Book Chapter Conflicting Aspects in the Flat Plate Boundary Layer Conventional Solution

The solution of the steady-state isothermal gas ﬂow with zero pressure gradient over a ﬂat plate is classical and was presented by Blasius in 1908. Despite being reduced to a third order Ordinary Di ﬀ erential Equation, nobody has solved it using just three boundary conditions. In consequence, the “solutions” (exact, or numerical), are not unique, do not satisfy Prandtl’s boundary layer concept and give rise to imprecise criteria of the boundary layer thickness deﬁnition (0.99.U ∞ where U ∞ is the free stream velocity) and to idealizations such as the displacement thickness, δ ∗ and the momentum thickness, δ I . Even though η, the similarity parameter, is deﬁned as a function of x, it is surprising that it is seen as a constant, η ∞ , at the limit of the boundary layer, being valid for the entire plate. It is shown that uncountable “solutions” satisfying the classical equation and its three natural boundary layer conditions can be built. The reduction technique of the Boundary Value Problem, BVP, to an Initial Value Problem, IVP, and the Perturbation Analysis Technique, used as an attempt to incorporate a fourth boundary condition to the differential equation, are discussed. A more general and direct method to deduce the classical Blasius’s flat plate equation is considered, as one of the steps to rectify its solution, and shed light on the origin of the conflicting issues there involved.


Introduction
In 1908, Blasius [1] presented a solution for simultaneous equations of motion and continuity on a flat plate. This system is a simplification of Prandtl's boundary layer equations characterized by the admission of incompressible flow, the steady-state regime, the isothermal gas flow and the zeropressure gradient all along the boundary layer. Conventionally, solving the resultant system of partial differential equations www.videleaf.com involves the use of the stream function, which allows the original differential partial equation to be transformed into a third-order non-linear ordinary differential equation, ODE, expressed in Eq. (1), with "natural" boundary conditions: ) , where η is the similarity parameter and f(η) the dimensionless stream function.
Although not in a realistic way, is Eq. (1) solved, without exception, with four boundary conditions in the specialized literature. This fact is bizarre because it is incompatible with the ODE rules, discussed in all books for undergraduate level that deal with ODE, see for example Courant [2] or Cebeci and Bradshaw [3]. In fact, a survey carried out in the literature that deals with solutions of Eq. (1) reveals the existence of different methods to incorporate a fourth boundary condition to this equation. Together with this fact, there is also misinterpretation of the generated results and the false idea that these solutions are in accordance with Prandtl's concept of boundary layer, even when the result contradicts the formulated hypothesis. Of course, at first glance, one admits that Eq. (1) should be solved with just three boundary conditions. Schlichting [2], for example, p. 127, explains: "The resulting differential equation is non-linear and of third order. The three boundary conditions (7.29), are, therefore, sufficient to determine the solution completely.". It is interesting to recall that Prandtl [4] commenting on the solution of Eq. (1) says that Blasius's solution, known as an exact solution for the flat plate boundary layer, was in fact solved by approximation, p. www.videleaf.com 87 in Prandtl [4]. He explains the used solution mechanism, thus: "We may first ask for a simple solution. Of these we find only one, which corresponds to the potential flow; it can, however, be useful and is ".
He also argued that the power series f should be asymptotic, having all its coefficients a formation law. He shows that f may be built from the approximation given by Eq. (5), where the constant is noted by -β, i.e. a negative number, which gives and as a second approximation, , yielding: (Eq. (5), copied verbatim from the original).
