Transient natural convection in a 2D enclosure with a heat source at the bottom

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The rollers used in drafting textile fibres are disk-shaped, the lower one made of steel, the upper with a metal centre and a fairly thin elastic cover. The length over which the rollers are in contact, and the pressure distribution over this length, are factors which affect their performance as drafting agents. The purpose of this paper is to show how these quantities vary with the material and thickness of the cover, the pressure between the rollers, anil the roller size. The effect of allowing slippage at the inner boundary is also considered.

The system can be considered mathematically as one of generalized plane stress in an elastic layer, with given displacement conditions on its inner boundary (the interface between metal and cover will be termed the ‘inner boundary’ of the cover) and subject to pressure by a body of given shape on its free face. The layer is sufficiently thin for the inner boundary conditions to affect the stresses in the contact zone.

The analysis of contact stresses was first carried out by Hertz (1), and quoted in Love (2). The application to the two-dimensional case is given by Thomas and Hoersch (3)—their results, which are for plane strain, may be converted by the usual modification of the Poisson's ratio to those of generalized plane stress. These analyses, however, only hold if the contact stresses are the only effective forces over the contact zone.

The effect of the boundary conditions on the bolution for a single isolated force may be found straightforwardly by a method given in Coker and Filon (4). The displacement due to any pressure distribution over the contact zone can then be determined, and the actual pressure distribution may be found by imposing the condition of known displacement over this zone. It has been found most convenient, in practice, to determine the difference between the pressure distributions for an infinite and finite thickness; this difference can be expressed as a Fourier series, and a sufficient number of the coefficients can be found to give any desired accuracy.
Original languageEnglish
Pages (from-to)94-108
Number of pages14
JournalJournal of Theoretical and Applied Mechanics
Publication statusPublished - 1996


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