# On Solving the Nonlinear Falkner–Skan Boundary-Value Problem: A Review

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Analytical Treatments

#### 2.1.1. The Pohlhausen Flow

#### 2.1.2. The Blasius Flow

#### 2.1.3. The Crocco–Wang Transformation

#### 2.1.4. Other More General Analytical Approaches

#### 2.2. Numerical Treatments

#### 2.2.1. Shooting

- The work of Cebeci and Keller [45] is the first to employ the shooting approach to solve (5) and (6). They work directly with the physical domain $\eta \in [0,\infty )$, but march their solution from $\eta =0$ to $\eta ={\eta}_{\infty}$, where they assume ${\eta}_{\infty}$ is “sufficiently large.” They use the Newton’s method to adjust the initial condition.
- Cebeci and Keller [45] also employ the idea of parallel shooting to obtain the solution. This approach involves dividing the domain into several subintervals and applying shooting in each subinterval. The motivation for this approach comes from the desire to avoid any instabilities that may be caused while marching the solution using an initial-value method. The authors report successfully obtaining three significant digits in two iterations when shooting was carried out over three subintervals.
- Asaithambi [47] uses the classical, fourth-order Runge-Kutta method to march the solution from $\xi =0$ to $\xi =1$ and the secant method to iteratively obtain improved values for $\alpha $. Two initial approximations for $\alpha $ were needed.
- Salama [48] uses a single-step method of order 5 to march the solution from $\xi =0$ to $\xi =1$, coupled with a third-order rootfinding method called the Halley’s method [49] to compute $\alpha $ and the secant method to compute ${\eta}_{\infty}$. It is also important to note that the solutions of (5) and (6) are obtained for $\beta $ values as large as 40.
- In another work, Asaithambi [22] uses recursive evaluation of Taylor coefficients to march the solution from $\xi =0$ to $\xi =1$ and the secant method to iteratively obtain improved values for $\alpha $. Two initial approximations for $\alpha $ were needed, and higher orders of accuracy, as high as 7, were achieved with ease. Moreover, this work explores (5) and (6) for $\beta $ values as large as 40 as well and compares the results obtained previously by Salama [48].
- Zhang and Chen [50] used the Runge-Kutta (4,5) formula to march the solution from $\xi =0$ to $\xi =1$, and the Newton’s method to improve the guess for $\alpha $, and hence required only one initial guess to get started.
- Sher and Yakhot [51] shoot from the end corresponding to $\eta \to \infty $ instead of from $\eta =0$. They choose an arbitrary value for $f\left(\eta \right)$ as $\eta \to \infty $, a value close to 1 for ${f}^{\prime}\left(\eta \right)$ as $\eta \to \infty $ and estimate ${f}^{\u2033}\left(\eta \right)$ as $\eta \to \infty $ using a simple analysis of the asymptotic behavior of the solution.

