In blade coating, which is widely used for sheet materials such as paper, a substrate passes through an opening of fixed height as it is pulled from a pool of the coating liquid. The opening can have various shapes. For the one in Fig. P8.5, the gap between blade and substrate decreases linearly from h0 to h_L over a distance L, as in Eq. (8.2-2). The substrate velocity is U. Of interest is the final coating thickness h_∞, which generally differs from h_L. The pressure P₁ in the upstream liquid may exceed the pressure P₀ in the air. Static pressure variations over the small heights involved can be ignored. You may assume that the lubrication approximation is applicable and that surface tension is negligible.

(a) Explain why there is plug flow far from the opening, such that for all y when h = h_∞.

(b) Relate h_∞ to the dimensions, U, and the pressure drop (|ΔP| = P₁ − P₀), and show that it can be less than, equal to, or greater than h_L.

(c) Under certain conditions, flow reversal in part of the gap will carry some of the liquid back to the upstream reservoir. This recirculation might greatly increase how long some of the liquid stays in the system, which might in turn degrade the coating material. How should the dimensions and operating conditions be chosen to ensure that v_x≥0 for all y?

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Liquid P= P₁ no X=0 y Figure P8.5 Blade coating. h(x) իլ X=L Air P= Po th U