On the Origin of the Bump in the Profile of Surface-Tension-Gradient-Driven Spreading Films
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ON THE ORIGIN OF THE BUMP IN THE PROFILE OF SURFACE-TENSION-GRADIENT-DRIVEN SPREADING FILMS
P. CARLES AND A.M. CAZABAT College de France, Physique de la Mati~re Condens6e, 11 place Marcelin Berthelot, 75231 Paris Cedex 05, France
ABSTRACT We show that the presence of a bump, in the profile of spreading films driven by forces such as surface tension gradients, can be explained by a simple analysis of the steady-state velocity, without taking into account contact line effects. INTRODUCTION A thin liquid film can be made to climb up a vertical plate against gravity, provided there exists a downwardly directed temperature gradient, as has been demonstrated experimentally [1,2]. The driving force for the spreading is the resulting surface tension gradient, which appears at the liquid surface and induces a flow towards the cold end of the plate, where the surface tension is higher (Marangoni effect [3,4]). The edge of the film thus made to grow exhibits different behaviours, according to the precise experimental conditions : in reference [1], the contact line remains straight throughout the spreading process, whereas a fingering instability appears in reference [2]. This striking difference has been attributed to the fact that the profile of the film, in the experiments of reference [1], exhibits a bump behind the contact line, whereas, in reference [2], the thickness of the film decreases monotonically. We address here the origin of the presence of this bump, and show that it can be explained by hydrodynamics only, and that the diffusion processes invoked previously [5] might not be necessary. A REMINDER ON THE EXPERIMENTS Experiments from reference [11 The first experiments that dealt with thin wetting films, climbing against gravity, under a surface tension gradient induced by a temperature difference, were conducted by Ludviksson and Lightfoot [1]. The liquid used was squalane, climbing on a silver wall. The spreading was initiated by pulling the wall out of a bath of squalane, thus entraining a rather flat film on the silver surface. Above the position of this first film, which drained under gravity, there appeared another film, the climbing of which was due to the surface tension gradient resulting from the imposed temperature gradient : the subject of their study is this second film. This film is essentially flat, except near the contact line where it has to curve to meet the bare substrate. In that region, the thickness of the film decreases monotonically to zero, i.e., there is no bump in the profile. Correlatively, the contact line remains straight. It is shown, both theoretically and experimentally, that the thickness h0 of the flat film is asymptotically given by h0 = eM(r), where eM(r)=-
(1)
pg is the thickness selected by the competition between the surface tension gradient r and the gravitational force per unit volume pg. The corresponding velocity v of the climbing front is Mat. Res. Soc. Symp. Proc. Vol. 248. 01992 Materials Research Society
520
v =3 a'C 16
(2)
Tlpg
where "l is the viscosity of
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