#### 2.2.2. Finite Differences

- Asaithambi [52] truncates the infinite domain at $\eta ={\eta}_{\infty}$, uses the transformation $\xi =\eta /{\eta}_{\infty}$ as done previously, and sets $u=\frac{1}{{\eta}_{\infty}}\frac{df}{d\xi}$, to transform the problem (5) and (6) on the physical domain $\eta \in [0,\infty )$ to the system$$\begin{array}{cc}\hfill \frac{df}{d\xi}& ={\eta}_{\infty}u,\hfill \end{array}$$$$\begin{array}{cc}\hfill \frac{{d}^{2}f}{d{\xi}^{2}}& =-{\eta}_{\infty}f\frac{du}{d\xi}-{\eta}_{\infty}^{2}\beta (1-{u}^{2}),\hfill \end{array}$$$$\begin{array}{cccccccccc}\hfill f& =0,\phantom{\rule{1.em}{0ex}}u\hfill & & =0\phantom{\rule{1.em}{0ex}}\hfill & & \mathrm{at}\phantom{\rule{1.em}{0ex}}\hfill & & \xi \hfill & & =0,\hfill \end{array}$$$$\begin{array}{cccccccccc}\hfill u& =1,\phantom{\rule{1.em}{0ex}}\frac{du}{d\xi}\hfill & & =0\phantom{\rule{1.em}{0ex}}\hfill & & \mathrm{at}\phantom{\rule{1.em}{0ex}}\hfill & & \xi \hfill & & =1.\hfill \end{array}$$In order to discretize (32)–(35), the interval $\xi \in [0,1]$ is subdivided into N subintervals $[{\xi}_{j},{\xi}_{j+1}]$ for $j=0,1,\cdots ,N-1$ at ${\xi}_{j}=jh$ with $h=1/N$. Letting ${f}_{j}$ and ${u}_{j}$ denote the approximations to $f\left({\xi}_{j}\right)$ and $u\left({\xi}_{j}\right)$ respectively, Equation (32) is discretized as$$\frac{{f}_{j}-{f}_{j-1}}{h}-{\eta}_{\infty}\left[\frac{{u}_{j}+{u}_{j-1}}{2}\right]=0,\phantom{\rule{1.em}{0ex}}j=1,2,\cdots ,N,$$$$\frac{{f}_{j-1}-2{f}_{j}+{f}_{j+1}}{{h}^{2}}+{\eta}_{\infty}{f}_{j}\frac{{u}_{j+1}-{u}_{j-1}}{2h}+{\eta}_{\infty}^{2}\beta (1-{u}_{j}^{2})=0,\phantom{\rule{1.em}{0ex}}j=1,2,\cdots ,N-1.$$The boundary conditions (34) translate to$${f}_{0}=0,\phantom{\rule{1.em}{0ex}}{u}_{0}=0,$$$${u}_{N}=1,\phantom{\rule{1.em}{0ex}}\frac{3{u}_{N}-4{u}_{N-1}+{u}_{N-2}}{2h}=0.$$Newton’s method is used for the iterative solution of the nonlinear system (37), Equation (36) is used to update the values of ${f}_{j}$, and (39) is used to update ${\eta}_{\infty}$. The solution of the nonlinear system by Newton’s method involves the repeated solution of a linear system with a tridiagonal coefficient matrix. In related work, Asaithambi [53] uses the transformation $\xi ={e}^{-\eta}$ to map the physical domain $[0,\infty )$ to the computational domain $[0,1]$ and discretizes the transformed equations and boundary conditions using second-order finite differences. This approach does not truncate the physical domain but still requires the iterative solution of a nonlinear system that involves the solution of a tridiagonal linear system during each iteration.
- Salama and Mansour [54] develop an unconditionally stable, fourth order finite difference method that involves only four grid points and apply it to the nonlinear Falkner–Skan problem (5) and (6). Their development uses the method of undetermined coefficients involving the unknown function values at points ${\eta}_{j},{\eta}_{j-1},{\eta}_{j-2},{\eta}_{j+1}$ to derive the discretization for the third derivative by setting the local truncation errors of orders 0, 1, 2, and 3 to zero. The details of the derivation are too technical to be included here and the reader is referred to the original publication [54]. In related work, Salama and Mansour [55] develop a finite difference method of order 6 for solving steady and unsteady two-dimensional laminar boundary-layer equations. This method is also unconditionally stable. The authors illustrate the effectiveness of this method by solving the unsteady separated stagnation point flow, the Falkner–Skan equation and Blasius equation.
- Duque-Daza et al. [56] express (24) as$$\frac{{d}^{3}f}{d{\xi}^{3}}+{\eta}_{\infty}f\phantom{\rule{0.166667em}{0ex}}\frac{{d}^{2}f}{d{\xi}^{2}}-\beta {\eta}_{\infty}{\left(\frac{df}{d\xi}\right)}^{2}+\beta {\eta}_{\infty}^{3}=0,$$$$\begin{array}{cc}\hfill {f}_{i}^{\prime}& =\frac{{f}_{i-2}-8{f}_{i-1}+8{f}_{i+1}-{f}_{i+2}}{12h},\hfill \end{array}$$$$\begin{array}{cc}\hfill {f}_{i}^{\u2033}& =\frac{-{f}_{i-2}+16{f}_{i-1}-30{f}_{i}+16{f}_{i+1}-{f}_{i+2}}{12{h}^{2}},\hfill \end{array}$$$${f}_{i}^{\u2034}=\frac{-{f}_{i-2}-{f}_{i-1}+10{f}_{i}-14{f}_{i+1}+7{f}_{i+2}-{f}_{i+3}}{4{h}^{3}}$$$${f}_{1}^{\prime}=0,\phantom{\rule{1.em}{0ex}}{f}_{N}^{\prime}={\eta}_{\infty}$$$$\begin{array}{cc}\hfill \frac{48{f}_{2}-36{f}_{3}+16{f}_{4}-3{f}_{5}}{12h}& =0,\hfill \end{array}$$$$\begin{array}{cc}\hfill \frac{3{f}_{N-4}-16{f}_{N-3}+36{f}_{N-2}-48{f}_{N-1}+25{f}_{N}}{12h}& ={\eta}_{\infty}.\hfill \end{array}$$Next, plugging ${f}^{\prime}={\eta}_{\infty}$ and ${f}^{\u2033}=0$ into (40) for $\xi =1$ yields$${f}_{N}^{\u2034}=0,$$$$\frac{15{f}_{N-6}-104{f}_{N-5}+307{f}_{N-4}-496{f}_{N-3}+461{f}_{N-2}-232{f}_{N-1}+49{f}_{N}}{8{h}^{3}}=0.$$Then the Equation (40) itself is evaluated at $\xi =0$ to obtain ${f}_{1}^{\u2034}+\beta {\eta}_{\infty}^{3}=0$, which is discretized as$$\frac{232{f}_{2}-461{f}_{3}+496{f}_{4}-307{f}_{5}+104{f}_{6}-15{f}_{7}}{8{h}^{3}}+\beta {\eta}_{\infty}^{3}=0.$$Finally, the condition ${f}_{N}^{\u2033}=0$ is used to solve for ${\eta}_{\infty}$. For this purpose, the second-order derivative ${f}_{N}^{\u2033}$ is discretized as$${f}_{N}^{\u2033}=\frac{35{f}_{N}-104{f}_{N-1}+114{f}_{N-2}-56{f}_{N-3}+11{f}_{N-4}}{12{h}^{2}},$$

