0.8 Multidimensional laminar flow  (Page 3/8)

The boundary conditions are

The partial differential equations do not have an explicit dependence on time as time only enters thorough the boundary conditions. Thus the variables will be made dimensionless with respect to the time dependent boundary conditions for the purpose of solving the PDE.

The dimensionless equations and boundary conditions with the $*$ dropped are now

Integration of $v_r$ with respect to $z$ in the equation of motion and applying the boundary conditions results in the velocity profile across the film thickness.

Integration of the continuity equation over the film thickness gives,

The velocity profile across the thickness is substituted into the above equation and the integration preformed.

Integration and application of the boundary conditions give

The pressure and radius can now be converted to dimensional variables so we can see the dependence of the parameters.

The pressure distribution has a maximum at the center of the disk and decreases to zero at the outer radius of the disk. The pressure integrated over the area of the disk gives the force required to bring the disks together, each disk with a velocity $U$ , when each disk is a distance $h$ from the midplane.

This expression can be turned around to express the velocity of each disk approaching each other when a force $F$ is applied.

This result is the classical Reynolds (1886) velocity for the thinning of two parallel disks.

If the applied force is constant, the above equation can be integrated to determine the time it takes to thin down from some initial thickness, $h_i$ .

It the initial thickness is large but unknown, then it can be assumed to be infinity with only a small error in the time to thin down to a small thickness. An explicit expression for the time to thin from infinite thickness to a thickness $h$ is

From this expression, we see that it will take an infinite time to thin to zero thickness. In reality, as the film becomes very thin, surface forces (disjoining pressure) will become important in accelerating or retarding the rate of thinning. If the surfaces are solid surfaces, contact will be made at high points (roughness) and the contact stresses may limit the thinning.

Transient Drainage of a Vertical Film Earlier we treated the steady film flow along an inclined plane. Here we will consider the transient drainage of a film that has zero flux at the upstream boundary. This corresponds to the transient behavior immediately after the flow if liquid is shut off in the problem of the steady flow along an incline plane. We will treat the wall as if it was vertical. If it is inclined from the vertical, the acceleration of gravity in the solution will need to be multiplied by the cosine of the angle from the vertical. It is assumed that the film is thin enough for the Reynolds number to be negligible and there are no ripples. Also, it is assumed that the thickness is large enough that surface forces (disjoining pressure) are negligible. The fluid in the film is assumed to be incompressible and Newtonian and the surrounding fluid is assumed to have zero density and viscosity. The surface tension and surface viscosity are neglected. The initial thickness of the film is assumed to be a constant value, $h_i$ . Let $z$ be the coordinate in the direction of the film flow and $x$ the direction perpendicular to the wall. The equations of motion and continuity equation for thin films, discussed in Chapter 6 have several terms that can be neglected. The resulting equations and boundary conditions follows.