Rivulet flow of generalized Newtonian fluids

Steady unidirectional gravity-driven flow of a uniform thin rivulet (i.e., a rivulet with small transverse aspect ratio) of a generalized Newtonian fluid down a vertical planar substrate is considered. The parametric solution for any generalized Newtonian fluid whose viscosity can be expressed as a function of the shear rate and the explicit solution for any generalized Newtonian fluid whose viscosity can be expressed as a function of the extra stress are obtained. These general solutions are used to describe rivulet flow of Carreau and Ellis fluids, highlighting the similarities and differences between the behavior of these two fluids. In addition, the general behavior of rivulets of nearly Newtonian fluids and of rivulets with small or large prescribed flux as well as the behavior of rivulets of strongly shear-thinning Carreau and Ellis fluids are also described. It is found that whereas the monotonic dependence of the viscosity of a Carreau fluid on its three nondimensional parameters and of an Ellis fluid on two of its three nondimensional parameters leads to the expected dependence of the behavior of the rivulet on these parameters (namely, that increasing the viscosity of the fluid leads to a larger rivulet), the nonmonotonic dependence of the viscosity of an Ellis fluid on the nondimensional parameter that measures the degree of shear thinning leads to a more complicated dependence of the behavior of the rivulet on this parameter. In particular, it is also found that when the maximum extra stress in the rivulet is sufficiently large a rivulet of an Ellis fluid in the strongly shear-thinning limit in which this parameter becomes large comprises two regions with different viscosities. In the general case of nonzero viscosity in the limit of large extra stress the two regions have different constant viscosities, whereas in the special case of zero viscosity in the limit of large extra stress one region has constant viscosity and the other has a nonconstant power-law viscosity, leading to a pluglike velocity profile with large magnitude in the narrow central region of the rivulet.

Figure 1

Steady unidirectional gravity-driven flow of a uniform thin rivulet of a generalized Newtonian fluid with constant semiwidth $a$, contact angle $\beta$, and volume flux $Q$ down a vertical planar substrate.

Figure 2

Plots of the semiwidth $a$ as a function of the flux $Q$ for a Carreau fluid with $\beta =1$ when (a) ${\mu }_{\infty }=1/10$ and $N=1/2$ for $\lambda =0$, 1, $\cdots$, 5, 10, 20, $\cdots$, 100, (b) $N=1/2$ and $\lambda =1$ for ${\mu }_{\infty }=0$, $1/4$, $\cdots$, 1, and (c) ${\mu }_{\infty }=1/10$ and $\lambda =1$ for $N=0$, $1/4$, $\cdots$, 1. In (a) the dashed curve shows the leading-order asymptotic solution in the limit $\lambda \to \infty$ given by (32), namely $a={(105{\mu }_{\infty }\overline{Q}/4{\beta }^3)}^{1/4}={(21\overline{Q}/8)}^{1/4}$.

Figure 3

Plots of the flux $Q$ as a function of $\lambda$ for a Carreau fluid with $a=1$ and $\beta =1$ when (a) $N=1/2$ for ${\mu }_{\infty }=0$, $1/4$, $\cdots$, 1 and (b) ${\mu }_{\infty }=1/10$ for $N=0$, $1/4$, $\cdots$, 1. The dashed lines show the asymptotic values in the limit $\lambda \to \infty$ given by (32), namely $Q=4{\beta }^3{a}^4/105{\mu }_{\infty }$.

Figure 4

Plots of the flux $Q$ as a function of ${\mu }_{\infty }$ for a Carreau fluid with $a=1$ and $\beta =1$ when (a) $N=1/2$ for $\lambda =0$, 1, $\cdots$, 5, 10, 20, $\cdots$, 100 and (b) $\lambda =2$ for $N=0$, $1/4$, $\cdots$, 1. In (a) the dashed curve shows the leading-order asymptotic solution in the limit $\lambda \to \infty$ given by (32), namely $Q=4{\beta }^3{a}^4/105{\mu }_{\infty }=4/105{\mu }_{\infty }$.

Figure 5

Plots of the flux $Q$ as a function of $N$ for a Carreau fluid with $a=1$ and $\beta =1$ when (a) ${\mu }_{\infty }=1/10$ for $\lambda =0$, 1, $\cdots$, 5, 10, 20, $\cdots$, 100, 200, $...$, 1000 and (b) $\lambda =2$ for ${\mu }_{\infty }=0$, $1/4$, $\cdots$, 1. In (a) the dashed line shows the asymptotic value in the limit $\lambda \to \infty$ given by (32), namely $Q=4{\beta }^3{a}^4/105{\mu }_{\infty }=8/21\simeq 0.3810$.

