Lubrication with Non-Newtonian Fluids

Hydrodynamic lubrication is studied for both shear thinning and viscoelastic polymer solutions. We find that elasticity, notably strong normal stresses, does not change the friction significantly for the range of parameters tested in this manuscript. Shear-thinning properties, on the other hand, do change the formation of the lubricating layer thickness and the dependence of friction on velocity relative to Newtonian fluids. A hydrodynamic model that includes shear thinning is developed and compared to experimental data. The model describes the dependence on lubrication parameters well, but underestimates the lubricating layer thickness by a constant factor of roughly 1.5. The theory allows us to define a Hersey-like number for shear-thinning fluids that describes the lubricating layer thickness as a result of the balance between normal load and viscous force. For each tested liquid it succeeds in collapsing friction measurements onto the same curve. The friction analysis for both lubrication theory and experiments then reveals that shear thinning mainly changes the layer thickness, which is the main determinant of the friction coefficient.

Figure 1

(a),(b) Schematics of the side view (a) and top view (b) of our setup, identical to Ref. [17]. The disk is pulled with velocity $U_0$ by a rheometer arm over a glass track covered in lubricant. Eight induction sensors just underneath the glass allow for the measurement of the central layer thickness $h_0$, i.e., the narrowest part of the gap in (a). The disk is slightly tilted in such a way that this point is shifted by an amount $f{R}_s$ in the x direction. (c),(d) Flow curves of the four concentrations PEO 4M (c) and of the PEO 2M and xanthan gum solutions (d) as measured by a rheometer. The dark blue lines show power-law fits $\eta (\dot{\gamma })=m{\dot{\gamma }}^{n-1}$ for shear rates $\dot{\gamma }$ from 30 to 600 s⁻¹. The corresponding values of m and n are shown in the legends. Inset in (d) shows the normal stress for the PEO $10\mathrm{g}/\mathrm{l}$ (4M), PEO $5\mathrm{g}/\mathrm{l}$ (4M), PEO $10\mathrm{g}/\mathrm{l}$ (2M), and xanthan gum concentrations measured by a rheometer; the color code is that of the main figures (c),(d).

Figure 2

Layer thickness results for (a) PEO $10\mathrm{g}/\mathrm{l}$ (4M), (b) PEO $10\mathrm{g}/\mathrm{l}$ (2M), and (c) xanthan gum $5\mathrm{g}/\mathrm{l}$. The forces next to the data denote the load on the disk and the error bars display the 5-μm measurement error. The stresses in square brackets denote the mean normal stress present underneath the disk. Despite their difference in viscoelasticity, all three liquids show similar response on layer thickness when velocity and load is varied. For the point marked with a brown box in (a) the lift due to normal stresses is estimated in the main text.

Figure 3

(a) Modeled Stribeck curves (representing the relation between Hersey number and friction coefficient) for several values of n as calculated by Eq. (6). (b) Lubricating layer thickness as function of Hersey number as calculated by Eq. (4). For both plots, the Hersey number as defined in Eq. (5) is used and the values $R_s=1.95\mathrm{cm}$ and $R=1.00\mathrm{m}$ are used to reflect the experimental situation. Data points represent values calculated by the finite-element implementation, and are connected by a spline for visual guidance.

Figure 4

(a)–(c) Layer thickness as determined by experiments (symbols) and model outcomes multiplied by constant C (solid lines) for (a) PEO 7.5 g/l (4M), (b) PEO 5.0 g/l (4M), and (c) PEO 10.0 g/l (2M). The forces next to the data denote the load on the disk. The stresses in square brackets denote the mean normal stress present underneath the disk. (d) Fitted values of C for all tested liquids.

Figure 5

Friction coefficient $\mu$ as a function of Hersey number as defined in Eq. (5), for PEO $2.5\mathrm{g}/\mathrm{l}$ (4M), PEO $5.0\mathrm{g}/\mathrm{l}$ (4M), PEO $7.5\mathrm{g}/\mathrm{l}$ (4M), and PEO $10.0\mathrm{g}/\mathrm{l}$ (2M). (A plot of the other liquids can be found in Fig. 9 in [21] and in the pp1.) Colors represent series of measurements with different normal force, and are equivalent to the colors used in Fig. 4. For each liquid all data points fall onto a single curve, highlighting the usefulness of the Hersey number as defined in Eq. (5).

Figure 6

(a) The flow curve of a shear-thinning liquid [PEO $10.0\mathrm{g}/\mathrm{l}$ (2M)], a Newtonian liquid [polydimethylsiloxane (PDMS) oil], and a shear-thickening liquid (cornstarch in water) as measured by a rheometer. The dark blue lines show power-law fits for shear rates between $30$ and $600{\mathrm{s}}^{-1}$, of which the corresponding parameters m and n are displayed in the legend. (b) Layer thickness as a function of velocity as measured in the sliding setup for the same three liquids. The dashed lines visualize the trend. For the shear-thinning and Newtonian liquid, this trend is sublinear, whereas the shear-thickening liquid displays a superlinear trend. The normal forces applied on the disk are $0.27\mathrm{N}$ for the PEO, $0.41\mathrm{N}$ for the oil, and $0.23\mathrm{N}$ for the cornstarch.

Figure 7

(a) Experimental friction coefficient $\mu$ as function of central layer thickness. The values of the dashed line have been subtracted for all data points to compensate for noise friction, as further explained in the text. Two theoretical curves ($n=0.25$ and $n=1.00$) coming from the model are plotted for the sake of comparison. The data points of the experiments with shear-thinning fluids lie above those of the Newtonian fluids, which is in agreement with the prediction of the model. (b) Model predictions of the friction coefficient as a function of layer thickness for several values of n. The shaded areas in both figures display the narrow range that all results fall into. The values $R_s=1.95\mathrm{cm}$ and $R=1.00\mathrm{m}$, representing the experimental setup, are used in Eqs. (4) and (6) to obtain the model results.

Figure 8

Dimensionless functions (a) ${\mathrm{\Lambda }}_n({\tilde{R}}_s)$ and (b) $K_n({\tilde{R}}_s)$ as defined in Eqs. (A13a) and (A13b) for various values of power-law index n. The points denote values calculated by the finite-element simulation and are connected by a spline curve. The blue points correspond to $n=1$ and are equivalent to the curves in Fig. A1 in Ref. [17], which were calculated by Mathematica.

Figure 9

Friction coefficient $\mu$ as a function of Hersey number as defined in Eq. (5), for PEO $10.0\mathrm{g}/\mathrm{l}$ (4M) and xanthan $5\mathrm{g}/\mathrm{l}$ (the absent liquids in Fig. 5 in the main text). Colors represent series of measurements with different normal force, and are equivalent to the colors used in Fig. 4 in the main text. For PEO $10\mathrm{g}/\mathrm{l}$ (4M) the data points do not exactly align, which is ascribed to the large amount of drag friction the disk experiences at high velocity when ploughing through the thick yield-stress-exhibiting liquid.