Figure P$7-21.$ Wire$-$coating process.

$7\mathrm{.}21.$ Wire Coating

Suppose that in a continuous coating process a wire is pulled from a reservoir of the coating material through a tapered die, as shown in Fig. P$7-21.$ In general, the final coating thickness $(h_{})$ will depend on the wire velocity $(U),$ the die dimensions, the pressure difference across the die $)=(P_0-P_L,$ and the liquid properties. Assume that the lubrication approximation is applicable and that the coating is thin compared with the radius of the wire. Rectangular coordinates can be used.

$($a$)$ Relate $h_{}$ to $P,U,$ and the die geometry. Obtain a general result for $h_{},$ and one for the special case where $h(x)$ is linear.

$($b$)$ If $h_0$ is too large, some of the liquid will recirculate instead of being drawn out with the wire, an undesirable situation. Assuming that $h_{}$ is given, determine the maximum opening for which there will be no recirculation $)=(h_{\text{m}\text{a}\text{x}}.$ Also, sketch several streamlines for a case where $h_0>h_{\text{m}\text{a}\text{x}}.$ To guide your sketch, find the equation for the streamline that separates the forward$-$flow and recirculating regions.