Across an interface
by Yi Zhang
I was reading this interesting paper by Idelsohn et al when I noticed that for the Navier-Stokes equation the velocity boundary condition in normal and tangential direction have different physical meanings. We know that the NSE requires complete (against normal) velocity profile to determine the problem. Through a point in incompressible flow domain, we draw an arbitrary straight (to eliminate tension’s effect) line as interface separating two sides of the domain. In order to determine the problem on side A, velocity at interface should be applied as boundary condition. Now, normal component of velocity at that point should be continuous across the interface (since it’s arbitrary instead of physical), and it corresponds to the incompressibility condition: what goes in should equal to what goes out. It is easy to see when we imagine a slice of spacial volume (against material volume) along the interface, and its thickness is much less than its span. The match of normal velocity indicates the conservation of mass within that volume.
On the other hand, the match of tangential component of the velocity means something else. Imagine that they does not match across the interface, what would happen? Well, the gradient of tangential velocity would be infinite (remember that interface has no thickness, so the dx in the derivative goes to zero), and the shear stress in that direction would be infinite (Newtonian fluid), and apparently it’s physically impossible. This is like a boundary layer with zero thickness. For a boundary layer, viscous effect dominates and causes relatively large viscous force. In our imagined case this means the infinite viscous force and unbalance the momentum conservation.
So here are two conservation laws hidden in one boundary condition. Mathematically it suffices to see this when we degenerate from NSE to Euler equation. By losing one order of the PDE, the equation can not meet both normal and tangential boundary condition, and this corresponds to the slip condition when only normal velocity is prescribed (usually zero) out side the boundary layer.