End effects in rotational viscometry I. No-slip shear-thinning samples in the Z40 DIN sensor

Abstract

Neglect of end effects in Couette rotational viscometry introduces a 10–30% error in the estimate of shear stress at the spindle surface. Actual deviations depend on the shear-thinning level of a given sample. We tackle the end effect for the standard sensor Z40 DIN according to the ISO 3219 by solving the related 2D boundary-value problem for a class of shear-thinning viscosity functions. The pseudosimilarity method of treating the primary data leaves an error of about 0.5% in shear stresses. Further reduction in the errors needs a full numerical simulation for each point of the primary data based on a suitable wide-range representation of the viscosity function. To support a high accuracy of torque calibrations, the effect of inertia on torque for Newtonian liquids in standard sensor Z40 DIN at Re < 500 is calculated using the FLUENT 6.2 commercial software.

List of symbols

C :

total torque, Pa m³

c _L :

correction on end effects, c _L  = T / T _L , see also Eq. 2

D  = d / R :

normalized distance between a rim and an axial border, Fig. 1

E = η _∞  / η ₀ :

viscosity ratio in the normalized Cross model, Eq. 22

G = |∇ V| = [(∂ _X V)² + (∂ _Z V)²]^1/2 :

normalized module of velocity gradient

h :

gap thickness, m

H = h/R :

normalized gap thickness, Fig. 1

K :

consistency coefficient for the power-law viscosity function, Pa sⁿ

k _H :

estimate of the transient region length within the gap, Fig. 1

l :

length of the gap between coaxial cylinders, m

L = l / R :

normalized length of the gap, Fig. 1

n :

global flow index for the power-law viscosity function

n′:

local flow-behavior index, Eq. 4

N :

intermediate flow index in the normalized Cross model, Eq. 22

$q = {{\mathop \gamma \limits^. }} \mathord{\left/ {\vphantom {{{\mathop \gamma \limits^. }} {{\mathop \gamma \limits^. }_{I} }}} \right. \kern-\nulldelimiterspace} {{\mathop \gamma \limits^. }_{I} }$ :

normalized shear rate

Q = R _shaft/R :

normalized shaft radius, Fig. 1

R :

radius of the spindle (inner rotating cylinder), m

Re = ΩR ²/ν :

Reynolds number, ν -kinematic viscosity, m² s⁻¹

s = σ / σ _I :

normalized shear stress

s[q] = Y[q] q :

normalized viscosity function

S = σ _R  / σ _I :

normalized shear stress at the spindle surface for ideal Couette flow

T = C / 2πσ _I R ³ :

normalized torque

T _L :

T for ideal Couette flow, T _L  = S L

T _k,p :

contributions to T (k = C , G , A; p = srf, btm); Fig. 1, Eq. 10

V = ω / Ω :

normalized velocity field

$W = \Omega \mathord{\left/ {\vphantom {\Omega {{\mathop {\gamma _{I} }\limits^. }}}} \right. \kern-\nulldelimiterspace} {{\mathop {\gamma _{I} }\limits^. }}$ :

normalized speed

X = r/R, Z = z/R :

normalized meridional coordinates

$Y = {\eta {\mathop \gamma \limits^. }_{I} } \mathord{\left/ {\vphantom {{\eta {\mathop \gamma \limits^. }_{I} } {\sigma _{I} }}} \right. \kern-\nulldelimiterspace} {\sigma _{I} }$ :

normalized viscosity

Y[q] = s[q] / q. :

normalized viscosity function

α :

angle of the spindle cone, Fig. 1

${\mathop \gamma \limits^. }$ :

viscometric shear rate, s⁻¹

${\mathop {\gamma _{I} }\limits^. }$ :

characteristic shear rate (a model parameter), s⁻¹

Γ :

spindle contour in normalized meridional plane (X, Z)

$\eta = \sigma \mathord{\left/ {\vphantom {\sigma {{\mathop \gamma \limits^. }}}} \right. \kern-\nulldelimiterspace} {{\mathop \gamma \limits^. }}$ :

viscosity, Pa s

κ = R / (h + R):

ratio of radii of the inner to outer cylinder

σ :

viscometric shear stress, Pa

σ _I :

characteristic shear stress (a model parameter), Pa

σ _R :

shear stress at the spindle surface for ideal Couette flow, Pa, Eq.  (2)

ω(r, z):

meridional field of angular speed in liquid, rad s⁻¹

Ω :

angular speed of the spindle, rad s⁻¹

[..]:

any representation of the viscosity function

Subscripts, superscripts:

 

 L, C, G, A :

contours of the sensor in the (X, Y) plane, Fig. 1: ideaL gap, Cylinder faces, Gap entrance, Axis or shAft;

 srf, btm:

dividing the overall domain at the surface and bottom parts, Fig. 1