Thin Film Flow Down an Inclined Plane - MATLAB Captures Bifurcation Scenarios B.Uma Department of Mathematics College of Engineering, Guindy, Anna University Chennai 600 025, India
[email protected]
I. Introduction The flow of liquids in thin films exists in a wide variety of naturally occurring phenomena as well as in practical operating situations of importance. One of the important classes of problems in Fluid Mechanics is the flow in a thin film down an inclined plane / a vertical wall in the presence of a free surface. The development of flow in such a fluid film presents certain peculiar features and displays a variety of interesting dynamical behaviour. The formation of waves involves a variety of spatiotemporal patterns and transitions. As a result, the flow down an inclined plane / a vertical wall has served as a paradigm in the investigation of Newtonian and non-Newtonian free surface instabilities. The dynamics of thin film waves has attracted the attention of various industries due to its dramatic effect on transport rate of mass, heat and momentum. Further, the stability of a film flow system along an inclined plane under the action of gravity is of practical significance in many chemical and nuclear engineering applications. The investigations on the stability characteristics of thin films down an inclined plane / a vertical wall have shown that the wavy structure on the free surface of the film flow is sensitive to various factors such as the flow rate, angle of inclination and longitudinal length of the test section and so on. The wave structure on the free surface of the film flow system can be described qualitatively as follows. Once the flow rate exceeds the instability threshold, small ripples of high 1
frequency first emerge in the inception region. As the ripples propagate downstream, they grow both in amplitude and wavelength. The free surface is eventually covered with irregular and chaotic waves of large amplitude and long wavelength. Further, the experiments by Liu and Gollub (1993) demonstrate that, farther downstream, the film flows produced by either regular high frequency forcing or by natural choice are eventually dominated by a small number of irregularly spaced solitary humps, which emerge through phenomena of period doubling and wave merging. The experiments by Chu and Duckler (1974, 1975) and Takahama and Kato (1980) indicate the predominance of irregular chaotic waves sufficiently downstream. It is of interest to capture physical phenomena exhibited by the experiments using a theoretical model and with this in view, attention is focused on the stationary waves of finite-amplitude and long wave length on a thin viscous film down an inclined plane at high Reynolds numbers and moderate Weber numbers.
II. Mathematical Formulation The two dimensional flow of a thin layer of an incompressible Newtonian fluid down an inclined plane (Fig. I) is considered. The non-dimensional equations and boundary conditions are obtained as
∂u ∂t ∂v 2 ∂t
∂u ∂v + = 0 ∂x ∂y ∂u ∂u ∂p 3 ∂2u 1 ∂2u +u +v = − + + + ∂x ∂y ∂x Re Re ∂x2 Re ∂y 2 ∂v ∂v ∂p 3 cot θ 3 ∂ 2 v ∂2v +u +v = − − + + ∂x ∂y ∂y Re Re ∂x2 Re ∂y 2 u = 0, v = 0
p+
2 Re
on y = 0 !" 2 2 #−1 ∂u ∂H ∂v ∂H ∂u ∂H ∂v ∂H + 2 − 2 − 1 + 2 ∂y ∂x ∂x ∂x ∂x ∂x ∂y ∂x " 2 #−3/2 2 ∂ H ∂H +2 W e 2 1 + 2 = 0 on y = H ∂x ∂x 2
(1) (2) (3) (4)
(5)
∂u ∂v + 2 ∂y ∂x
"
1−
2
∂H ∂x
v=
2 #
+ 2
2
∂H ∂H +u ∂t ∂x
∂v ∂u − ∂y ∂x
∂H =0 ∂x
on y = H
at y = H
(6) (7)
¯3 g sin θ h σ 0 ¯ 0 is the is the Reynolds number, W e = 2 ¯ is the Weber number, h 2 3ν ρ¯ u 0 h0 ¯2 g sin θ h 0 unperturbed uniform flow depth, u ¯0 = . The two equation model describing the 3ν temporal and spatial evolution of the local flow rate Q(x, t) and the local flow depth H(x, t) where Re =
is obtained from equations (1) - (7) using EIM with self-similar velocity profile assumption by including terms upto O(2 ), when Re ' O(−1 ) and W e ' O(1). The exact solution for the uniform primary flow described by the parabolic velocity profile 3Q u= 2H is assumed, where Q =
RH 0
2y y2 − 2 H H
(8)
u dy.
