0.7 Laminar flows with dependence on one dimension  (Page 5/6)

Correlation of friction factor versus Reynolds number appear in the literature with all three definitions of the friction factor and usually without a subscript to denote which definition is being used.

Non-Newtonian fluids. The velocity profiles above were derived for a Newtonian fluid. A constitutive relation is necessary to determine the velocity profile and mean velocity for non-Newtonian fluids. We will consider the cases of a Bingham model fluid and a power-law or Ostwald-de Waele model fluid. The constitutive relations for these fluids are as follows.

The power-law model is an empirical model that is often valid over an intermediate range of shear rates. At very low and very high shear rates limiting values of viscosity are approached.

Assignment 8.1

1. Express dimensionless velocity as a function of the dimensionless radius and dimensionless groups. Plot the following cases:

   Table of cases to plot
   Case	$R_1/R_2$	$\partial P/\partial z$	$v_{z1}$	$v_{\theta 1}$	$q$
   1	0	$\ne 0$			$\ne 0$
   2	0.5	$\ne 0$	0	0	$\ne 0$
   3	0.5	$=0$	$\ne 0$	0	$\ne 0$
   4	0.5	$\ne 0$	$\ne 0$	0	0
   5	0.5	$\ne 0$	$\ne 0$	$\ne 0$	$\ne 0$

2. What is the net flow if the inner cylinder is translating and the pressure gradient is zero?
3. What is the pressure gradient if the net flow is zero? Plot the velocity profile for this case.

Assignment 8.2

1. Calculate and plot the velocity profile, normalized with respect to the mean velocity for $n=1,0.67,0.5,\text{and}0.33$ .
2. Derive an expression corresponding to Poiseuille law.
3. Derive the same relation between friction factor and Reynolds number as for Newtonian flow by defining a modified Reynolds number for power-law fluids.

Steady film flow down inclined plane

Steady film flow down an inclined plane corresponds to case 4 (Churchill, 1988) or Section 2.2; Flow of a Falling Film (BSL, 1960). These flows occur in chemical processing with falling film sulfonation reactors, evaporation and gas adsorption, and film-condensation heat transfer. It is assumed that the flow is steady and there is no dependence on distance in the plane of the surface due to entrance effects, side walls, or ripples. The Reynolds number must be small enough for ripples to be avoided. The configuration will be similar to that of BSL except $x=0$ corresponds to the wall and the thickness is denoted by $h$ rather than $\delta$ .

It is assumed that the gas has negligible density compared to the liquid such that the pressure at the gas-liquid interface can be assumed to be constant. The potential gradient in the plane of the film is constant and can be expressed either in term of the angle from the vertical, $\beta$ , or the angle from the horizontal, $\alpha$ .