= Solution
{c}
The held-flat interface imposes the <no-penetration boundary condition>, while its specified tangential <traction> gives a <shear stress> condition:
$$
\boxed{w(x,0)=0,\qquad u_z(x,0)+w_x(x,0)=\beta x.}
$$
A differentiable boundary value $w(x,0)\equiv0$ implies $w_x(x,0)=0$, so the second <boundary condition> is also $u_z(x,0)=\beta x$. There is no imposed <no-slip boundary condition> at this interface; tangential <velocity> is determined by the <shear stress> and the fluid equations.
The solution in part (a) has zero tangential <traction>, and so remains valid for the whole interface \b[if and only if $\beta=0$]. At the single point $x=0$ the imposed <shear stress> vanishes for any $\beta$, but this does not satisfy a <boundary condition> that must hold for every $x$.
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