*2.2.1 Case without suction (vp = 0)*

When replacing in Eq. (7) the value of *vp* = 0, the differential equation of *h x*ð Þ is written as

$$U\_{\infty}a\_2 \frac{dh(\mathbf{x})}{d\mathbf{x}} = \frac{\nu}{h(\mathbf{x})}a\_3\tag{10}$$

The resolution of Eq. (10) with the boundary condition in Eq. (7) leads to the solution:

$$h(\mathbf{x}) = 2\sqrt{\mathbf{1} + \sqrt{2}} \sqrt{\frac{\nu \mathbf{x}}{U\_{\infty}}} \tag{11}$$

So, we obtain the profile of the boundary layer velocity of the flow on the impermeable flat plate:

$$\frac{u}{U\_{\infty}} = \text{erf}(\mathbf{0}, \mathbf{32}\eta) \tag{12}$$

These results enable us to determine the various characteristics of the boundary layer as the boundary layer thickness, the displacement thickness, the momentum thickness, and the friction coefficient:

$$\frac{\delta}{\mathbf{x}} = \frac{5,66}{\sqrt{\text{Re}\_{\mathbf{x}}} \mathcal{X}} \frac{\delta}{\delta} = \frac{5,66}{\sqrt{\text{Re}\_{\mathbf{x}}}} ; \frac{\delta\_1}{\delta} = 0, \mathfrak{A} 1; \frac{\delta\_2}{\delta} = 0, 128; \frac{1}{2} \mathcal{G} \mathcal{f} = \frac{0,36}{\sqrt{\text{Re}\_{\mathbf{x}}}} \tag{13}$$

*Boundary Layer Theory: New Analytical Approximations with Error and Lambert Functions… DOI: http://dx.doi.org/10.5772/intechopen.88637*

### *2.2.2 Case with uniform suction*

where *erf <sup>y</sup>*

*Aerodynamics*

where

written as

solution:

impermeable flat plate:

δ <sup>x</sup> <sup>¼</sup> <sup>5</sup>*,* <sup>66</sup> ffiffiffiffiffiffiffiffi Rex p

**124**

thickness, and the friction coefficient:

*δ <sup>x</sup>* <sup>¼</sup> <sup>5</sup>*,* <sup>66</sup> ffiffiffiffiffiffiffiffi Re*<sup>x</sup>* p ;

**2.2 Analytical solutions**

uniform suction when *vp* = �*v*<sup>0</sup> 6¼ 0.

*2.2.1 Case without suction (vp = 0)*

*h x*ð Þ � �

is the Error function and *h x*ð Þ is the unknown scale function to

*h x*ð Þ *<sup>α</sup>*<sup>3</sup>

ffiffi 2 <sup>p</sup> � <sup>1</sup> ffiffiffi

*<sup>π</sup>* <sup>p</sup> *,* (8)

*<sup>π</sup>* <sup>p</sup> *:* (9)

*h x*ð Þ *<sup>α</sup>*<sup>3</sup> (10)

¼ *erf*ð Þ 0*;* 32*η* (12)

*Cf* <sup>¼</sup> <sup>0</sup>*,* <sup>36</sup> ffiffiffiffiffiffiffiffi Re*<sup>x</sup>*

p (13)

(7)

(11)

be determined. The choice of the Error function has the advantage of a good approximation of the exact solution of Blasius which will be proven below.

*dh x*ð Þ

differential equation of *h*(*x*), with a boundary condition as below:

*U*∞*α*<sup>2</sup>

8 < :

ð<sup>∞</sup> 0

*α*<sup>2</sup> ¼

The insertion of Eq. (6) in Eq. (5), with the conditions Eqs. (3) and (4), gives a

*dx* � *vp* <sup>¼</sup> *<sup>υ</sup>*

*h*ð Þ¼ 0 0

*erf z*ð Þð Þ 1 � *erf z*ð Þ *dz* ¼

The analytical resolution of the differential Eq. (7) depends inevitably on the boundary conditions, in particular, the value of suction velocity *vp*, since we consider two cases: an impermeable flat plate when *vp* = 0 and a porous flat with

When replacing in Eq. (7) the value of *vp* = 0, the differential equation of *h x*ð Þ is

