AIAA JOURNAL Vol. 44, No. 4, April 2006 Study of Wetting in an Asymmetrical Vane–Wall Gap in Propellant Tanks Yongkang Chen ∗ and Steven H. Collicott † Purdue University, West Lafayette, Indiana 47907-2023 Wetting of an asymmetrical vane–wall gap geometry is investigated. The geometry is common in vane-type liquid propellant management devices. Critical wetting conditions are determined by applying the method by Concus and Finn (Concus, P., and Finn, R., “On Capillary Free Surfaces in the Absence of Gravity,” Acta Mathematica, Vol. 132, 1974, pp. 177–198). The critical wetting condition is expressed in terms of critical contact angle as a function of gap size, with the vane thickness and obliquity angle as relevant parameters. It is found that the increase of both the vane thickness and the obliquity angle improves the critical wetting conditions. The analytical results are confirmed with Surface Evolver numerical computations, which also provide graphical descriptions of capillary surfaces. Results from drop tower experiments confirm the analysis and also reveal that the advance rate of the meniscus tip decreases with the gap size when other parameters are fixed. Nomenclature L = meniscus tip location, mm R 0 = radius of arc Ŵ 0 of C-singular solution R γ = radius of arc Ŵ ˜ r = inner radius of the test cell, mm t = time, s α = vane obliquity angle, deg Ŵ = name and length of arc Ŵ Ŵ cr = name and length of critical arc Ŵ Ŵ 0 = name and length of arc Ŵ of C-singular solution γ = contact angle, deg γ cr = critical contact angle, deg δ, ˜ δ = dimensionless, dimensional, mm, gap size δ cr = critical gap size ǫ, ˜ ǫ = dimensionless, dimensional, mm, vane thickness θ = one-half of angle spanned by arc Ŵ  = perimeter of   ∗ ,  0 = name and length of subarcs of   = functional in method by Concus and Finn 3  = name and area of problem domain  ∗ ,  0 = name and area of subdomain of  Introduction P ERFORMANCE of vane-type propellant management devices (PMD) in weightlessness is vital to positioning, controlling, and transporting the liquid propellant in fuel tanks for successful operation of spacecraft. Proper performance, such as reorienting the propellant following a maneuver, is enabled by a critical contact angle condition. For contact angles below this critical value, a finite height single-valued capillary surface fails to exist in a cylindrical container. 1 As a result, a capillary-driven flow will ensue that is im- portant to successful PMD operation. Similarly, for a given contact Presented as Paper 2003-4893 at the AIAA/ASME/SAE/ASEE 39th Joint Propulsion Conference, Huntsville, AL, 20–23 July 2003; received 24 January 2005; revision received 27 October 2005; accepted for publica- tion 3 November 2005. Copyright c  2005 by Purdue University. Published by the American Institute of Aeronautics and Astronautics, Inc., with per- mission. Copies of this paper may be made for personal or internal use, on condition that the copier pay the $10.00 per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923; include the code 0001-1452/06 $10.00 in correspondence with the CCC. ∗ Postdoctoral Fellow, School of Aeronautics and Astronautics; currently Postdoctoral Fellow, P.O. Box 751-ME, Mechanical and Materials Engi- neering Department, Portland State University, Portland, OR 97207-0751. Member AIAA. † Associate Professor, School of Aeronautics and Astronautics. Associate Fellow AIAA. angle, a gap between the vane and the wall must be sufficiently small to create the necessary wicking flows. In a previous study, 2 critical wetting conditions have been deter- mined for the symmetric configurations in which the vane is perpen- dicular to the tank wall, thus, with α = 0, in Fig. 1. Figure 1 shows a comparison of the menisci in a corner and in the vane–wall gap geometry. The reason for making such a comparison is that the ex- istence of the gap can substantially change the properties related to capillarity, as presented by Chen and Collicott. 2 The critical wetting conditions are determined by applying the method by Concus and Finn (see Ref. 2 and references therein for more details). There are several geometric parameters involved in the problem such as the gap size, the vane thickness, and the contact angle. The effects of these parameters have been examined by allowing only one param- eter to vary at a time. The results were presented in the form of the critical contact angle or the critical gap size. Most spacecraft PMD vanes are flexible and likely never perfectly perpendicular to the wall, and, therefore, the symmetric configura- tion is only an ideal condition. Thus, it is important to study asym- metric configurations in which the vane is not perpendicular to the tank wall to determine how a tilt of the vane affects the wicking power of the geometry. Consequently, there is an extra geometric parameter compared to the symmetric configuration, that is, the vane obliquity angle α as shown in Fig. 1. In the following text, the effects of the relevant geometric parameters on the critical wet- ting conditions are determined. This is followed by Surface Evolver computations to determine the critical contact angle to confirm the analysis and to provide graphic descriptions of the capillary sur- faces. In the end, results from drop-tower experiments that confirm the analysis and show the wicking rates are reported. Analysis The foundations for the analysis are described by several authors. 3,4 The notation and methodology from Concus and Finn 3 are followed in this paper. Consider a cross section of a cylindrical container with its bottom completely covered with liquid, as shown in Fig. 2.  denotes the entire domain and  denotes the boundary of . The domain  of type 1 has an interior corner, whereas in the type 2 domain a reentrant corner  exists on the boundary of  with an angle greater than 180 deg measured from inside . Assume that the free surface intersects the cross section along an arbitrary circular arc Ŵ of radius R γ given by R γ = / cos γ (1) Ŵ must meet the boundary wall at the contact angle γ . (There are two such arcs in the type 2 domain of Fig. 2.) In the presence of a reentrant corner , the angle at which Ŵ meets the edge of the 859