Equation (6) gives, thus, the searched asymptotic formula. Prandtl [4] complements by saying that the resulting approximation may "be refined still further putting f = f 1 + f 2 + f 3 , where f 3 is to be small compared to f 2 , and proceeding as before". However, also according to Prandtl, this new approach is not required "because the power series converges comparatively rapidly". Blasius, in fact, used just f 1 and f 2 , considering that the values of f, and of its first and second derivatives, and , are the same for the two solutions, giving the condition to determine, algebraically, ,  and , (integration constants, where " occurs in the power series developed at  = 0 … and  and  appear in the asymptotic solution...". At the end Prandtl [4] states: "The solution of this problem is not in practice very convenient. It leads to the following values,  = 0.332,  = 1.73,  = 0.231. The above calculations are due to Blasius [1].". Guran and Gwinner [5] in a 2012 publication, commenting on this solution, say on p. 335: "The problem was first solved by Blasius using a series expansions method. But the proposed analytic series solution does not converge at all.". On the other hand, the y-direction component velocity, v, in the flat plate boundary layer, which results from the continuity www.videleaf.com equation in differential form, deduced by Prandtl is given by Eq. (7): Then, admitting in Eq. (7), the condition settled by Eq. (4), the only way to match the potential flow approach, considering Eq. (7) free from error, is that Comparison between Eqs. (5) and (8) forces  in Eq. (5) to become null, instead of being a negative number, as affirmed by Prandtl [4]. On the other hand, v at any ( ) must be zero, i.e. all along the x -coordinate. However, we have to be aware that no one has used nor referred to Eq. (7) as a boundary condition, although, apparently, it could be chosen as another natural condition at . What is the problem if Eq. (7) is not admitted as a boundary condition? The fact is that when v is considered null at to induce the existence of potential flow, it should cause some x-velocity components inside the boundary layer to exceed in magnitude the free stream velocity, U  , that is in more than 43%, as discussed in Jaguaribe [6], and showed later in this paper. And, if v > 0 at this means there will be a flow crossing the boundary layer to counteract the effect of the fluid deceleration caused by the fluid viscosity. In other words, outside the boundary layer there will not be potential flow. Rohsenow and Choi [7], page 39, even though apparently using Eq. (7) to show normal parabolic profiles for the x-direction velocity components, u, do not admit the nonexistence of the potential flow, justifying the mathematical result in the exact solution of Blasius equation as "an anomaly". In spite of this, these authors, as well as Sissom and Pitts [8], emphasize that the solution of Eq. (1) as has been posed, is in agreement with the classical boundary layer theory. There are many other discrepancies in the analysis of the classical Blasius's solution theory whose results, supposedly unquestionable, are impossible to give support or to correspond to some mathematical rules or www.videleaf.com existing physical principles in Fluid Mechanics, as will be shown later.
A more general and direct method to deduce the classical Blasius's flat plate equation will be presented, as one of the steps towards correctly solving Blasius equation, and bringing some explanation for the origin of these conflicting results.

The Fourth Boundary Condition
Equation (1) being a third order nonlinear ordinary differential equation, corresponding to a two-point boundary value problem, should require just three boundary conditions. Thus, an attempt to introduce an additional boundary condition should result in a mathematical error. However, without fail, for solving the Blasius equation, either numerically or analytically, an extra boundary condition has always been used, such as which, with the help of a heuristic trial-and-error method, unveils the "solution". On the other hand, the common view of Blasius's solution is similar to the one by Criminale [9]: "No known analytical solutions exist and therefore the mean profile must be determined numerically", or to that by Parveen [10] "The non-linear mathematical model of the problem prohibits the use of the analytical methods". This belief has brought about through decades a great number of different methods to solve the centennial problem. Among them a mathematical technique of reducing the Boundary Value Problem, BVPformed by Eqs.
(1) to (3) -to an Initial Value Problem, IVP. Such a method, as mentioned by Weill [11] and by Fazio (2002) [12], was first used by Topfer [13], and seems to have dramatically influenced mathematicians and engineers since the 18 th century as a means of solving a non-linear ordinary differential equation. Indeed, as commented by Compte [14], the method applies to linear and non-linear problems. In some circumstances, it is easy to apply, and when it does converge, it is generally the most efficient method among others.
For solving Blasius equation, however, it becomes a hard task since it requires that variables be transformed before the application of the Euler method, see Fazio (2013) [15]. To carry out this transformation one needs to admit a function, such as F(C, ), where C is an arbitrary constant to be determined in terms of [ ( )] , for a large value of . The goal is to evaluate ( ) To do this the initial value problem for F( ), given by Eq. (1) is numerically evaluated with the initial conditions F(0) = ( ) and ( ) From this point, Rubel [16], and Fazio (2008) [17] indicate that the request to evaluate ( ) cannot be fulfilled. Therefore, an approximation is required, i.e. a suitable truncate boundary layer to be used instead of , where is large enough to guarantee also that ( ) where and as , Fazio, 2008 [17]. Thus, the three boundary conditions are replaced by the asymptotic conditions, i.e.
