Laminar Boundary layer (2) the Blasius equation

Considering an uniform flow past a fixed plate, the developing boundary layer is laminar for low Reynolds numbers Rex, where Rex is the Reynolds number defined in terms of the distance from the upstream edge of the plate. In the simple case of a laminar boundary layer over of flat plate in absence of pressure gradient, the fundamental equations may be simplified since the pressure gradient is zero everywhere, ∂P/∂x = ∂P/∂y = 0, the plate is assumed horizontal (∂zo/∂x=0) and the flow is steady (∂Vx/∂t=0).
Since the flow is incompressible, the momentum equation may be expressed in terms of the stream function. With a change of variable, the momentum equation may be transformed as a differential equation in terms of the dimensionless stream function: i.e., the Blasius equation.
The Blasius equation was developed by Paul Richard Heinrich Blasius (1883-1970) as part of his Ph.D. thesis under the supervision of Ludwig Prandtl at Göttingen university (Germany) (Blasius 1907). Blasius' career later spanned over the period of two World Wars, and his key contributions included the Blasius equation and the Blasius friction factor formula for smooth turbulent pipe flows.
The Blasius equation is seen as an important contribution to fluid mechanics. It demonstrated the validity of the boundary layer concept for a laminar flow. Practical applications include blood flows, slow geophysical flows (e.g. mud flows), lubrication and other forms of industrial flows.
The Blasius equation was later extended, and some applications may include boundary layer flows with a temperature-dependant property.
The boundary layer is a fundamental concept in applied fluid mechanics and hydrodynamics, valid for both laminar and turbulent flows. Herein, the laminar boundary layer was reviewed. A basic understanding of the fluid dynamics processes is critical in many real-word applications and discussed in a number of relevant RUclips video movies in the same Playlist at:
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Acknowledgements
Professor Colin J. APELT
References
BLASIUS, H. (1907) "Grenzschichten in Flūssigkeiten mit kleiner Reibung." PhD Dissertation, University of Göttingen.
BLASIUS, H. (1908). "Grenzschichten in Flüssigkeiten mit kleiner Reibing." Z. Math. Phys., Vol. 56, pp. 1-37 (in German) (also NACA Tech. Memo. No. 1256).
SCHLICHTING, H. (1979). "Boundary Layer Theory." McGraw-Hill, New York, USA, 7th edition.
LIGGETT, J.A. (1994). "Fluid Mechanics." McGraw-Hill, New York, USA.
CHANSON, H. (2014). "Applied Hydrodynamics: An Introduction." CRC Press, Taylor & Francis Group, Leiden, The Netherlands, 448 pages & 21 video movies (ISBN 978-1-138-00093-3).