Viscous fluid boundary condition Consider an incompressible viscous ﬂuid of kinematic viscosity
$ν$ , dynamic viscosity
$µ$
and density
$ρ$ . A viscous boundary layer is located over a solid surface at $y = 0$ and $x &gt; 0$. The ﬂow in the boundary layer must match with the constant inviscid bulk velocity $U$. The steady two-dimensional boundary-layer equations are $$\frac{∂ u}{ ∂ x} + \frac{∂ v}{ ∂ y} = 0$$
$$ρ
\bigg( u
\frac{∂ u }{∂ x}
+v
\frac{∂ u }{∂ y} \bigg)=
µ \frac{∂ ^2u }{∂ y^2}$$ where $u(x,y)$ is the horizontal velocity and $v(x,y)$ the vertical velocity.

(i) State the boundary conditions on $u$ and $v$ along $y = 0$ and $x &gt; 0$. 

(ii) State the far-ﬁeld condition on $u$ as $y→∞$. 



It says "The ﬂow in the boundary layer must match with the constant inviscid bulk velocity $U$" so does this mean that the resultant of the $u$ and $v$ velocity components should be equal to $U$? 

Then for (i) $u(x,y=0)=v(x,y=0)=U/\sqrt2$?

For (ii), is the far field condition just $u \rightarrow 0$ as $y \rightarrow \infty$?
'It says "The ﬂow in the boundary layer must match with the constant inviscid bulk velocity U'

It means that as you approach the boundary from within the boundary layer, the velocity must approach the bulk velocity U.  It does NOT mean that the velocity within the boundary layer is constant.
'It says "The ﬂow in the boundary layer must match with the constant inviscid bulk velocity U'

It means that as you approach the boundary from within the boundary layer, the velocity must approach the bulk velocity U.  It does NOT mean that the velocity within the boundary layer is constant.