) (10) and where is a guess value when using the Runge-Kutta method, generally, with a shooting method, cf. Parveen [10], Bataller [18], and Criminale [10]. In practical terms, , as In fact, the classical Blasius's solution can be found, admitting straight away , see Asaithambi [19], or for a better mathematical explanation Brighi and Hoernel [20]. We should be aware, however, that the technique of reducing BVP to IVP, cannot always be employed. As pointed out by Na [21] the boundary conditions at the initial point have to be homogeneous for this transformation method to be applied. Klamkin [22], also citing Na [21], comments that this condition is not always necessary, but he adds that for all BVP's and IVP's to be considered, the existence and the uniqueness of the latter should be tacitly assumed, which in turn will have implications for the www.videleaf.com former. In general, though, this is not true, see Zeldovich and Myshkis [23], pp. 32 and 33, or Nickel [24] who affirm that the boundary conditions (u = v = 0 on y = 0, and u U  as y ) ensure the existence of a solution with no guarantee, however, of uniqueness. In reality, no matter the technique to solve the classical Blasius's problem, the "solution" will not be unique, if in Eq. (11) is admitted equal to zero. Such a stratagem, or any other considering ( ) where and as yields just an unique value for ( ( )), independent of its position on the plate and there will also be a unique value for . Furthermore, not only will ( ) )  become null, but all derivatives of order higher than two as well, see Eq.
(1), thus consolidating the existence of multiple "solutions" for the classical Blasius equation. These solutions recognize Eq. (7) as correct, i.e. in conformity with the physical reality, without accepting the nullity of the y-direction component velocity, v, at This fact is another example of incorrectness in this theory. In addition, discarding the possibility of f( ) to be equal to implies that v is not null, but constant. This can be demonstrated by deriving Eq. (7) in terms of , at  = concluding that v at is a constant, which prevents the possibility of the existence of potential flow, and does not fulfill Prandtl´s expectations of the boundary layer concept.
As well known, all analytical or numerical solutions found in the technical literature, reproduce the results originally presented by Blasius, see for example: Schlichting [2], Ahmad and Faiz [25], Asaithambi [19], Guran and Gwinner [5], Motsa et al. [26], Brighi and Hoernel (2008) [27], Houghton et al. [28], Katopodes [29], Hesselgreaves et al. [30], all of them with the same particularity: not allowing the existence of the potential flow, external to the boundary layer. The fact remains that, independent of the traditional method used, it is impossible to guarantee the uniqueness of the classical Blasius's solution when www.videleaf.com a complementary condition, such as ( ) where when as for example, in using the Perturbation Analysis Technique. This technique was also employed to solve Blasius equation, Van Dyke (1964) [28]. In discussing the solution of Eq. (1) "free of factors of 2", and using f1 instead of Van Dyke (1975) [29], comments "This is a two-point boundary-value problem, which would ordinarily require repeated guessing of ( ) to satisfy the condition (7.21c) for large ". As we have pointed out above, this corresponds to adding a supplementary constraint (a "fourth boundary condition") without any guarantee of uniqueness. In fact, Van Dyke (1975) [29], in section 7.6. Uniqueness of the Blasius Solution, comes to this conclusion: "The above solution of the problem (7.17) is not unique from a mathematical viewpoint.". And we can add that it also prevents the nullity of the y-direction component velocity, v, at , setting up just a unique value of for the entire flat plate, and also giving rise to the displacement effect. Van Dyke (1967) [30] p. 2, justifies the existence of the displacement effect in flat shapes as being a result of the change of the outer tangential speed induced by the boundary layer itself. The same view is shared by Kaplun (1957) [31], or by Kaplun (1954) [32], p. 111 :"The flow field is divided into two separate regions, that is, a boundary layer region where the flow field is obtained from boundary layer equations, and an outer region where the Euler equations are used to obtain an external flow, corrected for the displacement effect". This explanation does not hold, because as we know, all the difficulty originates in the way Eq. (7) was generated, causing an unbalanced mass computation in the chosen elemental differential volume.