*dh x*ð Þ *dx* <sup>¼</sup> *<sup>υ</sup>*

The resolution of Eq. (10) with the boundary condition in Eq. (7) leads to the

So, we obtain the profile of the boundary layer velocity of the flow on the

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> <sup>þ</sup> ffiffi 2 p q ffiffiffiffiffiffiffi *υx*

These results enable us to determine the various characteristics of the boundary layer as the boundary layer thickness, the displacement thickness, the momentum

*<sup>δ</sup>* <sup>¼</sup> <sup>0</sup>*,* 128; <sup>1</sup>

2

*U*<sup>∞</sup> r

*z*¼0

¼ 2 ffiffiffi

*<sup>α</sup>*<sup>3</sup> <sup>¼</sup> *derf dz* � �

*U*∞*α*<sup>2</sup>

*h x*ð Þ¼ 2

*u U*<sup>∞</sup>

*δ*1

*<sup>δ</sup>* <sup>¼</sup> <sup>0</sup>*,* 31; *<sup>δ</sup>*<sup>2</sup>

We considered the case of a flow on a permeable flat plate with uniform suction *vp* = �*v*<sup>0</sup> 6¼ 0. To simplify the resolution of the differential Eq. (7), we imposed the particular solution:

$$h(\mathbf{x}) = A\_1 + A\_2 \mathcal{W}(\mathbf{g}(\mathbf{x})) \tag{14}$$

where *A*<sup>1</sup> and *A*<sup>2</sup> are constant parameters, *W* is the Lambert function, and *g*(*x*) is a function of *x* to determine.

By inserting Eq. (14) in the differential Eq. (7), we supplied the parameters of the scaling function:

$$A\_1 = A\_2 = \frac{b\_1}{\frac{v\_p}{U\_\infty}}; \mathbf{g}(\mathbf{x}) = -\frac{1}{2\nu} \exp\left(-\frac{b\_2}{\left(\frac{U\_\infty}{v\_p}\right)^2} \mathbf{x} + b\_3\right) \tag{15}$$

where

$$b\_1 = -\frac{a\_3 \nu}{U\_\infty} = -\frac{2\nu}{\sqrt{\pi} U\_\infty}, b\_2 = \frac{(1+\sqrt{2})\pi U\_\infty}{2\nu}, \text{and } b\_3 = \ln\left(2\nu\right) - 1 \tag{16}$$

Thus, the profile of the boundary layer velocity of the flow on the permeable flat plate with uniform suction is

$$\frac{u}{U\_{\infty}} = \text{erf}\left(\mathbf{y} / \frac{b\_1}{\frac{v\_p}{U\_{\infty}}} \left[\mathbf{1} + W\left(-\frac{\mathbf{1}}{2\nu} \exp\left(-\frac{b\_2}{\left(\frac{U\_{\infty}}{v\_p}\right)^2} \mathbf{x} + b\_3\right)\right)\right]\right) \tag{17}$$

These results enable us to determine the various characteristics of the boundary layer as the boundary layer thickness, the displacement thickness, the momentum thickness, and the parietal friction coefficient:

$$\delta(\mathbf{x}) = \mathbf{1}, \Re 2h(\mathbf{x}); \delta\_1(\mathbf{x}) = \mathbf{0}, \dots \\ \text{564h}(\mathbf{x}); \delta\_2(\mathbf{x}) = \mathbf{0}, \text{231h}(\mathbf{x}); \mathbf{G}^\sharp(\mathbf{x}) \\ \text{Re}\_\mathbf{x} = \frac{4}{\sqrt{\pi}} \frac{\mathbf{x}}{h(\mathbf{x})} \tag{18}$$

We can rewrite this friction coefficient in the universal form of law recommended by Iglisch (1949) [11].

$$\frac{C f(t)}{2\frac{v\_p}{U\_\infty}} = f\left(t = -\frac{v\_p}{U\_\infty}\sqrt{\text{Re}\_\mathbf{x}}\right) \tag{19}$$

Thus

$$\frac{\mathcal{G}f(t)}{2\frac{v\_p}{U\_o}} = -\frac{1}{\mathbf{1} + \mathcal{W}\left(-\frac{1}{2\nu}\exp\left(-\frac{(1+\sqrt{2})\pi}{2}t^2 + b\_3\right)\right)}\tag{20}$$
