Abstract. The approximate Lie group method is used to investigate the evolution of the free surface of a thin liquid drop on a slowly dropping flat plane. Surface ...
Nonlinear Dynamics 28: 167–173, 2002. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.
An Approximate Lie Group Investigation into the Spreading of a Liquid Drop on a Slowly Dropping Flat Plane E. MOMONIAT Centre for Differential Equations, Continuum Mechanics and Applications, Department of Computational and Applied Mathematics, University of the Witwatersrand, Private Bag 3, Wits 2050, Johannesburg, South Africa (Received: 16 February 2001; accepted: 7 September 2001) Abstract. The approximate Lie group method is used to investigate the evolution of the free surface of a thin liquid drop on a slowly dropping flat plane. Surface tension effects are ignored. A group classification is performed to determine the rate at which the plane drops. An approximate group invariant solution is then calculated for the free surface of an evolving liquid drop on the slowly dropping flat plane. An important parameter in the solution is the initial angle of the plane. For small angles there is no significant difference in the drop profile. For larger angles, changes in the drop profile and rate of spreading are significant. Keywords: Thin film, slowly dropping flat plane, approximate Lie group method.
1. Introduction In this paper we investigate the evolution of the free surface of a thin viscous liquid drop on a slowly dropping flat plane. The flat plane is fixed at one end to a horizontal surface, as can be seen in Figure 1. Research in this area is important in determining the effects of sloping on coating flows (see, e.g., [12]). Another application is in modelling the spread of droplets on an aircraft wing that is being flattened at a rate determined in this paper. We also obtain an initial tilt angle for which the results derived in this paper are valid. The free surface equation is obtained by considering the classic problem of a liquid drop flowing down a fixed slope. This problem can be found in many textbooks on viscous flow (see, e.g., [1, 13]). The free surface equation for a liquid drop spreading down an inclined plane is given by ht +
∂ g g sin αh2 hx = cos α (h3 hx ), ν 3ν ∂x
Figure 1. Sketch of slowly dropping inclined plane with a liquid drop.
(1)
168 E. Momoniat (subscripts denote differentiation unless otherwise indicated) where the free surface is given by h = h(t, x). The right-hand side of (1) is not included in the analysis of Acheson [1]. In deriving (1) it is assumed that the angle of the flat plane is fixed. Equation (1) has been used to investigate the stability of flow down an inclined plane (see, e.g., [7]) and also flow over a gently sloping viscous dry bed (see, e.g., [4]). In this paper, we use (1) to model the spreading of a liquid drop on a slowly dropping flat plane by choosing α = εF (t),
ε 1,
(2)
where F is an arbitrary function of t. The rate at which the flat plane is dropping is then given by −ε dF (t)/dt. To first-order in ε (1) can be written as ht + λεF (t)h2 hx =
λ ∂ 3 (h hx ), 3 ∂x
(3)
where g . ν The unperturbed equation from (3) is given by λ=
(4)
λ ∂ 3 (h hx ). (5) 3 ∂x The nonlinear diffusion equation (5) is part of the larger class of nonlinear diffusion equations: ∂ ∂h n ∂h = h . (6) ∂t ∂x ∂x ht =
Kath and Cohen [8] present a similarity solution to (6) as well as other similarity solutions that exhibit waiting-time behaviour. Lie point symmetry generators admitted by (6) have been determined by Ovsiannikov [10, 11] and listed in the handbook by Ibragimov [6]. For the case n = 3 the Lie point symmetry generators are given by Y1 = ∂t , Y4 =
