In the coating process in Fig. P7.11 a liquid is extruded through a slot in a die onto a solid sheet moving horizontally at velocity V. The pressure at the slot exit is P₁ and there is a constant gap thickness H between the die and sheet. Within the gap there is forward flow of liquid in a region of length L₁ and no net flow in a region of length L₂. Both regions are long enough that the flow in them may be approximated as fully developed. After exiting the die the liquid thickness reaches another constant value, H_∞. The upstream and downstream air pressures are uniform at P₀ and surface tension, viscous stresses in the air, and static pressure variations are all negligible. Assume that P₁, V, H, and L₁ are given, whereas L₂ and H_∞ are unknown.

(a) Determine v_x(y) in the region of length L₁.

(b) Determine v_x(y) in the region of length L₂ and show how to predict L₂.

(c) Explain why there is eventually plug flow (a uniform velocity) in the liquid beyond the die, such that v_x = V. Use that information to find H_∞.

(d) The force needed to pull a sheet of width W past the die will be affected by the shear force F that the liquid exerts on the sheet. Calculate F⁄W. Does the force from the liquid assist or retard the movement of the sheet?

Transcribed Image Text:

Ľ X Air Po H Die P₁ Die Liquid Air Po Hoo 4₂ 4₁ Figure P7.11 A solid sheet moving at velocity Vis coated by extruding a liquid through a slot. V