Neglect hydrostatic pressure variation in the air.
The usual boundary condition in a liquid at the interface between this liquid and a gas is the so-called shear-free condition. This assumes that the shear force on the liquid by the gas is negligible. The interfacial shear stress in the honey is μ dw/dx. So, the boundary condition at the interface is μ dw/dx = 0 at x = δ.
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