Y2 = ∂x ,
Y3 = 2t∂t + x∂x ,
3 x∂x + h∂h. 2
We define X0 as a linear combination of (7) of the form X0 = X0 = (k1 + 2k3 t)∂t + (k2 + x(k3 + 3/2k4 ))∂x + k4 h∂h ,
(7) 4
i=1 ki Yi ,
i.e. (8)
where the ki are constant. An optimal system has been calculated and hence all group invariant solutions admitted by (5) have been classified [10, 11]. We modify the similarity solution presented by Kath and Cohen [8] and write it as h = ∗ h (t, x), where h∗ is given by 1/3 x2 1 ∗ 1− 2 (9) h (t, x) = R(t) R (t) and where
1/5 10 . R(t) = 1 + λt 9
(10)
Approximate Lie Group Investigation 169 is the radius of the drop at time t. The similarity solution (9) can be determined from (8) by choosing k2 = 0,
k3 =
5λ k1 , 9
k4 = −
2λ k1 9
(11)
and solving the quasi-linear partial differential equation X0 (h − h∗ (t, x))|h=h∗ = 0,
(12)
i.e. (9) is a group invariant solution admitted by (5) when (11) holds. The paper is divided up as follows: in Section 2 approximate Lie point symmetry generators and an approximate group invariant solution admitted by (3) is determined. A functional of the arbitrary function, F (t), is calculated. Concluding remarks are made in Section 3. 2. Approximate Symmetries and Invariant Solution Baikov et al. [2, 3] have conducted pioneering work on approximate Lie point symmetries admitted by differential equations. The interested reader is also referred to [5]. A brief summary of the calculations done to determine approximate Lie point symmetries admitted by (3) is included in this paper. An approximate Lie point symmetry generator is given by X ≈ X0 + εX1 ,
(13)
where X0 , the Lie point symmetry generator admitted by the unperturbed equation, is given by (8). The generator X1 is written as X1 = ξ 1 (t, x, h)∂t + ξ 2 (t, x, h)∂x + η(t, x, h)∂h. The functions ξ1 , ξ2 and η are found by solving the determining equation g g X [2] ht + εF (t)h2 hx − (h3 hx ) = O(ε 2 ). ν 3ν (3)
(14)
(15)
The second prolongation, X [2] , of X is defined by X [2] = X + ζ1 ∂ht + ζ2 ∂hx + ζ11 ∂htt + ζ12 ∂htx + ζ22 ∂hxx ,
(16)
where ζi = Di (η) − hk Di (ξ k ), ζj = Dj (ζi ) − hik Dj (ξ k ),
i = 1, 2, i, j = 1, 2,
(17) (18)
with summation over the repeated index k from k = 1 to k = 2 and D1 = Dt = ∂t + ht ∂h + ht t ∂ht + ht x ∂hx + · · · ,
(19)
D2 = Dx = ∂x + hx ∂h + ht x ∂ht + hxx ∂hx + · · · .
(20)
The coefficients ζi and ζij depend on ht which is eliminated using (3). Equation (15) may be separated according to the derivatives of h. Since the symmetry generator admitted by the
170 E. Momoniat unperturbed equation is already known, we separate (15) by derivatives of h which also have a coefficient ε. The resulting system of equations that need to be solved is given by 1 1 2 2 1 hξx − η + ah4 ξxx − hξti = 0, 3 9 3 1 1 = 0, ξh2 + ah2 ξx1 + ah3 ξxh 3
(21a) (21b)
ξx1 = 0,
(21c)
1 ηt − ah3 ηxx = 0, 3
(21d)
ξh1 = 0,
(21e)
1 1 = 0, ξh1 + hξhh 3 1 4 2 ah F (t) k3 + k + ah2 F (t)(k1 + 2k3 t) 2 2 1 2 − ξt2 = 0, − 2ah2 ηx − ah3 ηxh + ah3 ξxx 3 3 1 2 1 2 1 + ah4 ξxx − hξt1 = 0, −2η − hηh − h2 ηhh + 2hξx2 + h2 ξxh 3 3 3 1 2 2 1 + 2ah2 ξx1 + ah3 ξxh = 0. ξh2 + hξhh 3 3 Solving the system (21) we find that 1 η = (2b3 − b1 )h, 3 where the bi are constant. Also, the arbitrary function, F (t), must satisfy ξ 1 = b2 + b1 t,
ξ 2 = b4 + b3 x,
(2k3 + k4 )F (t) + 2(k1 + 2k3 t)F (t) = 0,
(21f)
(21g) (21h) (21i)
(22)
(23)
whence F (t) = β(k1 + 2k3 t)−(2k3 +k4 )/(4k3 ) ,
(24)
where β is an arbitrary constant. An approximate group invariant solution h ≈ h∗ (t, x) + εφ(t, x),
(25)
admitted by (3) is calculated by solving X(h − h∗ (t, x) − εφ(t, x))|h≈h∗ +εφ1 ≈ O(ε 2 ).