Integration of the continuity equation (1) with respect to y from y = 0 to y = H along with the boundary condition (7) yields Ht + Q x = 0
(9)
Multiplying the x-momentum equation (2) with u and integrating the resulting equation with respect to y from y = 0 to y = H, gives Z
H 0
∂u ∂u ∂u ∂p 3 ∂2u 1 ∂2u u +u +v +u − u− u − u dy = 0 ∂t ∂x ∂y ∂x Re Re ∂x2 Re ∂y 2
Substituting for u from (8) in (10), the governing equation is obtained as
3
(10)
6Qt 3Ht Q 81QQx 54Hx Q2 3 1 3Q 3 cot θ − + − − + + Hx 2 2 3 5H 5H 35H 35H Re Re H 3 Re 33Hx Qxt 33HQxxt 107Ht Qxx 107Hxt Qx 107Hxx Qt 2 − + + + + −W eHxxx − 140 140 280 140 280 2 107Hxxt Q 9Hx Ht Qx 123Ht Hxx Q 87Hx Qt 9Hx Hxt Q + − − + − 280 35H 140H 140H 35H 39Ht Hx2 Q Qx Qxx QQxxx 533Hxx QQx 241Hxxx Q2 − + − + + 70H 2 3 3 448H 448H 2 2 3 2 2 333Hx Hxx Q 113Hx QQx 23Hx Q 73Hx Qx 95Hx QQxx − − + + − 448H 2 112H 2 112H 3 112H 224H 2 27Qxx 27Hx Qx 36Hxx Q 24Hx Q − − + =0 (11) − Re 5H 5H 2 5H 2 5H 3
Equations (9) and (11) describe the spatial and temporal evolution of Q(x, t) and H(x, t).
III. Linear Instability of the Uniform Flow If η and q denote the infinitesimal disturbances from the uniform flow, H = 1 + η, Q = 1 + q, then linearisation of equations (9) and (11) and elimination of q gives 6 102 54 3 3 cot θ 27 36 ηtt + ηxt + ηxx + (ηt + 3ηx ) − ηxx − ηxxt + ηxxx 5 35 35 Re Re Re 5 5 33 601 241 2 ηxxtt + ηxxxt + ηxxxx = 0 − −W eηxxxx + 140 840 448
(12)
By considering a wave like disturbance η = ei(x−ct)
(13)
and substituting (13) in (12), a characteristic equation for the complex phase velocity c = cr + ici is obtained as
11 2 2 5 9 17 601 2 1+ c + i + − − c 56 2Re 2Re 7 1008 15 6 9 5 cot θ 5 2 1205 2 + −i + + − − We + =0 2Re Re 7 2 Re 6 2688
The phase velocity and growth rate of the potentially unstable mode is given by q √A2 +B 2 +A q √A2 +B 2 −A 17 601 2 5 9 + 1008 + − 2Re + 2Re + 7 2 2 cr = ; c = i 2 1 + 11 2 2 1 + 11 2 56 56 4
(14)
(15)
The threshold for linear stability or neutral stability is obtained from (15) by setting ci = 0. It is observed that on the plane of Re/ cot θ versus , there are three branches of neutral curves, for a given Weber number W e. They are the Re/ cot θ-axis, the -axis and the curve given by Re = cot θ 1−
5 6
+ 32 5 W e2 + 18
5 5 16093 5 2 + 3− We − + O(4 ) 16093 2 = 6 6 18 94080 94080
(16)
The disturbances of the infinitely long waves are obtained from (16) by taking = 0 and this yields the critical condition for instability as cot θ/Re = 6/5. Therefore, the region of linear instability is 0 ≤ cot θ/Re < 6/5 while cot θ/Re > 6/5 is the region of linear stability.
