Problem Blasius

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 ... 

But this is just the Blasius problem, and /(y) is therefore the Blasius function. Clearly we need solve only a single trial problem with initial value /"(0) = A to obtain the asymptotic value f (oo) = B, and we can transform directly to the Blasius solution, which is obtained as an initial-value problem with /"(0) = A / IF/2. 

Problem 10-12. Higher-Order Approximations for the Blasius Problem. The classical boundary-layer theory represents only the first term in an asymptotic approximation for Re 1. However, in cases involving separation, we do not seek additional corrections because the existence of a separation point signals the breakdown of the whole theory. When the flow does not separate, we can calculate higher-order corrections, and these provide useful insight and results. In this problem, we reconsider the familiar Blasius problem of streaming flow past a semi-infinite flat plate that is oriented parallel to a uniform flow. 

Statement of the Blasius problem. We consider the steady-state problem on the longitudinal zero-pressure-gradient flow (VP = 0) past a half-infinite flat plate (0 < X < oo). We assume that the coordinates X and Y are directed along the plate and transverse to the plate, respectively, and the origin is placed at the front edge of the plate. The velocity of the incoming flow is U. ... 

Reversed statement of the Blasius problem. In applications [72], one often deals with the reversed statement of the Blasius problem, in which a halfinfinite plate moves in its plane at a velocity U. In this case, the boundary value problem (1.7.5) is replaced by... 

Equation 6.46 with its boundary conditions is the Blasius problem again. The energy equation is satisfied by... 

Solve the Blasius problem using the shooting method in Mathcad. 

Figure 2.2 Solution of the Blasius problem by the shooting method.

Figure 11.42. Plots of solution variables for the Blasius problem in fluid flow.

The Blasius boundary layer problem was solved in all these references for a parallel mean flow at Re = 1000 excited at the wall. In terms of the wall modes and 3, the disturbance stream function can also be written... 

The problem (10—64)—(10—67) was first solved by Blasius, in his doctoral thesis (1908), using the similarity transformation14... 

The solution for f(q ) cannot be obtained analytically. Although the similarity transformation has reduced the set of PDEs, (10-64), to a single ODE, (10-75), the latter is still nonlinear. In fact, Blasius originally solved (10-75) by using a numerical method, but with the algebra carried out by hand Fortunately, today accurate numerical solutions can be obtained with a computer. The main difficulty in solving (10 75) numerically is that most methods for solving ODEs are set up for initial-value problems. 

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. 

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... 

The problem of start-up flow for a circular cylinder has received a great deal of attention over the years because of its role in understanding the inception and development of boundary-layer separation. An insightful paper with a comprehensive reference list of both analytical and numerical studies is S. I. Cowley, Computer extension and analytic continuation of Blasius expansion for impulsive flow past a circular cylinder, J. Fluid Mech. 135, 389-405 (1983). 

This problem has also been solved numerically, and the function j(rf) is tabulated in [424]. We point out that in this case the solution differs from the corresponding Blasius solution. Thus, although physical consideration suggests that the inversion of flow is possible, the solution shows that it is impossible from the mathematical viewpoint. This is due to the fact that problems (1.7.5) and (1.7.11) are nonlinear. 

Since the forcing terms in equation (34) all vanish in this problem, we obtain equation (39), in which for simplicity we shall introduce the further assumption that C = 1—that is, p/t = [see equation (30)]. This assumption (that pp does not vary across the boundary layer) often is reasonable for gases if changes in the average molecular weight are negligible, then— because of the constancy of the pressure—the ideal-gas law implies that p 1/T, in which case constancy of pp corresponds to p T, a dependence close to the kinetic-theory predictions discussed in Appendix E. With C = 1, equation (39) is the Blasius equation [4], F " -F FF" = 0, and in view of equation (28), the boundary conditions implied by equations (48) and (49) are F co) = 1 and F (0) = 0. Use may be made of the present formula for p, C = 1, F (0) = 0, and equations (27) and (29) to ascertain the boundary condition implied by equation (50) the calculation results in... 

Pohlhausen [2] utilized the Blasius coordinate system and velocity distribution to evaluate the convective heating processes within the constant-property boundary layer on a flat plate. He solved two problems ... 

In the case of the other boundary-value problem mentioned, we can demonstrate mathematically that such a substitution is correct here we cannot make such a demonstration, so the resulting solution rests on this additional assumption. This assumption converts the set of two partial differential equations to a single, ordinary differential equation, which Blasius was able to solve in numerical form. The details of this calculation are shown by Schlichting [1, p. 135] see Prob. 11.4. The result is in the form of a curve of VJV versus r, shown in Fig. 11.3. 

It appears that our measurement follows the Smith model. But it is noted that the Reynolds number of our data for low Gortler parameter is considerably small. Then, it seems that the Blasius flow no longer holds good for those data. In connection with this problem Ragab and Nayfeh computed the neutral stability curves taking into account of the effect of displacement thickness of boundary layer. According to their results the curves approach to that of the Smith model departing from that of modified Smith model at low wave number when R becomes small. 

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