Ideal Fluids in Motion — Question 7

Question 7

An ideal wall forbids penetration but can allow tangential slip

A stationary straight wall lies along $y=0$, with fluid in $y>0$. Candidate velocity fields near it are A: $(U,0)$, B: $(U,V)$ with $V\ne0$, and C: $(ay,0)$. Take all constants finite. An inviscid model imposes no normal flow through the wall; a viscous no-slip model additionally matches the wall's tangential velocity.

1. Which fields satisfy impermeability? Which satisfy no slip at the wall?

2. Why is field A permissible in an ideal inviscid calculation even though a real stationary wall often has nearly zero adjacent fluid velocity?

Question 7 - Solution

An ideal wall forbids penetration but can allow tangential slip

(a) Derivation and calculation

At the stationary wall, impermeability requires $v_y(x,0)=0$:$$A:(U,0)\ \checkmark,\qquad B:(U,V)\ \text{fails},\qquad C:(0,0)\ \checkmark.$$$$\boxed{\text{impermeability: A and C}}.$$No slip additionally requires $v_x(x,0)=0$, giving$$\boxed{\text{no slip: C only}}.$$

(b) Result and interpretation

For an ideal impermeable wall,$$\mathbf v\cdot\hat{\mathbf n}=0$$$$\text{does not require}\quad\mathbf v\cdot\hat{\mathbf t}=0.$$An inviscid outer model can permit tangential slip. Real viscous flow can adjust near the wall; accuracy depends on the observable and on whether that near-wall region matters.

Physical check

Field B crosses a solid wall and fails even the weaker inviscid boundary condition.