Figure 6

Plots of the viscosity $\mu$ of an Ellis fluid given by (6) scaled according to (8) and (9) as a function of the shear rate $q$ when (a) ${\mu }_{\infty }=1/10$ and $\alpha =3$ for ${\tau }_{\mathrm{av}}=1/10$, $1/5$, $\cdots$, $9/10$, 1, 2, $\cdots$, 10, 15, 20, $\cdots$, 50, (b) $\alpha =3$ and ${\tau }_{\mathrm{av}}=1$ for ${\mu }_{\infty }=0$, $1/10$, $\cdots$, 1, and (c) ${\mu }_{\infty }=1/10$ and ${\tau }_{\mathrm{av}}=1$ for $\alpha =1$, 2, $\cdots$, 10. In (a) the dotted and dashed lines show the asymptotic values in the limits ${\tau }_{\mathrm{av}}\to 0$ and ${\tau }_{\mathrm{av}}\to \infty$, $\mu ={\mu }_{\infty }=1/10$ and $\mu =1$, respectively.

Figure 7

Plots of the semiwidth $a$ as a function of the flux $Q$ for an Ellis fluid with $\beta =1$ when (a) ${\mu }_{\infty }=1/10$ and $\alpha =2$ for ${\tau }_{\mathrm{av}}=1/5$, $2/5$, $3/5$, $4/5$, 1, 2, 3, 4, 5, 10, 15, 20, (b) $\alpha =2$ and ${\tau }_{\mathrm{av}}=1$ for ${\mu }_{\infty }=0$, $1/4$, $\cdots$, 1, and (c) ${\mu }_{\infty }=1/10$ and ${\tau }_{\mathrm{av}}=1$ for $\alpha =1$, 2, $\cdots$, 5. In (a) the dotted and dashed curves show the leading-order asymptotic solutions in the limits ${\tau }_{\mathrm{av}}\to 0$ and ${\tau }_{\mathrm{av}}\to \infty$ given by (32) and (20), respectively, namely $a={(105{\mu }_{\infty }\overline{Q}/4{\beta }^3)}^{1/4}={(21\overline{Q}/8)}^{1/4}$ and $a={(105\overline{Q}/4{\beta }^3)}^{1/4}={(105\overline{Q}/4)}^{1/4}$.

Figure 8

Plots of the flux $Q$ as a function of ${\tau }_{\mathrm{av}}$ for an Ellis fluid with $a=1$ and $\beta =1$ when (a) $\alpha =2$ for ${\mu }_{\infty }=0$, $1/4$, $\cdots$, 1 and (b) ${\mu }_{\infty }=1/10$ for $\alpha =1$, $\cdots$, 5. In (b) the dashed line shows the asymptotic value in the limit ${\tau }_{\mathrm{av}}\to \infty$ given by (20), namely $Q=4{\beta }^3{a}^4/105=4/105\simeq 0.0381$.

Figure 9

Plots of the flux $Q$ as a function of ${\mu }_{\infty }$ for an Ellis fluid with $a=1$ and $\beta =1$ when (a) $\alpha =2$ for ${\tau }_{\mathrm{av}}=1/20$, $1/10$, $\cdots$, $1/4$, $1/2$, 1, and 5, (b) ${\tau }_{\mathrm{av}}=1>{\tau }_{\mathrm{m}}=1/2$ for $\alpha =1$, $\cdots$, 5, and (c) ${\tau }_{\mathrm{av}}=1/10<{\tau }_{\mathrm{m}}=1/2$ for $\alpha =1$, $\cdots$, 5. In (a) the dotted and dashed curves show the leading-order asymptotic solution in the limit ${\tau }_{\mathrm{av}}\to 0$ given by (32), namely $Q=4{\beta }^3{a}^4/105{\mu }_{\infty }=4/105{\mu }_{\infty }$, and the leading-order asymptotic value in the limit ${\tau }_{\mathrm{av}}\to \infty$ given by (20), namely $Q=4{\beta }^3{a}^4/105=4/105\simeq 0.0381$, respectively.

Figure 10

Plots of the flux $Q$ as a function of $\alpha$ ($\ge 1$) for an Ellis fluid with $a=1$ and $\beta =1$ when (a) ${\mu }_{\infty }=1/10$ for ${\tau }_{\mathrm{av}}=1/20$, $1/10$, $\cdots$, $1/4$, 1, and 5, (b) ${\tau }_{\mathrm{av}}=1>{\tau }_{\mathrm{m}}=1/2$ for ${\mu }_{\infty }=0$, $1/4$, $\cdots$, 1, and (c) ${\tau }_{\mathrm{av}}=1/10<{\tau }_{\mathrm{m}}=1/2$ for ${\mu }_{\infty }=0$, $1/4$, $\cdots$, 1. In (a) and (c) the dashed lines show the asymptotic values in the limit $\alpha \to \infty$ given by (20) and (32), respectively, namely $Q=4{\beta }^3{a}^4/105=4/105\simeq 0.0381$ and $Q=4{\beta }^3{a}^4/105{\mu }_{\infty }=4/105{\mu }_{\infty }$.