IV. The Governing Equations for Stationary Waves A second-order theory for permanent waves which propagate at a constant speed without any change in form is pursued. While wave transitions in real-life involve complex spatio-temporal dynamics and many of these transitions lead to chaotic waves that are not stationary travelling waves, in what follows, bifurcation of stationary travelling waves has been examined as a preliminary study of the more complex transitions. The governing equations are obtained from (9) and (11) by transforming to a moving coordinate system defined 1 by ξ = (x − ct), where c is the propagation speed. Using the relations ∂ ∂ ∂ ∂ = , = −c and Q = c(H − 1) + 1 ∂x ∂ξ ∂t ∂ξ
(17)
obtained from (9), the third order ordinary differential equation for the flow depth is obtained as ¯ Hξ = H ¯ ¯ Hξ = H (18) ¯ ¯ ¯ = F (H, H, H; cotθ/Re, Re, c) H ξ D(H; W e, c) ¯ = H represents the slope, H ¯ = H represents the where H represents the flow depth, H ξ
ξξ
curvature of the free surface and D = W eH −
391 2 2 1211 241 c H + cH(c − 1) − (1 − c)2 6720 3360 448 5
(19)
6 54 3 cotθ 6 2 2 2 3 ¯ c H − cH(1 − c) − (1 − c) + H F = H 35 35 35 Re ¯ 9 H 36 2 + cH + H(1 − c) Re 5 5 379 21 333 2 3 2 2 ¯ ¯H − +H c H − cH (1 − c) − H(1 − c) 1680 2240 448 ¯ 2 3 H 24 23 2 2 27 23 3 2 ¯ + cH − (1 − c) + H cH + cH(1 − c) + (1 − c) Re 5 5 560 560 112 3 2 − (H − 1)(H + H + 1 − c) (20) Re
The fixed points are obtained by equating the right hand side of equation (18) to zero and √ −1 + 4c − 3 ¯ ¯ are given by HI = (H, H, H) = (1, 0, 0) and HII = , 0, 0 . The fixed point 2 HI corresponds to the uniform primary flow. The fixed point HII , is a function of c and is real and positive only for c > 1. Therefore, this corresponds to an asymptotic part of a non-uniform profile propagating at a speed c > 1. The fixed points are independent of θ, Re and W e and at c = 3, HI and HII cross each other, implying that a transcritical bifurcation exists at c = 3 at which the two fixed points exchange their stability properties. The parameter c is taken as the the bifurcation parameter for chosen values of other physical parameters. Fig. II shows the eigenvalue behavior of the two fixed points in the c versus cot θ/Re plane.
V. Bifurcation Scenarios - Numerical Solution By continuously varying the transcritical bifurcation parameter c about c = 3, either heteroclinic orbits or other nonlinear attractors are obtained for each fixed value of W e, cot θ/Re (W e ≤ 10 and Re = 13.33 and 100). No nonlinear attractor has been found in the regime of linear stability, cot θ/Re ≥ 1.2 for any Weber number and Reynolds number. In the linear regime of instability, the three dimensional dynamical system is solved numerically using ODE15s which is available in MATLAB5.3 with bounds for relative error as 10−10 . ODE15s in MATLAB is a variable order solver based on the numerical differentiation formulas. Optionally, it uses the backward differentiation formulas, also known as Gear’s method. It is a 6
multistep solver.
A. Heteroclinic Orbits For a typical value of wave speed c = 2.5, which lies above the Hopf threshold for HI but below c = 3.0, there is a heteroclinic orbit from HII to HI and its phase portrait, wave profile are given in Figures IIIa and IIIb. For any c above 3.0, which lies below the singularity boundary, there is a heteroclinic orbit from HI to HII and the phase portrait, wave profile are presented in Figures IIIc and IIId. Figure III also shows the corresponding results obtained by MIM. When c is slightly decreased from the Hopf bifurcation threshold, supercritical Hopfbifurcation occurs for the first fixed point. In this case, integration is carried out by taking the initial phase point in the neighborhood of this attractor and its subsequent bifurcation is then pursued. The computation is terminated at the point where no attractor could be detected. Integration is carried out for sufficiently long time until the trajectories finally settle down on the attractor. The bifurcation diagram for the permanent wave is obtained by choosing all the local maxima Hm of the time series for the flow depth H as the representative points and they are plotted against the bifurcation parameter c. In this bifurcation diagram, a single representative point denotes a limit cycle, two points at different heights denote a period-2 limit cycle and so on.
