ON THE FLOW OF A THIN LIQUID LAYER IN AN INCLINED CHANNEL OF TRIANGULAR TRANSVERSE SECTION DRIVEN BY A SURFACE TENSION GRADIENT. NUMERICAL AND ASYMPTOTIC ANALYSIS

. In this paper we investigate the locally unidirectional ﬂow of a thin liquid layer conﬁned to an inclined channel driven simultaneously by a surface tension gradient. MSC 2000. 76D08, 76D45, 76M55.


INTRODUCTION
We consider the steady behavior of a thin layer (trickle, rivulet).We follow the lubrication approximation to obtain an analytical solution for a rivulet of Newtonian fluid moving down a slowly varying substrate.We study the special case when the transversal sectional profile of the rivulet is triangular and we include the effects of a surface tension gradient.We are interested in the study of the Marangoni effect.We take into account a non-zero tangential stress boundary condition.The extra terms introduced by this non-zero shear stress have a reasonable physical significance.

PROBLEM FORMULATION
We choose Cartesian system Oxyz, with the x axis in the direction of flow and the y axis transverse to the direction of flow and so the fluid velocity is given by − → u = u(y, z) − → i .Here (L): z = (H + h)(y) denotes the unknown transverse profile of the free surface, (S): z = H(y) denotes the known transverse profile of the substrate and h is thickness of the layer.
We consider the unidirectional flow of a thin layer of a constant width a of Newtonian fluid with constant density ρ and constant viscosity µ down a substrate, which is inclined at an angle ϕ, 0 ≤ ϕ ≤ π/2, to the horizontal.
Employing the familiar lubrication approximation [1], [7] the governing equations for this unidirectional flow are simply They are to be integrated subject to the boundary conditions u = 0, on z = H(y) Here p = p(y, z) denotes the pressure in the fluid, p 0 is the constant pressure in the ambient gas above the fluid, Γ is the references value of surface tension, g is the acceleration due to gravity, Γ x is the surface tension gradient, and β is constant contact angle between the fluid and the substrate at the unknown position of the three-phase contact line.
Since the flow is unidirectional, kinematic condition is identically satisfied.
With the respect to the angle ϕ to the horizontal we consider the case 0 < ϕ < π/2.In this case with a triangular section we have Then the solution is as follows: • the velocity • the free-surface profile z = (H + h)(y), with H(y) = A |y|, is given by where we have introduced the notation m = |cos ϕ| .The volume flux of liquid running the substrate is At this point it is convenient to introduce appropriate nondimensional variables defined by: and so, dropping the star superscripts, we yield the following solution for Q: + 3 3ma cth 2 ma − 3 cth ma − ma .
If Γ x = 30.888689(mN/m)then we obtain for the volume flux the numerical value Q * = 0.0006.
In this case we compare the value of the volume flux given by [4], namely Q * = 0.0007, with our values.We observe that the error is O(10 −4 ).

ASYMPTOTIC ANALYSIS FOR H = 0
A) The limit a → 0. In the limit of a narrow rivulet, a → 0, gravity effects become insignificant and the behavior of Q depends only on the local behavior of H near y = 0.
In particular, if H(y) = O(y), as a → 0, then h(y) ∼ a 2 −y 2 2a , as a → 0, and hence 15 , in agreement with the corresponding result in the special case H ∼ 0.

DISCUSSION
If H(y) = A |y| then the profile is always convex when A < 1, and always concave when A > 1.
The case A = 1 is exceptional.In this case (H + h)(y) = H(a) = a, i.e, the transverse rivulet profile is identically flat for all values of ϕ.As a consequence, the exact expression for Q is particulary simple: 60 .If Γ x = 0 then we can write an explicit expression for a, namely a = 210Q 207 sin ϕ 1/4 .

CONCLUSIONS
In the lubrication approximation we have considered the flow in an inclined channel of triangular section.Consequently, the flow is driven simultaneously by gravity and a surface tension gradient.This gradient implies a non-zero tangential stress boundary condition (Marangoni effect).We have estimated the respond of the fluid to such stress.The extra terms that correspond to this boundary condition are in perfect agreement with the physics of the flow.More than that, in some particular situations we have estimated them numerically.There is a fairly good agreement between our computed values of the volume flux of the fluid and those reported in [4].