Slow Steady Viscous Flow of Newtonian Fluids in Parallel-Disk Viscometer With Wall Slip

Abstract

The parallel-disk viscometer is a widely used instrument for measuring the rheological properties of Newtonian and non-Newtonian fluids. The torque-rotational speed data from the viscometer are converted into viscosity and other rheological properties of the fluid under test. The classical no-slip boundary condition is usually assumed at the disk-fluid interface. This leads to a simple azimuthal flow in the disk gap with the azimuthal velocity linearly varying in the radial and normal directions of the disk surfaces. For some complex fluids, the no-slip boundary condition may not be valid. The present investigation considers the flow field when the fluid under test exhibits wall slip. The equation for slow steady azimuthal flow of Newtonian fluids in parallel-disk viscometer in the presence of wall slip is solved by the method of separation of variables. Both linear and nonlinear slip functions are considered. The solution takes the form of a Bessel series. It shows that, in general, as a result of wall slip the azimuthal velocity no longer linearly varies in the radial direction. However, under conditions pertinent to parallel-disk viscometry, it approximately remains linear in the normal direction. The implications of these observations on the processing of parallel-disk viscometry data are discussed. They indicate that the method of Yoshimura and Prud’homme (1988, “Wall Slip Corrections for Couette and Parallel-Disk Viscometers,” J. Rheol., 32(1), pp. 53–67) for the determination of the wall slip function remains valid but the simple and popular procedure for converting the measured torque into rim shear stress is likely to incur significant error as a result of the nonlinearity in the radial direction.

Non-Newtonian flow in parallel-disk viscometers in the presence of wall slip

“…These results are compared against the corresponding series solution reported by Yeow et al [12]. The test case reported here is for a Newtonian fluid with viscosity η new that follows the square dependence slip law S(v slip ) = k 2 v 2 slip .…”

“…1(a) shows the plot of the fluid velocity v θ (r, z) as a function of r at the upper disk surface (U), the lower disk surface (L) and at the mid plane of the disk gap (M). The continuous curves are the results of finite element computation and the discrete points are that given by the series solution of Yeow et al [12]. These results are for h/R = 0.15 and k 2 ωR 2 /η new = 5.…”

“…The general consensus appears to be that while the analysis of Yoshimura and Prud'homme [3] is not exact it is an acceptable approximation-particularly when the h:R ratio is small. Recently, Yeow et al [12] obtained a series solution for v θ (r, z) for Newtonian fluids that exhibit wall slip. This solution shows explicitly that, for the S(v slip ) investigated, the azimuthal velocity remains approximately linear in z.…”

“…In the finite element computation a three-constant Bird-Carreau equation is used to describe the general shear thinning behavior exhibited by non-Newtonian fluids. For comparison, the same set of slip functions S(v slip ) considered by Yeow et al [12] will be investigated. In each case the variation of v θ (r, z) and the shear stress with r and z will be examined and their departure from linearity quantified.…”

“…The series-solution approach of Yeow et al [12] relies on the linearity of the slow viscous equation for v θ (r, z). It is difficult to extend their approach to allow for the non-linearity introduced by non-Newtonian fluids that exhibit wall slip.…”