A similarity solution for viscous source flow on a vertical plane

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38 B. R. Duffy and H. K. Moffatt FIGURE 1. A trickle of fluid emitted on a near-vertical plate. effects of surface tension cannot be ignored; we consider this case here, again taking the film to be thin and the <strong>flow</strong> to be slow. Suppose a thin film of Newtonian liquid, of constant density ρ and viscosity µ, is <strong>flow</strong>ing down a plane inclined at an angle α to the horizontal, α being defined such that the liquid is on the ‘topside’ of the plane when α π and is on the ‘underside’ of the plane when α π (see Figure 1). We shall take the plane to be vertical or nearly vertical, so that α π is zero or is small. We use Cartesian coordinates Oxyz as in Figure 1, with Ox down a line of greatest slope and with Oz normal to the plane. A standard thin-film analysis shows that the (generally unsteady) free-surface profile z h(x, y, t) satisfies 3µh t [h(ρgh cos αγ h)]3ρgh h x sin α, (2.1) where γ denotes the coefficient of surface tension (assumed constant), g denotes the acceleration due to gravity, and suffixes denote partial differentiation. The case of a rivulet of uni<strong>for</strong>m width is obtained by putting h h(y) and setting to zero the term in square brackets in (2.1): ρgh cos αγh yy constant. (cf. Towell & Rothfeld, 1966; Allen & Biggin, 1974). This is to be integrated subject to the boundary conditions h 0 and htan β at y a, where a is the semi-width of the trickle, and β is the contact angle at the air–liquid–solid contact line, taken to be constant. For small α π the <strong>solution</strong> is h(y) tan β 2a (ay).
A <strong>similarity</strong> <strong>solution</strong> <strong>for</strong> iscous <strong>source</strong> <strong>flow</strong> on a ertical plane 39 The velocity component down the plane is u (ρg sin α2µ)(2hzz), and the volume flux of liquid running down the plane is Q a −a h(y) udzdy, ρgsin α 3µ a −a h dy, 4ρga sin α tan β , 105µ independent of γ (in this approximation). In general, liquid issuing from a <strong>source</strong> (whether a point <strong>source</strong> or a distributed <strong>source</strong>) will not have this profile, and there will be some sort of adjustment zone be<strong>for</strong>e a constant profile can be attained. Smith (1973) has studied this adjustment <strong>for</strong> the case when α is not near π, with surface tension neglected (at leading order). We, on the other hand, are interested in a near-vertical surface, so that α π, and surface tension cannot be neglected. In the spirit of Smith’s analysis, we seek a steady <strong>solution</strong> <strong>for</strong> which the length scale down the plane is much greater than that across the stream, so that h y h x in general. Then equation (21) becomes approximately hh yyyy 3h y h yyy 3(ρg sin αγ) h x 0. (2.2) The self-consistency of this approximation is checked a posteriori (see equations (212) and (213)) by verifying that 1 h y h x <strong>for</strong> the <strong>solution</strong> obtained; a more <strong>for</strong>mal justification of the approximation is given in an appendix. Here we shall consider <strong>solution</strong>s of (22) that are symmetrical about y 0, or, to be more precise, <strong>solution</strong>s <strong>for</strong> which h is even in y. This means that h y (x,0)0, h yyy (x,0)0. (2.3) We take the free-surface to meet the plane z 0 in the curves y y e (x), which are there<strong>for</strong>e three-phase contact lines (the ‘edges’ of the trickle). Then h(x, y e ) 0. (2.4) Also, the volume flux of liquid down the plane is approximately ρg sin α Q 3µ y e(x) h dy, (2.5) −y e (x) and herein we take Q to be a prescribed constant. It is expected that the contact angle β at the (anchored) three-phase lines y y e (x) will satisfy a restriction of the <strong>for</strong>m β r β β a , where β r and β a are constants (the receding and advancing contact angles). However, there is no reason to expect β itself to be constant on these contact lines (and indeed, simple observations of film-drainage down an inclined plane suggest strongly that β will vary quite considerably, at least where the film width varies in a downstream direction). Presumably conditions near the lateral contact lines are dominated by <strong>viscous</strong> stresses, in which case it is legitimate to assume that the contact angle adjusts itself locally to the value dictated by these stresses. There may, of course, be an ‘inner region’ (perhaps on the scale of intermolecular <strong>for</strong>ces) within which an adjustment to the appropriate contact angle takes place; however, the behaviour in such an inner region lies outside the scope of the present analysis. We there<strong>for</strong>e leave open the question
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A similarity solution for viscous source flow on a vertical plane

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