Answer to the October 15, 2000 Question |
Dip Coating of a Bingham Fluid
For solution of the velocity field, we use the general form of the equation of motion. Neglecting the inertial terms and the interfacial tension this reduces to:
Clearly, the pressure is independent of x since all x positions are exposed to atmospheric pressure. Since the draining flow is vertical, the gravitational acceleration is g_{x}=g. The constitutive equation is used to obtain the components of the stress tensor. Due to the simplicity of the velocity field, the rate of deformation tensor and thus the stress tensor have only two non-zero components:
where C(t) is some function of time. Next, we apply the boundary conditions. There is negligible shear stress on the fluid/air interface so the appropriate boundary condition is:
This leads directly to the shear stress as a function of position and time:
Thus, the shear stress is simply a linear function of y. Note that with the sign convention used here, the shear stress is negative.
Based on Equation {10} we can conclude immediately that the velocity
distribution depends on how the critical shear stress compares with
the product of density, gravity, and film thickness.
Case I: In this case there is no point for which the shear stress exceeds
the yields stress, so:
Case II: In this case the fluid will flow, but the shear rate will be nonzero only at y locations where the shear stress exceeds the critical shear stress for flow.
The rate of change of the coating thickness is thus:
This can be integrated from t=0 at an initial coating thickness of H_{o} to obtain an implicit integral equation for the coating thickness as a function of time:
The final thickness of the coating is much more easily obtained from our analysis of the velocity distribution. In fact, you might have recognized back at Equation {10} that:
The fluid stops flowing when the film is thin enough that the yield stress is not exceeded. When you are painting a wall the paint will go on smoothly if you apply a thin coat. However, if you apply a thick coat it will begin to run. Why? |