$($i$)$ State the boundary condition for an inviscid fluid at an impermeable fixed boundary

$x_3-h(x_1,x_2)=0,$

where we do not permit normal velocity but permit non$-$zero tangential velocity.

$($ii$)$ State the boundary condition for an inviscid fluid at a prescribed moving boundary

$x_3-h(x_1,x_2,t)=0,$

where we do not permit normal velocity but permit non$-$zero tangential velocity.

$($iii$)$ State the normal stress boundary condition at an interface

$x_3-h(x_1,x_2,t)=0$

between an inviscid incompressible fluid and a vacuum. You may assume that the interface has a constant surface tension.

$($iv$)$ State the normal stress boundary condition at an interface

$x_3-h(x_1,x_2,t)=0$

between a viscous incompressible fluid and a vacuum. You may assume that the interface has a constant surface tension.