Blasius series

Assuming large Prandtl numbers, one can substitute only the first terms of Blasius series [Eqs. (68a)] for the velocity components u and v, and the algebraic evaluation leads to the expression ... 

For a liquid droplet in a gaseous medium, the densities differ significantly. Whereas the linearized treatment can still be extended to the liquid-phase analysis, for the gas phase the conventional boundary layer analysis should be used (82). For high Reynolds number flow over a solid sphere, approximate solutions have been obtained using both the Blasius series and the momentum integral techniques (82). 

The boundary-layer problem for the specific case of a circular cylinder is (10-40), (10 41), (10-43), and (10-47), with ue and 3p/dx given by (10-122) and (10-123). The first point to note is that a similarity solution does not exist for this problem. Furthermore, in view of the qualitative similarity of the pressure distributions for cylinders of arbitrary shape, it is obvious that similarity solutions do not exist for any problems of this general class. The Blasius series solution developed here is nothing more than a power-series approximation of the boundary-layer solution about x = 0. 

It is of primary interest in many cases to calculate du/dY at Y = 0. This quantity is proportional to the shear stress and can be used in the absence of separation to calculate the frictional force on a body. More importantly, however, condition (10-120) indicates that separation will occur if du/dY 0 at any point x on the body surface (other than the stagnation point x = 0). Furthermore, the point x where this occurs should provide an estimate of the position of the separation point. To calculate (du/dY)Y=0 from the Blasius series solution, we require numerical values for the second derivative of f (Y) at Y = 0, namely,... 

Figure 10-9. The dimensionless shear stress as a function of position on the surface of a circular cylinder as calculated with the approximate Blasius series solution. Note that x is measured in radians from the front-stagnation point. The predicted point of boundary-layer separation corresponds to the second zero of du/dY 0. and is predicted to occur just beyond the minimum pressure point atx = jt/2.

Thus the boundary-layer solution, obtained by means of the Blasius series, indicates that separation should be expected at about 9S 110°. Careful numerical solution of the boundary-layer equations in the form (10-41) with ue given by (10-122) yields a predicted separation point at... 

F. Streaming Flow Past Axisymmetric Bodies - A Generalization of the Blasius Series... 

F. STREAMING FLOW PAST AXISYMMETRIC BODIES - A GENERALIZATION OF THE BLASIUS SERIES... 

The general problem (10-170)—(10-174) can be solved by a generalization of the Blasius series approximation from Section E. Tet us suppose that the two functions r(x) and ue(x), which differentiate between bodies with different geometries, can be approximated in the form... 

Consider the Blasius problem discussed in example 3.2.10. Obtain series solutions for this problem. Can you obtain physically meaningful series solutions for this problem using Maple s dsolve command ... 

Show that an approximate solution can be obtained for 2(x, Y) in the form of a series expansion in x with coefficients that are functions of the Blasius similarity variable rj. Determine the governing DEs and boundary conditions for the first tow terms in this expansion. It can be shown that... 

Of much greater interest is the case where Sc= v D>, since our principal concern is with dilute solutions. For this situation the diffusion boundary layer is imbedded in the viscous boundary layer, and the velocity it sees is that close to the wall. Solution to the steady, Blasius, flat plate, viscous boundary layer equation shows the velocity components close to the wall (y[Pg.108]

The solution of this problem for laminar flow over a flat plate giving and Vy as a function of x and y was first obtained by Blasius and later elaborated by Howarth (Bl, B2, S3). The mathematical details of the solution are quite tedious and complex and will not be given here. The general procedure will be outlined. Blasius reduced the two equations to a single ordinary differential equation which is nonlinear. The equation could not be solved to give a closed form but a series solution was obtained. 

In this case, the diffusion boundary layer is embedded in the viscous boundary layer and the velocity it sees is that close to the wall. The Blasius solution for flat plate boundary layer in the series form is... 

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