Available online at www.sciencedirect.com International Journal of Non-Linear Mechanics 39 (2004) 1013–1026 Axisymmetric spreading of a thin liquid drop with suction or blowing at the horizontal base D.P. Mason a; b , E. Momoniat a; b; * a Centre for Dierential Equations, Continuum Mechanics and Applications, Johannesburg, South Africa b School of Computational and Applied Mathematics, University of the Witwatersrand, Private Bag 3, Wits 2050, Johannesburg, South Africa Received 20 November 2002; accepted 14 May 2003 Abstract The axisymmetric spreading under gravity of a thin liquid drop on a horizontal plane with suction or blowing of uid at the base is considered. The thickness of the liquid drop satises a non-linear diusion equation with a source term. For a group invariant solution to exist the normal component of the uid velocity at the base, vn, must satisfy a rst-order quasi-linear partial dierential equation. The general form of the group invariant solution for the thickness of the liquid drop and for vn is derived. Two particular solutions are considered. Each solution depends essentially on only one parameter which can be varied to yield a range of models. In the rst solution, vn is proportional to the thickness of the liquid drop. The base radius always increases even for suction. In the second solution, vn is proportional to the gradient of the thickness of the liquid drop. The thickness of the liquid drop always decreases even for blowing. A special case is the solution with no spreading or contraction at the base which may have application in ink-jet printing. ? 2003 Elsevier Ltd. All rights reserved. Keywords: Asymmetric thin liquid drop; Non-linear diusion equation; Lie point symmetries; Group invariant solution 1. Introduction There is now a large literature on the spreading of a thin uid lm on a horizontal plane. Comprehensive reviews have been given by Oron et al. [1] and by Davis [2]. Davis and Hocking [3,4] have considered the spreading of a viscous liquid on a porous base. As the authors observe this has application to the spread- ing of water on a textile fabric or on the surface of a powder, in ink-jet printing and in the analysis of * Corresponding author. School of Computational and Applied Mathematics, University of the Witwatersrand, Private Bag 3, Wits 2050, Johannesburg, South Africa. E-mail address: ebrahim@cam.wits.ac.za (E. Momoniat). runo of rainwater over soils. In this paper we will consider the spreading of an axisymmetric thin liquid drop under gravity on a xed horizontal base. Surface tension will be neglected. However, at the base we will allow for suction of uid from the liquid drop or blowing of uid into the liquid drop. Recently, Momoniat et al. [5] obtained the group-invariant solution for the axisymmetric spread- ing of a thin liquid drop on an impermeable horizontal base. The solution was obtained by considering a linear combination of the Lie point symmetries of the non-linear diusion equation for the surface pro- le of the liquid drop. The normal component of the uid velocity at the base vanished. We will consider the case in which the normal component of the uid 0020-7462/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0020-7462(03)00093-3