comput. 172 (2006) 1133–1144
[17] P.H. Gaskell, M.D. Savage, J.L. Summers, H.M. Thompson, Stokes flow in closed rectangular domains, Appl. Math. Modell. 22 (1998) 727–743. [18] P.N. Shankar, M.D. Deshpande, Fluid mechanics in the driven cavity, Annu. Rev. Fluid Mech. 32 (2000) 93–136. [19] D.D. Joseph, L. Sturges, The convergence of biorthogonal series for biharmonic and stokes flow edge problems: Part II, SIAM. J. Appl. Math. 34 (1978) 7–27. [20] L.D. Sturges, Stokes flow in a two-dimensional cavity with moving end walls, Phys. Fluids 29 (5) (1986) 1731–1734. [21] C.I. Robbins, R.C.T. Smith, A table of roots of Sin z = z, Phil. Mag. 39 (7) (1948) 1005. [22] R.C.T. Smith, The bending of a semi-infinite strip, Austral. J. Sci. Res. 5 (1952) 227–237. [23] F. Gu¨rcan, Flow bifurcations in rectangular, Lid-Driven, Cavity Flows, PhD Thesis, University of Leeds, 1997.