Blasius boundary layer
In physics and fluid mechanics, a Blasius boundary layer (named after Paul Richard Heinrich Blasius) describes the steady two-dimensional boundary layer that forms on a semi-infinite plate which is held parallel to a constant unidirectional flow .
The solution to the Navier–Stokes equation for this flow begins with an order-of-magnitude analysis to determine what terms are important. Within the boundary layer the usual balance between viscosity and convective inertia is struck, resulting in the scaling argument
where is the boundary-layer thickness and is the kinematic viscosity.
However the semi-infinite plate has no natural length scale and so the steady, incompressible, two-dimensional boundary-layer equations for continuity and momentum are
(note that the x-independence of has been accounted for in the boundary-layer equations) admit a similarity solution. In the system of partial differential equations written above it is assumed that a fixed solid body wall is parallel to the x-direction whereas the y-direction is normal with respect to the fixed wall, as shown in the above schematic. and denote here the x- and y-components of the fluid velocity vector. Furthermore, from the scaling argument it is apparent that the boundary layer grows with the downstream coordinate , e.g.
This suggests adopting the similarity variable
It proves convenient to work with the stream function , in which case
and on differentiating, to find the velocities, and substituting into the boundary-layer equation we obtain the Blasius equation
subject to on and as . This non-linear ODE can be solved numerically, with the shooting method proving an effective choice. The shear stress on the plate
can then be computed. The numerical solution gives .
Falkner–Skan boundary layer
We can generalize the Blasius boundary layer by considering a wedge at an angle of attack from some uniform velocity field . We then estimate the outer flow to be of the form:
Where is a characteristic length and m is a dimensionless constant. In the Blasius solution, m = 0 corresponding to an angle of attack of zero radians. Thus we can write:
As in the Blasius solution, we use a similarity variable to solve the Navier-Stokes Equations.
It becomes easier to describe this in terms of its stream function which we write as
Thus the initial differential equation which was written as follows:
Can now be expressed in terms of the non-linear ODE known as the Falkner–Skan equation (named after V. M. Falkner and Sylvia W. Skan).
(note that produces the Blasius equation). See Wilcox 2007.
In 1937 Douglas Hartree revealed that physical solutions exist only in the range . Here, m<0 corresponds to an adverse pressure gradient (often resulting in boundary layer separation) while m > 0 represents a favorable pressure gradient.
- V. M. Falkner and S. W. Skan, Aero. Res. Coun. Rep. and Mem. no 1314, 1930.
- Blasius, H. (1908). "Grenzschichten in Flüssigkeiten mit kleiner Reibung". Z. Math. Phys. 56: 1–37. (English translation)
- Parlange, J. Y.; Braddock, R. D.; Sander, G. (1981). "Analytical approximations to the solution of the Blasius equation". Acta Mech. 38: 119–125. doi:10.1007/BF01351467.
- Pozrikidis, C. (1998). Introduction to Theoretical and Computational Fluid Dynamics. Oxford. ISBN 0-19-509320-8.
- Lien-Tsai, Yu; Cha'o-Kuang, Chen (1998). "The solution of the blasius equation by the differential transformation method". Math. Comp. Model. 28: 101–111. doi:10.1016/S0895-7177(98)00085-5.
- He, Jihuan (1998). "Approximate analytical solution of Blasius' equation". Commun. Nonl. Sci. Num. Simul. 3 (4): 260–263. Bibcode:1998CNSNS...3..260H. doi:10.1016/S1007-5704(98)90046-6.
- Liao, S.J. (1999). "An explicit, totally analytic approximation of Blasius’ viscous flow problems". International Journal of Non-Linear Mechanics 34 (4): 759–778. Bibcode:1999IJNLM..34..759L. doi:10.1016/S0020-7462(98)00056-0. (see homotopy analysis method)
- Ropman-Miller, Lance; Broadbridge, Philip (2000). "Exact integration of reduced Fisher's equation, Reduced Balsisu equation and the Lorenz model". J. Math. An. Applic. 251 (1): 65–83. doi:10.1006/jmaa.2000.7020.
- He, Ji-Huan (2002). "A simple perturbation approach to Blasius equation". Appl. Math. Comput. 140 (2-3): 217–222. doi:10.1016/S0096-3003(02)00189-3.
- Liao, S.J.; Campo, A. (2002). "Analytic solutions of the temperature distribution in Blasius viscous flow problems". Journal of Fluid Mechanics 453: 411–425. Bibcode:2002JFM...453..411L. (see homotopy analysis method)
- Schlichting, H. (2004). Boundary-Layer Theory. Springer. ISBN 3-540-66270-7.
- Wang, Lei (2004). "A new algorithm for solving classical Blasius equation". Appl. Math. Comput. 157 (1): 1–9. doi:10.1016/j.amc.2003.06.011.
- Wilcox, David C. Basic Fluid Mechanics. DCW Industries Inc. 2007
- Abbasbandy, S. (2007). "A numerical solution of Blasius equation by Adomina's decomposition method and comparison with homotopy perturbation method". Chaos. Solit. Fract. 31 (1): 257–260. Bibcode:2007CSF....31..257A. doi:10.1016/j.chaos.2005.10.071.
- Parand, K; Deghan, Mehdi; Pirkhedri, A. (2009). "Sinc-collocation method for solving the Blasius equation". Phys. Lett. A 373 (44). doi:10.1016/j.physleta.2009.09.005,.
- Parand, K.; Taghavi, A. (2009). "Rational scaled generalized Laguerre function collocation method for solving the Blasius equation". J. Comp. Appl. Math. 233: 980–989. Bibcode:2009JCoAM.233..980P. doi:10.1016/j.cam.2009.08.106.