(26)
Choosing b1 =
10λ b2 , 9
b3 =
2λ 2 b , 9
b4 = 0
with (11) and separating (26) by coefficients of ε 1 we find that
(27)
Approximate Lie Group Investigation 171 φ1 (t, x) = (9 + 10λt)−1/5 H (ϕ),
ϕ=
x (9 + 10λt)1/5
when (11) holds. From (25) we obtain 1/3 x2 1 1− 2 + ε(9 + 10λt)−1/5 H (ϕ). h(t, x) = R(t) R (t)
(28)
(29)
The arbitrary function H (ϕ) is calculated by substituting (29) into (3) and separating by coefficients of ε (there are no coefficients of ε 0 present), to obtain the ordinary differential equation H (ϕ)(3ϕ 2 − 31/5 ) + l0ϕH (ϕ) + 4H (ϕ) −
6.34/5 2/5
k1
ϕ=0
(30)
Equation (30) can easily be solved to give H (ϕ) =
3.34/5 2/5
7k1
(31)
ϕ
and therefore
1/3 x2 3 x 1 1− 2 , + εκ h(t, x) = R(t) R (t) 7 R 2 (t)
(32)
where k=
β 2/5
(33)
.
k1
Imposing (11) on (24) we obtain κ F (t) = 2 . R (t)
(34)
Since, R(0) = 1, the initial slope is at an angle of εF (0) = εκ
(35)
radians. Equation (34) is sketched in Figure 2. We note that the solution (9) satisfies the property h(t, R(t)) = 0.
(36)
We impose the condition h(t, R ∗ (t)) ≈ O(ε 2 )
(37)
on (32). However, the solution (32) is not necessarily symmetric about the vertical axis, so we impose the conditions h(t, −R−∗ (t)) ≈ O(ε 2 ),
h(t, R+∗ (t)) ≈ O(ε 2 )
(38)
on (32). We make a further simplifying assumption that ∗ = R(t)(1 + ε m δ+/− ), R+/−
(39)
172 E. Momoniat
Figure 2. Sketch of (34).
∗ (t), R ∗ (t)] and h on the interval [−R(t), R(t)] for κ = 0.1, Figure 3. Plot of the h∗ (• • •) on the interval [−R− + λ = 0.1, ε = 0.01 at t = 0, t = 100 and t = 1000.
where δ+/− are both constants. We find that 3 3 3 3 1 1 ε 3 κ 3 , R+∗ = R(t) 1 + ε3κ 3 . R−∗ = R(t) 1 − 2 7 2 7
(40)
We note that the change in radius is significant when κ ≥ O(ε −1 ) since ε 1. This can be observed in Figure 4 where we have chosen κ = ε −1 . 3. Concluding Remarks An important parameter in the approximate invariant solution is the initial angle of the plane, εκ. We observe from Figure 3 that choosing κ 1, i.e. the initial slope is small, has no significant effect on the evolving profile of the liquid drop. When we choose κ = ε −1 as in Figure 4, we note that there is a significant steepening of the drop profile at the leading edge of the drop and a more pronounced flattening at the back (see also [1]). This is as a result of the initial angle being very steep. The rate of spreading is also affected as can be seen from
Approximate Lie Group Investigation 173
∗ (t), R ∗ (t)] for κ = 100, λ = 0.1, ε = 0.01 at t = 0, t = 100 and Figure 4. Plot of the h∗ on the interval [−R− + t = 1000.
(40). We can conclude from these results that a dropping plane may produce droplets with a very steep leading edge with a rate of spreading which is faster than spreading on a flat plane. Future work on this problem involves using the Navier–Stokes equation for a rotating frame of reference to determine a free surface equation that includes centrifugal effects. References 1. 2. 3.
4. 5.
6. 7. 8. 9. 10. 11. 12. 13.
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