# Boundary layer equation

I was trying to understand the derivation of the boundary layer equations at p.145 of http://www.unimasr.net/ums/upload/files/2012/Sep/UniMasr.com_919e27ecea47b46d74dd7e268097b653.pdf. : the derivation is completely given at that page.

I almost figured it all out, but I don't understand (12-8). With the approximations they made, I should get (12-8), but not the last term $\mu\frac{\partial^2v_y}{\partial x^2}$. I don't see how they get this term, when approximating $\tau_{xy} = \mu \frac{\partial^2 v_y}{\partial x^2}$ and $\sigma_{yy} = -P$ does not give the result shown. I wonder if I'm wrong, or the book is just mistaken? I hope someone can help me out here.

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That's obviously a typo. The last term should be $\mu \frac{\partial^2 v_y}{\partial y^2}$. Which would have the same order of magnitude as the rest of terms in the equation. – user23660 Aug 22 '13 at 3:38
@user23660 You should probably turn that into an answer -- maybe show that it is of the same order of magnitude for completeness. – tpg2114 Aug 22 '13 at 4:45

Usually the derivatives of $\sigma_{xx}$, $\sigma_{yy}$, $\tau_{xy}$ and $\tau_{yx}$ in equations (12-5) and (12-6) are treated in combination. For (12-6) the right hand side becomes $$\frac{\partial \tau_{xy}}{\partial x} + \frac{\partial \sigma_{yy}}{\partial y} = \mu \frac{\partial^2 v_x}{\partial x \, \partial y} + \mu \frac{\partial^2 v_y}{\partial x^2} - \frac{\partial P}{\partial y} + 2 \mu \frac{\partial^2 v_y}{\partial y^2}\\ = - \frac{\partial P}{\partial y} + \mu \frac{\partial^2 v_y}{\partial x^2} + \mu \frac{\partial^2 v_y}{\partial y^2} + \mu \left( \frac{\partial^2 v_x}{\partial x \, \partial y} + \frac{\partial^2 v_y}{\partial y^2} \right)$$ Then $$\frac{\partial^2 v_x}{\partial x \, \partial y} = \frac{\partial^2 v_x}{\partial y \, \partial x} \quad\Rightarrow\quad \frac{\partial^2 v_x}{\partial x \, \partial y} + \frac{\partial^2 v_y}{\partial y^2} = \frac{\partial}{\partial y} \left( \frac{\partial v_x}{\partial x} + \frac{\partial v_y}{\partial y} \right)$$ which is zero due to the continuity equation for incompressible fluids (12-10). Therefore, $$\frac{\partial \tau_{xy}}{\partial x} + \frac{\partial \sigma_{yy}}{\partial y} = - \frac{\partial P}{\partial y} + \mu \left( \frac{\partial^2 v_y}{\partial x^2} + \frac{\partial^2 v_y}{\partial y^2} \right) \quad\mbox{,}$$ which is the usual right hand side for the $y$ component of the Navier-Stokes equations.
The argument from Prantl is that the boundary layer thickness is small compared to the characteristic length of the object. For derivatives this implies that derivatives with respect to $x$ are smaller than derivatives with respect to $y$, and that second derivatives with respect to $x$ are even smaller than second derivatives with respect to $y$.
With this argument, the last term in (12-8) should read $$\mu \frac{\partial^2 v_y}{\partial y^2}$$ and not $$\mu \frac{\partial^2 v_y}{\partial x^2} \quad\mbox{.}$$ It's a typo as user23660 pointed out.