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Icicle spacing Thomas E. Lockhart and Tim Kline Two theories 1,2 have been proposed to account for the spacing between adjacent icicles. Both assume that the icicle spacing is determined by the spacing of liquid water drops that then freeze. I. Figure out the spacing by balancing the drop’s weight with surface tension.1 This assumes that the drops individually achieve a certain size (and therefore, shape) independently of each other. The spacing is then the center-to-center spacing of adjacent drops, which is equivalent to the diameter of each drop measured at the substrate. Makkonen and Fujii 1 calculate the spacing for an assumed drop shape to be 2.13 cm, which generally matched their experimental values. II. Attribute the spacing to a Rayleigh-Taylor instability2, which requires solving the fluid dynamics equations (Navier-Stokes equations) for a film of liquid water. The Rayleigh- Taylor instability occurs anytime a heavier fluid is located above a lighter fluid. The heavier fluid falls down into the lighter fluid, with a characteristic rippling of the interface between the fluids. One particular wavelength of ripple grows faster than the others, and gives rise to the drop spacing. This approach incorporates the interaction between the adjacent drops. de Bruyn 2 reports the spacing wavelength to be 2.47 cm, in slight disagreement with method I. We won’t be pursuing the mathematical development of this model due to its greater complexity (5 coupled partial differential equations). However, we should note that this analysis incorporates the interactions between the drops which are ignored in method I. Nothing prevents the drop spacing from exceeding these values, because water may accumulate on a surface at points determined by the surface itself or any channeling of water leading to the icicle-producing surface, for example. Our data indicates that icicle spacings are typically in the range from 1.6 cm to 2.3 cm, with even larger spacings occurring in decreasing frequency. We can account for this spacing range using method I alone, although we adopt a slightly different procedure.3 Rather than using force balance, mechanical equilibrium for the drop is expressed in terms of pressure, which leads to the Laplace form for pressure in a curved surface. The Laplace pressure occurs in any situation where a tensile force is balancing out a pressure, such as in our drops or in a balloon being blown up. This makes a better estimate of the spacing by allowing the drop shape and size change as each drop grows larger, so that the icicle spacing is determined by the drop profile at the time of freezing. For the largest drops, the surface curvature has to be large (small radius of curvature) in order to generate a bigger Laplace pressure. Counter to one’s intuition, the drops get closer together as their individual volumes increase. Icicle ripples A different instability gives rise to a set of ripples running along the length of any one icicle.4,5 These ripples have a very regular spacing (about 1 cm) which doesn’t change as the icicle ages, but the amplitude of the ripples can change. This rippling is again due to competition between two processes. Any little initial disturbance of a liquid film running down the outside of the icicle will cause a bulge and an accompanying concavity. The bulge’s convex surface will cool more quickly and is able to more rapidly dissipate the latent heat of fusion upon freezing than the concave region (this is called a Laplace instability). So water is more likely to freeze on the convex sections than on the concave ones, which means the bulges build up more rapidly than the spaces between them. The effect that counterbalances this is the flow of the liquid film itself which tries to even out the temperature differences caused by the convexities and concavities. The two effects come to equilibrium at a particular spacing between bulges: 0.8 cm or so, depending on thickness of the liquid water layer.4 Details of the icicle spacing problem: Method I: Drop shape determines spacing Water drops hanging from a surface are called “pendant drops”. Their shape is determined by a competition between the weight of the drop causing it to fall and the surface tension that resists changes in the surface area of the drop. According to Makkonen and Fujii,1 a pendant drop is supported by the force at it rim due to the vertical component of surface tension: (1) F,v = 2 r sin r where r is the radius of the dr drop’s base (where it contacts the substrate), is the surface dz ds tension and is shown in the accompanying figure 1. Figure 1: the pendant drop and its coordinate system Following the same model, each section of thickness dr z on the rim of the drop feels a force of gravity given by r dr (2) Fg - ρg 2π r z dr r where is the density and g is the acceleration due to gravity. Makkonen and Fujii then set F,v = Fg, and take a derivative of both sides with respect to r, which gives (3) 2 d(r sin )/dr = -2 g rz or In figure 1, note that sin = dz/ds and tan = dz/dr. If we make the same small angle approximation as Makkonen and Fujii so that sin tan (i.e., the very initial stages when the drops are still almost flat), we end up with (4) d(r dz/dr)/dr + grz = 0. This can be rewritten in terms of the following dimensionless variables: γ (5) λ ; r/; z/ ρg is often called the capillary length and sets the scale over which capillary forces are important. For water at 0°C, = 75.6 dyn/cm, = 1.00 g/cm3, and g = 980 cm/s2, so = = 0.2777 cm. Equation 4 then becomes d 2 1 d (6) 0 d 2 d which is the zero-order Bessel equation of the