Answer to the October 15, 2000 Question

We start with the selection of a coordinate system, as indicated in Figure 2 at right. The time dependent thickness of the film is given by the function H(t). The answers to the questions which were posed can be easily obtained from the velocity distribution vx(y,t). vy = 0. The z-coordinate is neutral. Thus, vz = 0 and all derivatives with respect to z are zero. The equation of continuity in rectangular coordinates for an incompressible fluid is: {2} which in this case reduces to {3} This indicates that the x-velocity is a not a function of x, which we have already recognized from the symmetry of the problem.

Figure 2.

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:

{4}

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:

{5}

Rearrangement of Equation {1} (from the problem statement) allows determination of the deformation rate: {6} This function of plotted in Figure 3 at right.

Figure 3.

Thus, the equation of motion reduces to a simple form which can easily integrated:

{7}

{8}

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:

{9}

This leads directly to the shear stress as a function of position and time:

{10}

Thus, the shear stress is simply a linear function of y. Note that with the sign convention used here, the shear stress is negative.

{11}

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.

{12}

The flow situation in this case is illustrated in Figure 4 at right. Applying the definition of y* to Equation {10} quickly yields: {13} In region A the velocity is finite. Integration of Equation {5} and application of the no-slip boundary condition at the substrate surface yields the velocity distribution in this region: {14} The velocity throughout region B is, of course, the same as the velocity at y*: {15}

Figure 4.

The rate of change of the coating thickness is simply related to the flow rate of fluid off of the substrate. The total volume of fluid in the film on the substrate is V = WLH(t) where W is the width and L is the height of the substrate. The rate of change of this total volume is the flow rate, which is in turn related to the velocity distribution which we have just obtained:

{16}

The rate of change of the coating thickness is thus:

{17}

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:

{18}

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:

{19}

The fluid stops flowing when the film is thin enough that the yield stress is not exceeded.