For laminar boundary layer flow over a flat plate, the 2D steady boundary layer equations can be simplified by introducing the similarity variable \(\eta = y \sqrt{\frac{u_\infty}{\nu x}}\).
Blasius Ordinary Differential Equation
Substituting the stream function \(\psi = \sqrt{\nu x u_\infty} f(\eta)\) into the momentum equation yields the Blasius equation:
\[2 f''' + f f'' = 0\]
With boundary conditions: \(f(0) = f'(0) = 0\), and \(f'(\infty) = 1\). Solving this numerically yields the boundary layer thickness at \(u/u_\infty = 0.99\):
\[\delta(x) = \frac{5.0 x}{\sqrt{Re_x}}\]
Thermal Boundary Layer Solution
For fluids with \(Pr > 0.6\), the local Nusselt number derived from the similarity temperature profile is:
\[Nu_x = \frac{h_x x}{k} = 0.332 Re_x^{0.5} Pr^{1/3}\]
References
- Schlichting, H. (1979). Boundary-Layer Theory. McGraw-Hill.
- Incropera, F. P., & DeWitt, D. P. (2006). Fundamentals of Heat and Mass Transfer. Wiley.