first kind. Boundary conditions for this problem are: dη (7) 0 at ξ 0 and at ξ α dξ where is the dimensionless radius of the drop’s rim. So the small angle approximation leads to a drop in the shape of a Bessel function. To find , you have to find the first nonzero value for which Bessel function = J0() has an extremum (i.e, a minimum in this case). This happens at = = 3.83 (Thanks, Lyle!!!). The drop spacing will be twice the drop radius, so S = 2 = 2.13 cm. Not surprisingly, this isn’t what we found in our data on icicles in nature. The reason is simple: the small angle approximation effectively eliminates any effect of curvature because the drop is considered nearly flat. Our data suggests that this is not the case with icicles formed in nature. The discrepancy can be resolved by altering the procedure to include more fully the role played by surface tension and surface curvature. To do this, we have to follow the procedure outlined by Boucher and Evans.3 Rather than explicitly consider forces, we will look at pressures. The mechanical equilibrium condition for the interface between two fluids is given by the Laplace formula: (8) p = (1/R1 + 1/R2) where p is the pressure difference between the two fluid, is the surface tension and the R’s are the principal radii of curvature for the interface. Using the coordinate system shown in Figure 2 (note that z is reversed from figure 1), we note that R will be a function of z. If we treat the air pressure as negligible compared to the liquid pressure, p = p, where p is the liquid’s pressure. In any fluid, the pressure at a depth is caused by the weight of the fluid above it, so p = g(h - z), where we have set the density of the air below the drop equal to zero and h is the maximum displacement of the drop below the substrate. z The Laplace equation h becomes: dr (9) g(h-z)(1/R1 + 1/R2) dz R1 and R2 are functions of x ds and z, so this equation determines the actual drop r Figure 2: the pendant drop shape. At any point (r, z) on coordinate system for the the drop’s surface, the tangent Laplace calculation to the surface makes angle with horizontal. We define s as the total distance along the profile from (0,0) to (r,z). Therefore dz/dr = tan and cos = dr/ds. The radii of curvature at that point are related to :3 (10) R1 = dr/d(sin) = dr/(cos d), R2 = r/sin Put all this together in equation 9 and we get (11) g(h-z)(d/ds + (sin )/r) Solving this for d/ds gives: (12) d/ds = g(h-z)/ - (sin )/r In addition, there are some boundary conditions: s = 0 and = 0 at (0,0). In addition, dr/ds = cos and dz/ds = sin . Now in principle, all we have to do is integrate this for each point on the profile (each s value) and we’d know the profile. The trouble is there is no closed form solution and the integral must be done numerically. Rather repeat the calculations, we’ll defer to Boucher and Evans3, who determine a number of profiles, corresponding to different drop volumes. Their results are scaled by a factor a such that a2 = 2/g, which means a 2 λ in the analysis of Makkonen and Fujii1. In particular, Boucher and Evans generated a table for the radius, depth and volume of drops below an infinite plane. One of the primary results of their numerical calculation is the existence of a maximum possible drop volume. The radius of such a drop is 2.2766 a, which corresponds to a drop spacing (assumed to be equal to the drop diameter) of 1.79 cm at 0 °C. The other interesting feature is that for smaller volume drops, the curvature is such that the drop radius at its base is larger than that of the maximum volume drop, with a maximum radius of 2.71 a, corresponding to a diameter of 2.13 cm. In other words, small volume drops with very small h values give a result exactly the same as Makkonen and Fujii. This isn’t surprising, because the small angle approximation they used is the same are requiring a small (and hence a tiny h value) in the Boucher and Evan calculation. Presumably, one could get exactly the same result by using equation 3 rather than the approximations that led to equation 4. What is perhaps a bit disturbing is that nothing in this calculation takes the interaction between adjacent drops into account, and yet the spacing can be maintained over quite large distances. Something has to enforce this spacing between drops and yet the model calculates the size of each drop individually, independent of all the others in the array of drops. A moment’s reflection may give a clue: this is analogous to the spacing of birds perching on a wire. Each bird stays just out of reach of the neighbors, so a minimum spacing is enforced just by requiring the birds to all coexist on the same wire. Similarly, the water drops get as big as they can before freezing, running out of water, or falling off, resulting in a more or less uniform spacing but not less than the minimum spacing required by the drop size. References: 1. Makkonen, Lasse and Fujii, Yutaka “Spacing of icicles”, Cold Regions Science and Technology 21, 317-322 (1993). 2. de Bruyn, John R., “On the formation of periodic arrays of icicles”, Cold Regions Science and Technology 25, 225-229 (1997). 3. Boucher, E.A. and Evans, M.J.B., “Pendent drop profiles and related capillary phenomena”, Proceedings of the Royal Society London A 346, 349-374 (1975). 4. Ogawa, Naohisa and Furukawa, Yoshinori, “Surface instability of icicles”, Physical Review E 66, 041202 (2002) 5. Ueno, K., “Pattern formation in crystal growth under parabolic shear flow”, Physical Review E 68, 021063 (2003)

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posted: | 5/22/2011 |

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