**4. Boundary layer**

Boundary layers are regions in which a rapid change occurs in the value of a variable. Some physical examples include "the fluid velocity near a solid wall", "the velocity at the edge of a jet of fluid", "the temperature of a fluid near a solid wall." Ludwig Prandtl pioneered the subject of boundary layer theory in his explanation of how a quantity as small as the viscosity of common fluids such as water and air could nevertheless play a crucial role in determining their flow. The viscosity of many fluids is very small and yet taking account of this small quantity is vital. The essential point is that the viscous term involves higher order derivatives so that its omission necessitates the loss of a boundary condition. The ideal flow solution allow slip to occur between a solid and fluid. In reality the tangential velocity of a fluid relative to a solid is zero. The fluid is brought to rest by the action of a tangential stress resulting from the viscous force.

Mathematically the occurrence of boundary layers is associated with the presence of a small parameter multiplying the highest derivative in the governing equation of a process. A straightforward perturbation expansion using an asymptotic sequence in the small parameter leads to differential equations of lower order than the original governing equation. In consequence not all of the boundary and initial conditions can be satisfied by the perturbation expansion. This is an example of what is commonly referred to as a *singular perturbation problem*. The technique for overcoming the difficulty is to combine the straightforward expansion, which is valid away from the layer adjacent to the boundary. The straightforward expansion is referred to as the *outer expansion*. The *inner expansion* associated with the boundary layer region is expressed in terms of a stretched variable, rather than the original independent variable, which takes due account of the scale of certain derivative terms. The inner and outer expansions are matched over a region located at the edge of the boundary layer. The technique is called the method of *matched asymptotic expansions*.

Consider the following two-point boundary value problem:

$$\begin{cases} \varepsilon \frac{d^2 u}{d\kappa^2} + \frac{du}{d\kappa} = 2\kappa + 1, \quad \varkappa \in (0, 1) \\\\ u(0) = 1, \quad u(1) = 4, \end{cases} \tag{38}$$

where *ε*>0 is a small parameter. If we assume that *u* possesses a straightforward expansion in powers of *ε*,

$$
\mu(\mathbf{x}; \boldsymbol{\varepsilon}) \sim \mu\_0(\mathbf{x}) + \varepsilon \mu\_1(\mathbf{x}) + \varepsilon^2 \mu\_2(\mathbf{x}) + \cdots,\tag{39}
$$

then the equations associated with powers of *ε* leads to

$$O(\mathbf{1}): \qquad \frac{du\_0}{d\mathbf{x}} = 2\mathbf{x} + \mathbf{1},\tag{40}$$

$$O(\varepsilon^n): \qquad \frac{du\_n}{d\mathfrak{x}} = -\frac{d^2 u\_{n-1}}{d\mathfrak{x}^2}, \quad \text{for } n = 1, 2, 3, \dots \tag{41}$$

and the boundary conditions require

$$\mu\_0(\mathbf{0}) + \varepsilon u\_1(\mathbf{0}) + \cdots \sim \mathbf{1} + \varepsilon \cdot \mathbf{0} + \cdots,$$

$$\mu\_0(\mathbf{1}) + \varepsilon u\_1(\mathbf{1}) + \cdots \sim \mathbf{4} + \varepsilon \cdot \mathbf{0} + \cdots,$$

which leads to

$$\begin{aligned} \mu\_0(\mathbf{0}) &= \mathbf{1}, \quad \mu\_0(\mathbf{1}) = \mathbf{4}, \\ \mu\_n(\mathbf{0}) &= \mathbf{0}, \quad \mu\_n(\mathbf{1}) = \mathbf{0}, \quad \text{for } n = \mathbf{1}, 2, \cdots \end{aligned} \tag{42}$$

Equation (42) require that each *un*ð Þ *x* satisfy two boundary conditions. This is in general impossible since Eqs. (41) and (42) governing each *un* are of first-order. Now the question is which one of the boundary condition has to be taken into account. We will find out that the boundary condition at *x* ¼ 0 must be abandoned and consequently the expansion (39) is invalid near *x* ¼ 0.

The general solution of (42) is *<sup>u</sup>*0ð Þ¼ *<sup>x</sup> <sup>x</sup>*<sup>2</sup> <sup>þ</sup> *<sup>x</sup>* <sup>þ</sup> *<sup>C</sup>*, using the boundary condition *u*0ð Þ¼ 1 4, we obtain

$$
\mu\_0(\mathbf{x}) = \mathbf{x}^2 + \mathbf{x} + \mathbf{2}.
$$

From (42), we obtain the equations

$$\frac{du\_1}{d\mathbf{x}} = -\mathbf{2}, \quad u\_1(\mathbf{1}) = \mathbf{0},$$

$$\frac{du\_2}{d\mathbf{x}} = \mathbf{0}, \qquad u\_2(\mathbf{1}) = \mathbf{0},$$

and its solutions are

$$
\mu\_1(\varkappa) = -2(\varkappa - 1), \quad \mu\_n(\varkappa) = 0, \quad n \ge 2.
$$

Therefore, the outer expansion is

$$u^{\rm out}(\varkappa;\varepsilon) = \left(\varkappa^2 + \varkappa + 2\right) + \varepsilon \, 2(1-\varkappa),\tag{43}$$

where 'out' label is used to indicate that the solution is valid away from the region near *<sup>x</sup>* <sup>¼</sup> 0. Clearly *<sup>u</sup>*out fails to satisfy the boundary condition at *<sup>x</sup>* <sup>¼</sup> 0. The reason why the outer solution is of use is that it closely follows the exact solution of the problem except in a narrow region near *x* ¼ 0, where the exact solution changes rapidly in order to satisfy the boundary condition.

