The stability of a viscous liquid in a vertical tube containing porous material By R. A. Wooding Emmanuel College, University of Cambridge ( Communicated by Sir Geoffrey Taylor, F.B.S.—Received 3 March 1959) [Plat© 2] If a long vertical tube filled with porous material contains a viscous solution, the density of which increases with height as a result of the presence of the dissolved substance, the equili brium of the liquid is stable provided that the density gradient does not anywhere exceed the value dp _ 3-390/wc dZ gkb2 Here ac, the diffusivity of the solute through the saturated porous medium, is defined to be the quantity of solute diffusing across unit area within the porous medium per unit time under unit density gradient.
Stability of a viscous liquid in a vertical tube 121 diameter of about 1 mm, filled with liquid of density and closed at the bottom, is connected at the top to a reservoir containing a miscible liquid of density greater Kan p0. Since the system is unstable, convection currents are set up in the tube until the density gradient no longer exceeds the critical value at any point. At equilibrium, it is found that the density decreases with very nearly the critical gradient, from the value at the reservoir to the value p0 at a point some distance down the tube. Below that point, the liquid has uniform density p0.
122 R. A. Wooding would differ from D, the molecular diffusivity of the same solute in solution, and thai f the dimensionless ratio k/D would be a constant depending upon the geometry oi the porous material.
Stability of a viscous liquid in a vertical tube 123 kk > Equations (7) and (8) may be transformed into cylindrical co-ordinates ( ,<j), bz), here 6 is a typical linear dimension for the tube filled with porous material (e.g. the , idius in the case of a circular tube), and where bz = Z. For these equations, with oundary conditions (9), separable solutions can be written in the form TF(z)<D(r,0)ew<, (10) & = 0(z)0(r, <fi)e"*, (11) provided that <D(r, <fi) is a bounded function which satisfies the differential equation (Vf + a2)0> = 0 (12) nth the boundary condition 80 /dn = 0 on the tube walls. In (12), va = — - - I — 1 8r2 r ■s the two-dimensional Laplacian operator. Evidently O obeys the same differential aquation as does the displacement of a thin membrane which can vibrate in sequence of normal modes, corresponding to an infinite sequence of characteristic H Lvalues of the wave number cl. These values of are determined uniquely by the shape and dimensions of the tube cross-section, and are readily shown from (12) and uhe associated boundary conditions to be real numbers. This analogy of the vibrating tmembrane has been observed by Pellew & Southwell (194®) ^ the classical Ray- sleigh instability problem. It should be noted that, in the present analogy, the thin emembrane is permitted to vibrate only in those modes for which the average {displacement at any instant over the total area of the membrane is zero. This corresponds to the continuity requirement that the integral of w over the total cross-sectional area of the tube should vanish.
124 R. A. Wooding right-hand side of (13), and assuming that W and 6 vary as sin sz for a very long tube, ] one obtains the consistency equation / kwW „ 2 eb2o)\ a2 .
125 Stability of a viscous liquid in a vertical tube there <xn is a root of the equation for given n. When 1 the factor cos n<j> ensures iiiat tlmcontinuity requirement is satisfied, as the integral of w over the tube cross- letion is zero. The case n = 0 is also admissible from continuity considerations, as f1 1 r J 0(ct0r)dr = — Ji(a0) Jo ao ifhich is zero by (20).
126 R. A. Wooding Equation (22) may be applied to an experiment, similar to Taylor’s experiment, in which the capillary tube is replaced by a tube of greater width filled with porous material. The tube is closed at the bottom, the porous medium being saturated with a liquid of density p0, and the upper end of the tube is connected to a reservoir containing a liquid of density px such that > p0. At equilibrium, the maximum depth of penetration Zp of the upper liquid into the tube is given approximately by „ gkb* P* dp (23)* p «*(KiD)JP'Mp)D(py where the ratio k/D is a dimensionless constant depending upon the properties of the porous material.
Stability of a viscous liquid in a vertical tube 127 ^liUpon the walls of the long tube, the boundary conditions will be the non-slip and filiation conditions w = 0 and 0 (28) JjBBectively, djdn signifying the normal derivative. However, it is not possible to mtisfy all the boundary conditions upon the ends of the tube, using a finite number ■# separable solutions for w and #. Adopting the argument used by Hales (1937), one 3Ly assume, by analogy with the principle of Saint Venant, that the effect of the ends ^inappreciable at distances of more than a few tube diameters, and maybe neglected Jr a long tube.
128 R. A. Wooding at r = 1, and two further conditions are implied by the fact that the solution must be I finite at r = 0. Equations (30) and (31) constitute a characteristic-value problem for A.
Stability of a viscous liquid in a vertical tube 129 4. An experiment on the stability of a liquid in a vertical tube CONTAINING POROUS MATERIAL t In order to compare the newly calculated Rayleigh number for a liquid in a vertical J DUbe filled with a porous medium with Taylor’s (1954) value for a liquid alone, an k experiment based upon Taylor’s original experiment was set up. Two vertical tubes, 1 each closed at the bottom, were connected to a common reservoir. The first was ■| a capillary tube of diameter about 1 mm, and the second was a tube of diameter C: about 1 cm filled with glass spheres having diameters ranging from 0-19 to 0-20 mm. if The greater diameter chosen for the second tube ensured that comparable equili brium density gradients were obtained in the two tubes, in spite of the considerable ^resistance to flow due to the packed spheres.