B. Summary of Bifurcation Scenarios for Re ≈ 13.33 and Re = 100 The results of the numerical experiment show four different types of bifurcation scenarios. Typical bifurcation sequences for W e = 1 for a set of values of cot θ/Re have been presented in the following figures. 1. Simple homoclinic bifurcation from the primary flow HI (cot θ/Re = 0.9; Figures IVa, V(i)a, V(i)b) 2. A period-doubling followed by a simple homoclinic bifurcation from the primary flow HI
7
(cot θ/Re = 0.55; Figures IVb, V(ii)a, V(ii)b) 3. Multiple hump homoclinic bifurcation from the primary flow HI (cot θ/Re = 0.48; Figures IVc, V(iii)a, V(iii)b) 4. Dominant scenario of period-doubling bifurcation leading to chaos from the primary flow HI (cot θ/Re = 0; Figures IVd, V(iv)a, V(iv)b) The bifurcation scenarios have been numerically determined for a wide range of Weber numbers, W e ≤ 10 The boundaries separating regimes of different bifurcation scenarios in terms of cot θ/Re in the regime of linear instability in the W e versus cot θ/Re plane are presented in Figure VI for limit cycles from HI for the cases Re = 13.33 and Re = 100. It is observed from Figure VI that for Re = 13.33, the Hopf-bifurcation threshold extends throughout the entire regime of linear instability for W e > W e(2) (= 0.513). In this region, the limit cycle either undergoes homoclinic bifurcations or period-doubling bifurcations. The region in which homoclinic bifurcations are observed dominates for smaller values of Weber numbers and this region has spread to the entire regime of linear instability around W e = W e(2) = 0.513. The period-doubling bifurcations exist for large Weber numbers. For W e < W e(2) , Hopf-bifurcation threshold exists and the limit cycle undergoes perioddoubling bifurcations for small values of cot θ/Re. As W e is further reduced, the limit cycles are the only attractors. When Re = 100, the homoclinic regime lies close to the linear instability threshold in W e > W e(2) , the limit cycle regime has diminished and lies close to the boundary of no attractor region in W e < W e(2) .
VI. Conclusion The finite amplitude waves of stationary form on a thin viscous film down an inclined plane at high Reynolds numbers and moderate Weber numbers have been considered using energy integral method (EIM). The two equation model obtained using EIM reduces to a
8
third order dynamical system in the frame of reference moving with the steady wave speed. Through numerical integration of the dynamical system, complex bifurcation scenarios have been captured using MATLAB.
VII References 1. J. Liu and J.P. Gollub, “Onset of spatially chaotic waves on falling films”, Phys. Rev. Lett. 70, 2289 (1993). 2. K.J. Chu and A.E. Duckler, “Statistical characteristics of thin wavy films: Part II. Studies of substrate and its wave structure” AICHE J. 20, 695 (1974). 3. K.J. Chu and A.E. Duckler, “Statistical characteristics of thin wavy films: Part III. Structure of the large waves and their resistance to gas flow” AICHE J. 21, 583 (1975). 4. H. Takahama and S. Kato, “Longitudinal flow characteristics of vertically falling liquid films without concurrent gas flow”, Int. J. Multiphase Flow 6, 203 (1980).
9
y¯ HH H
HH
* H
H j H
x¯ H
H * ¯ x, t¯) H H ¯ = h(¯ H Hy HH HH HH HH * HH HH HH H HH ¯ HH h HH0 HH HH HH HH HH HH HH HH HH ? HH HH g HH HH HH H HH
θ
HH
HH
HH
Figure I. Schematic Representation of a Thin Film Flow Down an Inclined Plane.
10
8
V
8
T (i)b
(i)a Hopf 7
7
6
* **
6
* ** 5
** *
5
c
* **
c 4
4
W M
** *
3 P
2Q
* * * Hopf
N R
3
X
0.6
0.8
1
1
S Z N
P
** *
2
0.2 O 0.4
0
W
* **
* ** 1
Z L
0
0.2
0.4
0.6
0.8
1
cot θ/Re
cot θ/Re 8
8
V
T
* **
(ii)a 7
(ii)b
7
* ** Hopf
* **
6
6
5
** *
5
c
c 4
4
W M 3 P Q 2
1
Z L
** * ** * * **
0O
X
W
N R
3 P
Hopf
* * *
2
* ** 0.2
0.4
0.6
0.8
1
1
1.2
Z S N
0
0.2
0.4
cot θ/Re
0.6
0.8
1
1.2
cot θ/Re
Figure II. Eigenvalue behaviour of the two fixed points in c versus cot θ/Re Plane for W e = 1 and (i) Re = 1/0.075 ≈ 13.33, (ii) Re = 100 (a) HI ; (b) HII . 11
1.1 c = 2.5
c = 3.3
1.1
1
1.05
H 0.9
H
0.8 0.01
0.95 5
(a)
−5
x 10
−3
x 10
−−−−− −−−−−
H
−−−−−
H
1.05
1
0
−3
0 −0.01
H
(c)
5
0 −−−−− −−−−−
1
0 −5
−3
x 10
−−−−−
−1
H
1.15 c = 2.5 MIM
1
c = 3.3
H
I
EIM
0.95
H
H
HII
MIM
1.1
EIM 1.05
0.9
H
I
1
HII
0.85
(b) 0.8
0
200
400
600
800
0.95
1000
ξ
(d) 0
1000
2000
3000
4000
5000
ξ
Figure III. A heteroclinic orbit, HII to HI ((a), (b)), HI to HII ((c), (d)), for W e = 1, Re = 100, cot θ/Re = 0.4 (a), (c) The phase trajectory; (b), (d) The profile. HII .