#### 2.2.3. Collocation Methods

- Parand et al. [57] consider (5) subject to the boundary conditions$$f\left(0\right)=\gamma ,\phantom{\rule{1.em}{0ex}}{f}^{\prime}\left(0\right)=0,\phantom{\rule{1.em}{0ex}}\phantom{\rule{4.pt}{0ex}}\mathrm{and}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{1.em}{0ex}}{f}^{\prime}\left(\eta \right)\to 1\phantom{\rule{4.pt}{0ex}}\mathrm{as}\phantom{\rule{4.pt}{0ex}}\eta \to \infty ,$$The rational Legendre functions $\left\{{R}_{n}\left(\eta \right)\right\}$ are defined by$${R}_{n}\left(\eta \right)={P}_{n}\left(\frac{\eta -L}{\eta +L}\right),$$$$w\left(\eta \right)=\frac{2L}{{(\eta +L)}^{2}}.$$Moreover, in (52) and (53), L is a constant known as the map parameter. While setting $L=1$ is most common, guidelines for the choice of value for L are given for rational Chebyshev functions in [58] that apply also to the Legendre family. The rational Legendre functions satisfy the relation$${R}_{n+1}\left(\eta \right)=\left(\frac{2n+1}{n+1}\right)\phantom{\rule{0.166667em}{0ex}}\frac{\eta -L}{\eta +L}\phantom{\rule{0.166667em}{0ex}}{R}_{n}\left(\eta \right)-\left(\frac{n}{n+1}\right){R}_{n-1}\left(\eta \right),\phantom{\rule{1.em}{0ex}}{R}_{0}\left(\eta \right)\equiv 1,\phantom{\rule{1.em}{0ex}}{R}_{1}\left(\eta \right)=\frac{\eta -L}{\eta +L},$$$$g\left(\eta \right)=\sum _{i=0}^{N}{a}_{i}{R}_{i}\left(\eta \right)$$$$({f}_{0}^{\u2034}\left(\eta \right)+{g}^{\u2034}\left(\eta \right))+({f}_{0}\left(\eta \right)+g\left(\eta \right))({f}_{0}^{\u2033}\left(\eta \right)+{g}^{\u2033}\left(\eta \right))+\beta (1-{({f}_{0}^{\prime}\left(\eta \right)+{g}^{\prime}\left(\eta \right))}^{2})=0,$$
- Kajani et al. [59] truncate the infinite domain $[0,\infty )$ to $[0,L]$ for a “large enough” L and use what they call shifted Chebyshev polynomials to solve (5) and (6). The shifted Chebyshev polynomials are defined by$${S}_{n}\left(\eta \right)={T}_{n}\left(\frac{2}{L}\eta -1\right),$$$${f}^{\u2033}\left(\eta \right)=\sum _{i=0}^{N}{a}_{i}{S}_{i}\left(\eta \right).$$They use (58) to obtain$${f}^{\prime}\left(\eta \right)=\sum _{i=0}^{N+1}{b}_{i}{S}_{i}\left(\eta \right),\phantom{\rule{1.em}{0ex}}f\left(\eta \right)=\sum _{i=0}^{N+2}{c}_{i}{S}_{i}\left(\eta \right),\phantom{\rule{1.em}{0ex}}{f}^{\u2034}\left(\eta \right)=\sum _{i=1}^{N}{a}_{i}{S}_{i}^{\prime}\left(\eta \right),$$$$\sum _{i=1}^{N}{a}_{i}{S}_{i}^{\prime}\left(\eta \right)+\left(\sum _{i=0}^{N+2}{c}_{i}{S}_{i}\left(\eta \right)\right)\left(\sum _{i=0}^{N}{a}_{i}{S}_{i}\left(\eta \right)\right)+\beta \left[1-{\left(\sum _{i=0}^{N+1}{b}_{i}{S}_{i}\left(\eta \right)\right)}^{2}\right]=0$$$${f}^{\prime}\left(L\right)=\sum _{i=0}^{N+1}{b}_{i}{S}_{i}\left(L\right)=1$$

