Rivulet flow of generalized Newtonian fluids

Abstract

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.

Non-Newtonian Pressure-Governed Rivulet Flows on Inclined Surface

“…Also, studies should be mentioned, where work [38] is perhaps the key paper on the rivulet flow of a viscoplastic fluid. Meanwhile, works [39][40][41][42][43][44][45][46][47][48] present a part of the contributions which stem from the large body of work on Newtonian and non-Newtonian rivulet flows carried out by Duffy and his research team in Scotland over the last couple of decades (not a complete list). In addition, handbook [25] concerns the problem under consideration, for which the theoretical basis has been described in detail, for solving a one-dimensional heat equation such as equations in (6) mentioned above.…”

Rivulet flow down a slippery substrate

“…In many of these situations the appropriate boundary condition at the solid-fluid interface is the classical no-slip boundary condition, and, as a consequence of this, all of the previous theoretical work on rivulet flow of which the authors are aware has adopted this condition without comment. In the absence of a recent review of rivulet flow, see, for example, the recent papers by Alekseenko et al [3], Herrada et al [4], Mahady, Afkhami and Kondic [5], Howell et al [6], and Al Mukahal, Duffy and Wilson [7,8], and the references therein, for a representative selection of these previous studies.…”

Waves in a rivulet falling down an inclined cylinder

“…Most theoretical studies, starting with the pioneering work 6 are devoted to stationary and smooth (without waves) rivulets falling over an inclined plane and on curved surfaces 7‐11 . The smooth rivulet profile and stationary modes of the rivulet flow were calculated both analytically in the approximation of the lubrication theory 6,7,9‐14 and using numerical methods 8,15‐17 based on Navier–Stokes Equations. A number of studies are related to the formation of stationary rivulets under the influence of wind on the liquid film flowing down the cable surface 18,19 .…”