Rohsenow and Choi [7] and Sissom and Pitts [8], in mentioning the three required conventional boundary conditions, go on emphasizing that the solution of Eq. (1) as has been posed, is in agreement with the classical boundary layer theory. However, an inconsistency seems to have called Rohsenow and Choi [7]'s, attention, making them affirm on p. 39: "Note an anomaly in this solution -in that, v, does not go to zero at large values of y". www.videleaf.com It is understandable that the flat plate boundary layer, treated as it has been, can only raise different questions about both its formulation and solution, chiefly among applied mathematicians. Weinan [33], for example, although believing that the boundary layer theory developed by Prandtl had become a standard tool, mentions, page 208, that "at the mathematical level, there is still a lack of fundamental understanding.".

A More General Method to Deduce the Series Solution of Eq. (1)
As already mentioned, the following partial differential equation is the classical flat plate boundary-layer expression: (12) that was solved by Blasius [1] in conjunction with the continuity equation: Let us consider a function  of x, and y, i.e.  = η(x,y) such as ( ) (14) where p, and A = A(U ∞ ,) are constants, U  being the freestream velocity. If A is chosen as q, being another constant, Eq. (14) turns into where, the Reynolds number is defined as Then, by dimensional analysis and, where  ∞ when y , i.e. y .

Solving the Ordinary Differential Equation (28)
The classical way of solving equation (28) requires the expansion of function z, and its derivatives into a Maclaurin series, which means that: with, the three natural boundary conditions: Replacing Eq. (28) with Eqs. (30), (32), and (33), and using the conditions (34) to (36), we may expand z() in terms of a Maclaurin series, Kreyszig [34], where only coefficients having index given by k = 3i + 2, are not null, i = 0,1,2,… On the other hand, considering a 2 = , where  = z  (0), all the other coefficients may be determined through the recurrent formula: Therefore,

Parametrization of the Series z() and its Derivatives
In terms of  and , the function z() and its derivatives can be expressed as: and so forth.
The use of the "fourth boundary condition", z η ( ∞ ) , and its consequences The "fourth boundary condition" written from equation (40) is expressed as: The value of X, a constant, is determined from Eq. (44). Choosing N = 467 in Eq. (44) results in X = 56.04955. On the other hand, considering the boundary condition, Eq. (34), together with Eq. (39),  is calculated as another constant, which means that the tangent at  = 0, for any x-velocity profile, is also a constant value, despite the fact that (x) is a variable. Thus, X = 56.0495 implies  = 0.33356. It is worth considering that, the value of  determined by distinct authors and using different procedures is situated around 0.332. Howarth (1938) [35], for example, considers  = 0.33206; for Asaithambi [18], depending on calculation accuracy, it situates between 0.342 and 0.33205733629; for Parlange et al. [36], 0.33205734; Parveen [10] mentioned it as equal to 0.332059. Consequently, given Eq.
(42), we can conclude that  ∞ is another invariant, i.e. 5.51821. It is also opportune to point out that Schlichting [2], p. 132, commenting on Blasius's solution, says: "The dimensionless boundary layer thickness, √ , remains constant as long as the boundary layer remains laminar.", no matter the value of x in the expression √ . As we know, the dimensionless boundary layer thickness is in reality  ∞ . This result, i.e.  ∞ as a constant, contrasts with the fact that in Eq. (19) is a function of x. Thus, the constancy of the dimensionless boundary layer thickness presents an incongruity, and it comes from the use of the "fourth boundary condition", ( ∞ ) , associated with the fact that  is a constant too, as we have just demonstrated. On the other hand, we should remember that , as well as that ( ) ( ∞ ) , and so on, see Eq. (28), generating different "solutions" for the Blasius equation, all of them featuring , and  ∞ , as constants.