#### 2.2.4. Other Methods

- Using a truncated domain at $\eta ={\eta}_{\infty}$ and imposing the asymptotic condition ${d}^{2}f/d{\eta}^{2}=0$ as $\eta \to \infty $, Asaithambi [60] solves (5) and (6) by formulating the problem in weak Galerkin form and with piecewise linear functions as basis functions and provides a finite-element solution. For this purpose, Asaithambi [60] rewrites (5) and (6) in the form$$\begin{array}{cccc}\hfill {g}^{\prime}& =u,\phantom{\rule{2.em}{0ex}}\hfill & & g\left(0\right)=0,\phantom{\rule{1.em}{0ex}}u\left(0\right)=0,\hfill \end{array}$$$$\begin{array}{cccc}\hfill {u}^{\u2033}& =-{\eta}_{\infty}^{2}g{u}^{\prime}-{\eta}_{\infty}^{2}\beta (1-{u}^{2}),\phantom{\rule{2.em}{0ex}}\hfill & & u\left(1\right)=1,\phantom{\rule{1.em}{0ex}}{u}^{\prime}\left(1\right)=0,\hfill \end{array}$$$$\begin{array}{cc}\hfill {\varphi}_{j}\left(\xi \right)& =\left\{\begin{array}{cc}{\displaystyle \frac{\xi -{\xi}_{j-1}}{h}},\hfill & \mathrm{for}\phantom{\rule{4.pt}{0ex}}\xi \in [{\xi}_{j-1},{\xi}_{j}];\hfill \\ {\displaystyle \frac{{\xi}_{j+1}-\xi}{h}},\hfill & \mathrm{for}\phantom{\rule{4.pt}{0ex}}\xi \in [{\xi}_{j},{\xi}_{j+1}];\hfill \\ 0,\hfill & \mathrm{otherwise},\hfill \end{array}\right.\hfill \end{array}$$Asaithambi [60] constructs the solution in the form$$g\left(\xi \right)=\sum _{j=0}^{N}{g}_{j}{\varphi}_{j}\left(\right]xi),\phantom{\rule{2.em}{0ex}}u\left(\xi \right)=\sum _{j=0}^{N}{u}_{j}{\varphi}_{j}\left(\xi \right)$$$$\begin{array}{cc}\hfill {\int}_{0}^{1}\left[{u}^{\u2033}+{\eta}_{\infty}^{2}g{u}^{\prime}+{\eta}_{\infty}^{2}\beta (1-{u}^{2})\right]{\varphi}_{j}\left(\xi \right)d\xi & =0,\hfill \\ \hfill \frac{{g}_{j}-{g}_{j-1}}{h}-\frac{{u}_{j}+{u}_{j-1}}{2}& =0.\hfill \end{array}$$Thus, the method uses both finite-elements and finite-differences to obtain a nonlinear system of algebraic equations which is solved using Newton’s method. However, this system has to be iteratively solved to adjust the value of ${\eta}_{\infty}$ so that the asymptotic condition ${u}^{\prime}\left(1\right)=0$ is satisfied. Asaithambi [60] handles this by using the secant method y setting$$w\left({\eta}_{\infty}^{\left(k\right)}\right)=\frac{3{u}_{N}^{\left(k\right)}-4{u}_{N-1}^{\left(k\right)}+{u}_{N-2}^{\left(k\right)}}{2h}$$$${\eta}_{\infty}^{(k+1)}={\eta}_{\infty}^{\left(k\right)}-w\left({\eta}_{\infty}^{\left(k\right)}\right)\phantom{\rule{0.166667em}{0ex}}\left[{\displaystyle \frac{{\eta}_{\infty}^{\left(k\right)}-{\eta}_{\infty}^{(k-1)}}{w\left({\eta}_{\infty}^{\left(k\right)}\right)-w\left({\eta}_{\infty}^{(k-1)}\right)}}\right].$$
- Recently, Hajimohammadi et al. [61] have developed a numerical learning approach to solving the Falkner–Skan problem (5) and (6). They cite several works from the literature that have used machine learning methods for the purpose of handling nonlinear models in the continuous domain, and demonstrate that a combination of numerical methods and machine learning methods could deal with nonlinearities much more easily and produce accurate solutions. In their work on the Falkner–Skan problem, they employ a combination of collocation using Rational Gegenbauer (RG) functions and the Least Squares (LS) Support Vector Machines (SM) and present results of their RG_LS_SVM method that compare well with results obtained previously by other researchers. The details of their method are too many to cover in this review and the reader is referred to the original publication [61].

