Foam application from a closed system – a study of machine and foam parameters
An attempt has been made to gain a greater insight into the interaction between foam and a moving textile substrate. The effects of changing wet pick–up, fabric velocity, liquid viscosity, foam density and mode of application on penetration have been studied. Application from a closed system makes it possible to apply an exact amount of liquid, and reflectance measurements revealed that between 10 and 60% wet pick–up the actual level of wet pick–up is of major importance. The amount of liquid available determines the likelihood of capillary transport. The results obtained in terms of fabric speed and foam density still leave questions to be answered on the mechanisms of foam flow and degradation in the textile.
The flow properties of foam
The rheological behaviour of foam, prepared from an aqueous solution with lauryl sulphate as surfactant, has been measured with a viscometer as a function of the blow ratio, bubble diameter distribution and concentration of the thickener used. The influence of wall slip and of hysteresis effects on the rheological behaviour have been determined. Calculations and measurements pertaining to the pressure drop in pipes have been performed, in which the foam was considered as a compressible pseudoplastic whose rheology depended on blow ratio. The distribution of the foam passing through a closed distribution unit has been studied. This distribution unit consisted of a pipe and a cross-head distribution slit. Complementary measurements of the flow of foam through narrow slits have been made. The design of a closed foam distribution unit is discussed. The consequences of the choice of several design parameters are examined.
Decidability of some intuitionistic predicate theories
Suppose T is a first order intuitionistic theory (more precisely, a theory of Heyting's predicate calculus, e.g., abelian groups, one unary function, dense linear order, etc.) presented to us by a set of axioms (denoted also by) T. Question. Is T decidable? One knows that if the classical counterpart of T (i.e., take the same axioms but with the classical predicate calculus as the underlying logic) is not decidable, then T cannot be decidable. The problem remains for theories whose classical counterpart is decidable. In [8], sufficient conditions for undecidability were given, and several intuitionistic theories such as abelian groups and unary functions (both with decidable equality) were shown to be undecidable. In this note we show decidability results (see Theorems 1 and 2 below), and compare these results with the undecidability results previously obtained. The method we use is the reduction-method, described fully in [12] and widely applied in [3], which is applied here roughly as follows: Let T be a given theory of Heyting's predicate calculus. We know that Heyting's predicate calculus is complete for the Kripke-model type of semantics. We choose a class M of Kripke models for which T is complete, i.e., all axioms of T are valid in any model of the class and whenever φ is not a theorem of T, φ is false in some model of M.