12
1.1
1.2
(a) cot θ/Re = 0.9
1.05
Hm
1.1
1
Hm 1
0.95
0.9
0.9 2.765
(b) cot θ/Re = 0.55
2.77
2.775
2.78
2.785
2.79
0.8 2.45
2.47
2.49
c 1.25
2.51
2.53
c 1.4
(c) cot θ/Re = 0.48
1.15
(d) cot θ/Re = 0
1.2
1.05 Hm
Hm 1 0.95 0.8
0.85 0.75 2.4
2.417
2.434 c
2.451
2.468
0.6 1.85
1.9
1.95
2
2.05
2.1
c
Figure IV. The bifurcation diagrams of limit cycles from HI : W e = 1, Re = 1/0.075 ≈ 13.33 (a) Simple homoclinic bifurcation, (b) Period-doublings + Simple homoclinic bifurcation, (c) Three hump homoclinic bifurcation, (d) Period-doubling route to chaos.
13
(i)a 1.1
H
(i)b 1.1
c = 2.768310771
c = 2.768310771 H
1
0.9 0.1 H 0
0.9
−−−−−
(ii)a 1.15
1
−−−−− −−−−−
0
−0.1 −0.1
H
0.1
0
50
(ii)b 1.15
H 0.95
−−−−−
H 0
−−−−− −−−−−
0
−0.5 −0.2
0.2
H
H 0.95
H
0
50
100 ξ
150
200
c = 2.4022259995
H 0.95
−−−−−
H 0 −0.5 −0.3
−0.1
−−−−− −−−−−
H
0.1
0.3
0.7
0
100
(iv)b 1.5
c = 1.87
ξ
200
300
c = 1.87
1
0.5 1
0.75
(iii)b 1.2
c = 2.4022259995
(iv)a 1.5
200
H 0.95
(iii)a 1.2
0.7 0.5
150
c = 2.4558356
c = 2.4558356
0.75 0.5
100 ξ
H 1
−−−−−
H 0
−1 −0.5
0
−−−−− −−−−−
0.5
H
0.5
0
100
200 ξ 300
400
500
Figure V. Selected phase portraits (a) and corresponding wave profiles (b) of attractors during: (i) Simple homoclinic bifurcation; cot θ/Re = 0.9, (ii) Period-doublings + Simple homoclinic bifurcation; cot θ/Re = 0.55, (iii) Three hump homoclinic bifurcation; cot θ/Re = 0.48, (iv) Period-doubling route to chaos; cot θ/Re = 0 for W e = 1, Re = 1/0.075 ≈ 13.33 from HI .
14
(1.14,10)
1
10
(a) Re ≈ 13.33
period−doubling
homoclinic
0
We 10
(0,0.513)
period−doubling (1.1733,0.51) (0.284,0.365)
(0,0.38)
(1.2,0.513) limit−cycle
−1
10
(cot θ / Re)
(0,0.18) 0
Γ
0.2 (0.3014,0.1) 0.4
0.6
no attractors 0.8
1
1.2
cot θ/Re (1.2,10)
1
10
(b) Re = 100 homoclinic
period−doubling We100
(1.06,0.51) limit−cycle
−1
10
0
(0.25,0.1)
(cot θ / Re)Γ
0.2 (0.3014,0.1) 0.4
0.6
(1.2,0.513) no attractors 0.8
1
1.2
cot θ/Re
Figure VI. Bifurcation scenarios delimited in the regime of linear SW-instability in the W e versus cot θ/Re plane for limit cycles from HI for (a) Re = 1/0.075 ≈ 13.33 (b) Re = 100.
15