About Eq. (27)
The classical Eq. (27), as well as Eq. (7) are also, as already shown, a result of the use of the continuity equation in differential form, i.e. Eq. (13). Equation (13), unconditionally used in many physical situations, does not take into account the particular aspect of the uniform velocity profile (uniform flow) intrinsic to the design of Blasius's flat plate flow. Therefore, all the correlations resultant from Eq. (28) cannot fit into what was ideally expected by Prandtl [4], namely, the occurrence of the potential flow. In fact, as noticed by Rohsenow and Choi [7], page 39, and by Bejan [37], page 58, the solution is not consonant with the potential flow, and it is not asymptotic, as we have also demonstrated above, in section 2. The impossibility of the classical Blasius equation solution to be in agreement with the potential flow is so evident that can be easily seen by evoking the solution in terms of the series z(), i.e. Eq. (38). To do so, we need to consider the natural conditions (34) to (36) Figure 1, which presents the dimensionless x and ydirection component velocities in terms of . In Figure 1, we see that the dimensionless y-component velocity profile allows the existence of the potential flow, when all of a sudden, at   , v goes to zero. As to the dimensionless x-direction component velocity, u/U  , we notice that Figure 1 shows dimensionless velocity magnitudes that overpass the constant reference line, drawn in parallel with axis-, which crosses the ordinate axis at u/U  = 1. Such evidence contrasts with the reality, and indicates that z(  ) cannot be, by any means, equal to   , and therefore, if Eq. (28) truly represents the y-direction component velocity, the potential flow will never exist, nor even the known boundary layer outline. In this respect, Rohsenow and Choi [7], page 39, show that the y-direction component velocity is given by which means that the asymptotic behavior does not exist, neither the potential flow Furthermore, these inconsistencies cause, as expected, a problem with mass and momentum balances, requesting the introduction of two artificial parameters, that is, the displacement thickness, *, and the momentum thickness,  I , in order to complement the quantities in deficit.

Conclusion
As we have seen, the classical flat plate Blasius's solution presents two kinds of inconsistency: mathematical and physical. A flagrant mathematical fault is committed when the Blasius's third order differential equation is solved using four boundary conditions. On the other hand, it is proved that the resultant velocity profiles, do not attend to the physical definition of boundary layer as defined by Prandtl. It may, therefore, come as something of a shock to realize that the classical image of the boundary layer of a flat plate is not reproduced under the classical mathematical model, despite the utilization of the three required standard boundary conditions. From the present discussion it follows that the reasons for these inconsistencies are: www.videleaf.com  the misinterpretation of the fact that  = z  (0), is a constant and an unknown parameter, z being the solution of the third-order non-linear ordinary differential equation, obtained by transformation from the original Blasius's partial differential equation, see Eq. (28). In consequence a fourth boundary condition is required to "solve" the ODE, resulting in a "solution" that is not unique and bringing about a series of other inconsistencies: just one value of η ∞ for all the entire flat plate; the need to lay hands on two artificial idealizations: the displacement thickness, δ * and the momentum thickness, δ I ; the non-existence of any explicit relation to the local Reynolds number, Rex.  the use of the continuity equation in the differential form to deduce the y-direction component velocity, which resulted, as we have seen, in the fact that v(η ∞ ) does not go to zero, thus, preventing the formation of the potential flow zone.  The determination of the standard form of the similarity solution by means of a transformation of the variables x, y and the stream function , so that the original Blasius's partial differential equation becomes independent of x. This implies that the dimensionless similarity variable, as y → ∞, i.e. η ∞ , remains invariant, in other words, not dependent on x, see Katopodes [29] on pages 661 and 662.
While the main goal of this paper is to shed some light on the inaccurate and erroneous aspects involved in the Blasius's classical solution, it also brings a new methodology that by avoiding the utilization of the stream function as the way to transform variables, it generates more general expressions for the solution of the flat plate boundary layer, under the same constraints as those used by Blasius. Taking into account Eq. (27) we notice that for p ≠ ½, the flat plate equation becomes conditioned to the values of Re x . This condition turns the correlation more susceptible to produce the solution of Blasius equation in conformity with the mathematical and physical requirements, and prevents the illogical deduction that results in η ∞ ≠ η ∞ (x).