## 3. Results

## 4. Discussion

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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Name of Flow | ${\mathit{\beta}}_{0}$ | $\mathit{\beta}$ | ${\mathit{f}}^{\u2033}\left(0\right)$ |
---|---|---|---|

Pohlhausen [21] | 0 | 1 | 1.154701 |

Blasius [3] | $\frac{1}{2}$ | 0 | 0.332058 |

Homann [32] | 2 | 1 | 1.311938 |

Hiemenz [4] | 1 | 1 | 1.232589 |

**Table 2.**Falkner–Skan solutions (${f}^{\u2033}\left(0\right))$ for selected values of $\beta \in [-0.1988,2]$.

$\mathit{\beta}$ | Asaithambi [22] | Zhang et al. [50] | Salama [48] |
---|---|---|---|

$-0.1988$ | 0.005225 | 0.005222 | 0.005226 |

$-0.1800$ | 0.128637 | 0.128636 | 0.128638 |

$-0.1500$ | 0.216361 | 0.216362 | 0.216362 |

$-0.1000$ | 0.319270 | 0.319270 | 0.319270 |

$0.0000$ | 0.469600 | 0.469600 | 0.469600 |

$0.5000$ | 0.927680 | 0.927680 | 0.927680 |

$1.0000$ | 1.232589 | 1.232587 | 1.232588 |

$2.0000$ | 1.687218 | 1.687218 | 1.687218 |

**Table 3.**Falkner–Skan solutions (${f}^{\u2033}\left(0\right))$ for select values of $\beta \in [10,40]$.

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Asaithambi, A.
On Solving the Nonlinear Falkner–Skan Boundary-Value Problem: A Review. *Fluids* **2021**, *6*, 153.
https://doi.org/10.3390/fluids6040153

**AMA Style**

Asaithambi A.
On Solving the Nonlinear Falkner–Skan Boundary-Value Problem: A Review. *Fluids*. 2021; 6(4):153.
https://doi.org/10.3390/fluids6040153

**Chicago/Turabian Style**

Asaithambi, Asai.
2021. "On Solving the Nonlinear Falkner–Skan Boundary-Value Problem: A Review" *Fluids* 6, no. 4: 153.
https://doi.org/10.3390/fluids6040153