Hydrodynamics of Falling Film Flow of Non-Newtonian Power-Law Fluids

Abstract

The new similarity analysis method has been applied to extensively study the gravity-driven flow of a non-Newtonian liquid film along inclined surface. The partial differential equations governing the hydrodynamics of the power-law fluid are transformed exactly into a set of two ordinary differential equations, which can be calculated numerically to an arbitrary degree of accuracy. The non-linearity of the momentum boundary layer problem for power-law fluid increases with increasing pseudo-plasticity $\left| 1- n \right|$ and the variable grid spacing is therefore increasingly important. The solutions of the system of dimensionless ordinary differential equations depends only on the single parameter $n$, and all other parameters, like the streamwise location $x$, the fluid properties $K/\rho$, and the component of the gravitational acceleration along the surface ${g}\cdot \text{ cos }\alpha$ have been combined into a generalized local Reynolds number Re$_{x }$ and dimensionless velocity $W_x$ and $W_y$. Various flow characteristics can thus be expressed only in term of n and Re$_{x}$. In order to determine $x_{0}$ the particular position $x_{0}$, at which the entire freestream has been entrained into the momentum boundary layer, and the associated critical film thickness $\delta _\mathrm{l} (x_0 )$, knowledge about the total mass flow rate$\rho Q$ within the film is also required, together with the new dimensionless mass flux parameter $\phi$. The latter quantity, which depends on the dimensionless boundary layer thickness$\eta _{\delta _\mathrm{l} }$ and the velocity components $W_{x,\delta _\mathrm{l} }$ and $W_{y,\delta _\mathrm{l} }$ at the edge of the boundary layer, is generally obtained as the numerical solution of the transformed problem and turned out to be function only of the power-law indexn. However, to facilitate rapid and accurate estimate of $\phi$, polynomial curve-fit formulas have been developed on the basis of the new similarity analysis model.