Figure 11

Plots of the viscosity $\mu$ of an Ellis fluid given by (6) scaled according to (8) and (9) with ${\tau }_{\mathrm{av}}=1$ and ${\mu }_{\infty }=1/4$: (a) as a function of $\tau$ and (b) as a function of $q$, for $\alpha =10$, 20, and 30. In (a) and (b) the dashed curves show the leading-order asymptotic expressions in the limit $\alpha \to \infty$ given by (60) and (61), respectively.

Figure 12

Sketch of the cross section of a rivulet of a strongly shear-thinning Ellis fluid with ${\mu }_{\infty }\ne 0$ in the limit $\alpha \to \infty$ when ${\tau }_{\mathrm{av}}<{\tau }_{\mathrm{m}}$. The shaded region has viscosity $\mu =1$ and the unshaded region has viscosity $\mu ={\mu }_{\infty }$; the two regions are separated from each other by the surface $z=H\left(y\right)=h\left(y\right)-{\tau }_{\mathrm{av}}$, shown with a dashed line. The darker shading denotes the two subregions in which the velocity is given by (20), and the lighter shading denotes the subregion in which the velocity is given by (64). The same sketch but with the label $\mu ={\mu }_{\infty }$ replaced with $\mu ={({\tau }_{\mathrm{av}}/\tau )}^{\alpha -1}$ also describes the special case ${\mu }_{\infty }=0$.

Figure 13

Contour plots of the velocity $u$ for an Ellis fluid with ${\mu }_{\infty }=1/3$, $\beta =1$ and $Q=1$: (a) the exact solution (33) for $\alpha =50$ and ${\tau }_{\mathrm{av}}=7/10$ ($<{\tau }_{\mathrm{m}}\simeq 0.9717$), for which $a\simeq 1.9433$, (b) the leading-order asymptotic solution in the limit $\alpha \to \infty$ given by (20), (32), and (64) for ${\tau }_{\mathrm{av}}=7/10$ ($<{\tau }_{\mathrm{m}}\simeq 0.9657$), for which $a\simeq 1.9313$, $b\simeq 1.0130$, and $H_{\mathrm{m}}\simeq 0.2657$, (c) the exact solution (33) for $\alpha =5$ and ${\tau }_{\mathrm{av}}=2$ ($>{\tau }_{\mathrm{m}}\simeq 1.1263$), for which $a\simeq 2.2525$, and (d) the leading-order asymptotic solution in the limit $\alpha \to \infty$ given by (20) for ${\tau }_{\mathrm{av}}=2$ ($>{\tau }_{\mathrm{m}}\simeq 1.1318$), for which $a\simeq 2.2635$. In (b) the surfaces $z=H$ and $y=\pm b$ are shown with dashed lines. The contours are drawn at intervals of $1/10$.

Figure 14

Plots of the viscosity $\mu$ of an Ellis fluid given by (6) scaled according to (8) and (9) with ${\tau }_{\mathrm{av}}=1$ and ${\mu }_{\infty }=0$: (a) as a function of $\tau$, and (b) as a function of $q$, for $\alpha =10$, 20, and 30. In (a) and (b) the dashed curves show the leading-order asymptotic expressions in the limit $\alpha \to \infty$ given by (66) with $\alpha =30$ and (67), respectively.

Figure 15

Contour plots of the velocity $u$ for an Ellis fluid with ${\mu }_{\infty }=0$, $\beta =1$ and $Q=1$: (a) the exact solution (37) for $\alpha =20$ and ${\tau }_{\mathrm{av}}=7/10$ ($<{\tau }_{\mathrm{m}}\simeq 0.8488$), for which $a\simeq 1.6977$, (b) the asymptotic solution in the limit $\alpha \to \infty$ given by (20), (68), and (69) for $\alpha =20$ and ${\tau }_{\mathrm{av}}=7/10$ ($<{\tau }_{\mathrm{m}}\simeq 0.8574$), for which $a\simeq 1.7148$ and $b\simeq 0.8131$, (c) the exact solution (37) for $\alpha =5$ and ${\tau }_{\mathrm{av}}=2$ ($>{\tau }_{\mathrm{m}}\simeq 1.1235$), for which $a\simeq 2.2470$, and (d) the leading-order asymptotic solution in the limit $\alpha \to \infty$ given by (20) for ${\tau }_{\mathrm{av}}=2$ ($>{\tau }_{\mathrm{m}}\simeq 1.1318$), for which $a\simeq 2.2635$. In (b) the surfaces $z=H$ and $y=\pm b$ are shown with dashed lines. The contours are drawn at intervals of $1/10$.