Stress.

Next: Boundary-layer Thickness, Skin friction, Up: Similarity Solution. Previous: Vorticity.

Stress.

The normal components of the stress perpendicular and parallel to the flat plate expressed non-dimensionally are

$\begin{displaymath}\frac{\tau_{xx}}{\rho U^{2}} = -\frac{p}{\rho U^{2}}+\underse... ...underbrace{2\frac{\nu}{U^{2}}\frac{\partial u}{\partial x}}} , \end{displaymath}$

(68)

$\begin{displaymath}\frac{\tau_{yy}}{\rho U^{2}} = -\frac{p}{\rho U^{2}}+\underse... ...underbrace{2\frac{\nu}{U^{2}}\frac{\partial v}{\partial y}}} . \end{displaymath}$

(69)

Therefore, in the limit $R \rightarrow \infty$ , we have that

$\begin{displaymath}\tau_{xx} = \tau_{yy} = -p \end{displaymath}$

(70)

The shearing stress over surfaces parallel to the wall is

$\begin{displaymath}\tau_{xy} = \mu\left(\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}\right), \end{displaymath}$

(71)

which is approximated in the same way as the vorticity, as follows.

$\begin{displaymath}\begin{split} \rho U^{2}\tau'_{xy} & = \rho \nu\left(\underse... ...}+\rho U^{2}R^{-3/2}\frac{\partial v'}{\partial x'} \end{split}\end{displaymath}$

(72)

In the limit $R \rightarrow \infty$ , the shear stress is given by

$\begin{displaymath}\tau'_{xy} \sim R^{-1/2}\frac{\partial u'}{\partial y'} \end{displaymath}$

(73)

or in terms of dimensional variables

$\begin{displaymath}\tau_{xy} = \mu\frac{\partial u}{\partial y}. \end{displaymath}$

(74)

For the Blasius laminar boundary layer similarity solution given by equation (3.47), the shear stress $\tau_{xy}$ is given by

$\begin{displaymath}\tau_{xy} \sim \mu\frac{\partial^{2} \psi}{\partial y^{2}} = \mu\left(\frac{U^{3}}{2\nu x}\right)^{1/2}f''(\eta), \end{displaymath}$

(75)

Next: Boundary-layer Thickness, Skin friction, Up: Similarity Solution. Previous: Vorticity.

Karl P Burr
2003-03-12