Scott Sciffer

A weak Asplund space whose dual is not weak* fragmentable

Under the assumption that there exists in the unit interval [0, 1] an uncountable set A with the property that every continuous mapping from a Baire metric space B into A is constant on some non-empty open subset of B, we construct a Banach space X such that (X∗,weak∗) belongs to Stegall’s class but (X∗,weak∗) is not fragmentable.

A phenomenological model of dynamic contact angle

The precise conditions determining dynamic angle contact behaviour are not well understood. At least three different mechanisms have been proposed previously, each predicting contact angle behaviour with some success. As in two of the three models discussed, the approach taken in this paper assumes that dynamic contact angle changes due to shear stresses modifying the force balance at the contact line. A linear phenomenological relationship is assumed between the liquid/solid surface tension and the shear stress generated by the flow. The shear stress predicted by Moffatts solution to the flow at some fixed distance from the contact line is assumed to be indicative of the shear stress experienced by the liquid/solid interface at the contact line. The force balance resulting from Youngs equation is then solved numerically to determine the dynamic contact angle. The capillary number is found to be the dominant parameter, but there is an additional parameter incorporating the effects of the solid surface. This is in line with the requirements of Dussan (1979) and the features of the other models. The resulting contact angle predictions are in good agreement with the existing models and with experimental observations at low to intermediate capillary numbers. At higher values of the capillary number the model predicts the correct trend but underestimates the dependence of contact angle on capillary number. This most likely indicates that the assumed linear relationship is non longer valid, or that the use of the classical solution to infer shear stresses breaks down.

SEPARABLE DETERMINATION OF FRECHET DIFFERENTIABILITY OF CONVEX FUNCTIONS

For a continuous convex function on an open convex subset of any Banach space a separability condition on its image under the subdifferential mapping is sufficient to guarantee the generic Frechet differentiability of the function. This gives a direct insight into the characterisation of Asplund spaces.

Locally Lipschitz functions are generically pseudo-regular on separable Banach spaces

For a locally Lipschitz function on a separable Banach space the set of points of Gâteaux differentiability is dense but not necessarily residual. However, the set of points where the upper Dini derivative and the Clarke derivative agree is residual. It follows immediately that the set of points of intermediate differentiability is also residual and the set of points where the function is Gâteaux but not strictly differentiable is of the first category.

CONTINUITY CHARACTERISATIONS OF DIFFERENTIABILITY OF LOCALLY LIPSCHITZ FUNCTIONS

Recently David Preiss contributed a remarkable theorem about the differentiability of locally Lipschitz functions on Banach spaces which have an equivalent norm differentiable away from the origin. Using his result in conjunction with Frank Clarkes non-smooth analysis for locally Lipschitz functions, continuity characterisations of differentiability can be obtained which generalise those for convex functions on Banach spaces. This result gives added information about differentiability properties of distance functions.

Hydrodynamic meniscus profiles in creeping flow

Abstract Many industrial coating processes involve drawing a thin strip or wire through a coating bath. A knowledge of the fluid dynamics at the entry is important for understanding the coating mechanism. In this note the fluid flow and meniscus profile generated by a strip being drawn vertically through a liquid surface are investigated under the assumption of creeping flow. It is known that the assumption of normal or shear stresses at the free surface and no-slip at the plate gives solutions with a force singularity on a microscopic length scale, and predicts a constant free surface velocity somewhat less than the strip velocity. It is shown that if the condition that the flow generates no normal stresses at the free surface is relaxed, then solutions of the creeping flow equations exist in which the free surface velocity decays to zero far from the contact line. The resulting normal stresses are then balanced against surface tension forces to determine the dynamic meniscus profile. Although the change of the dynamic contact angle with contact line speed remains the single biggest influence on the resulting meniscus, the normal stresses induced by the flow are a significant factor in determining the meniscus shape.

Flow fields with constant rate of strain along streamlines

In order to study the effect of strain rate on entrained bubbles it would be useful to produce a flow in which the bubbles are subjected to uniform rates of strain. However it is shown that the only steady two-dimensional or axisymmetric flows with constant components of the rate of strain tensor in streamline coordinates are the trivial cases of rigid body motion.

Concerning Differentiability Properties of Locally Lipschitz Functions

Preiss proved that on an Asplund space the Clarke subdifferential of a locally Lipschitz function can be expressed in terms of its Frechet derivatives. We extend this result for the wider class of Asplund generated spaces. Zajicek examined the relation between Frechet subdifferentiability and Frechet differentiability for a locally Lipschitz function on an Asplund space. We show that dense Frechet subdifferentiability implies generic uniform upper Dini subdifferentiability for a locally Lipschitz function on a Banach space and we characterise Asplund spaces by this property.

A Banach space where minimal weak cuscos are generically compact-valued

The space of continuous functions on the double arrow space has long been of interest in differentiability theory since many convex functions on this space are densely but not generically Gâteaux differentiable. We show that this space has the property that minimal weak* cuscos into its dual take compact values at the points of a denseGδ set.