The pressure loss at the entrance of a capillary tube was studied as a means of characterizing viscoelastic fluids. Measurements were made for four polymer solutions and correlated with an equation of the form
where D is the shear rate and where Hch and lambda ch are a characteristic stress and a characteristic time, respectively, determined independently from viscosity and normal stress measurements. Various theoretical analyses of capillary entrance flow are also compared.
The most fundamental experiments to measure stress behavior in viscoelastic fluids are those in which the fluid elements have undergone constant shear rate histories for a very long tine. These important viscometric flows are found in the basic viscometric geometries: capillary tube, rotational cylinder, cone and plate, etc. On the other hand, many industrially important flows are accelerative [non-viscometric] and must be analyzed mathematically using a constitutive equation or correlated empirically. The flow at the entrance of a capillary is this kind of flow. It is a particularly important one because of the wealth of data and experience from the capillary viscometer.
The measured pressure drop in a capillary viscometer is an over-all one from the upstream reservoir to the downstream reservoir or from the upstream reservoir to a free jet. The frictional losses [viscous dissipation include a loss upstream of the tube and inside the tube due to the developing flow, a loss during developed flow in the tube, and a loss at the exit. There is also a kinetic energy [Bernoulli] effect if either reservoir is not large compared with the tube. These various effects plus possible elastic effects may be summarized in terms of a mechanical energy balance as follows:
where F indicates irreversible dissipation and indicates differences between the two reservoirs. The quantity E is the elastic energy per unit mass and, in terns of the classical balance is part of the internal energy of the fluid.
For inelastic fluids [where PE/P = 0), analyses are available for the dissipation in each of the sections indicated in Eq. 1. For viscoelastic fluids, no rigorous analysis exists but if one assumes that the elastic effects may be added to the viscous effects [by no means certain], the elastic effects can be identified by difference and correlated experimentally. The analyses and procedures for the various terms will now be discussed in turn.
KINETIC ENERGY AND VISCOUS DISSIPATION EFFECTS
In a flow through a capillary from one stagnant reservoir to another, the net change in kinetic energy is zero.