The exact solution of the BVP (38) can be obtained as

$$u(\mathbf{x}) = A + B e^{-\mathbf{x}/\varepsilon} + \mathbf{x}^2 + \varkappa (\mathbf{1} - 2\varepsilon). \tag{44}$$

The constants *A* and *B* are determined from the boundary conditions:

$$\begin{cases} A + B = \mathbf{1}, \\ A + B e^{-1/\varepsilon} + \mathbf{2} - \mathbf{2}\varepsilon = \mathbf{4} \end{cases} \tag{45}$$

We know that *<sup>e</sup>*�1*=<sup>ε</sup>* <sup>¼</sup> *<sup>o</sup> <sup>ε</sup><sup>N</sup>* � �, as *<sup>ε</sup>* ! 0, for all *<sup>N</sup>*. This means that the exponential term tends to zero faster than any power of *ε*, as *ε* ! 0. It is called a *transcendentally small term* (T.S.T.) and can always be neglected since its contribution is asymptotically always less than any power of *ε*. Thus, (45) gives

$$A = \mathfrak{Z}(\mathfrak{1} + \mathfrak{e}), \quad B = -(\mathfrak{1} + \mathfrak{2}\mathfrak{e}),$$

and the exact solution is

$$
\mu^{\rm ex}(\mathbf{x}) = \mathbf{2}(\mathbf{1} + \boldsymbol{\varepsilon}) - (\mathbf{1} + \mathbf{2}\boldsymbol{\varepsilon})e^{-\mathbf{x}/\boldsymbol{\varepsilon}} + \boldsymbol{\varkappa}^2 + \boldsymbol{\varkappa}(\mathbf{1} - \mathbf{2}\boldsymbol{\varepsilon}),
\tag{46}
$$

after rearranging the terms in asymptotic order, we obtain

$$u^{\rm ex}(\mathbf{x}) = \left(\mathbf{x}^2 + \mathbf{x} + \mathbf{2}\right) - e^{-\mathbf{x}/\varepsilon} + \varepsilon \left[2(\mathbf{1} - \mathbf{x}) - 2e^{-\mathbf{x}/\varepsilon}\right].\tag{47}$$

Comparing the exact solution with the outer expansion shows that the terms involving *<sup>e</sup>*�*x=<sup>ε</sup>* are absent. The effect of these terms is negligible when *<sup>x</sup>* <sup>¼</sup> *<sup>O</sup>*ð Þ<sup>1</sup> . But, when *<sup>x</sup>* <sup>¼</sup> *<sup>O</sup>*ð Þ*<sup>ε</sup>* , then *<sup>e</sup>*�*x<sup>ε</sup>* <sup>¼</sup> *<sup>O</sup>*ð Þ<sup>1</sup> . It is clear that as *<sup>ε</sup>* ! 0 the region in which the outer solution departs from the exact solution becomes arbitrarily close to *x* ¼ 0 with a thickness *O*ð Þ*ε* . This region is called the *boundary layer*.

The behavior of the exact solution and the zeroth-order term of the outer expansion are plotted in **Figure 1** for various values of *ε*.

**Figure 1.** *Exact solution of (38) for various values of ε.*

*Perturbation Expansion to the Solution of Differential Equations DOI: http://dx.doi.org/10.5772/intechopen.94173*

By differentiating the leading order term *u*ex <sup>0</sup> , of the exact solution, we have

$$\begin{aligned} u\_0^{\rm ex} &= x^2 + x + 2 - e^{-x/\varepsilon} \\\\ \frac{d u\_0^{\rm ex}}{d x} &= 2x + 1 + \frac{1}{\varepsilon} e^{-x/\varepsilon} \\\\ \frac{d^2 u\_0^{\rm ex}}{d x^2} &= 2 - \frac{1}{\varepsilon^2} e^{-x/\varepsilon} \end{aligned}$$

Outside the boundary layer, i.e., for *<sup>x</sup>* <sup>¼</sup> *<sup>O</sup>*ð Þ<sup>1</sup> , we have *<sup>e</sup>*�*x=<sup>ε</sup>* <sup>¼</sup> *<sup>o</sup> <sup>ε</sup><sup>N</sup>* , <sup>∀</sup>*N*, so *ε*�1*e*�*x=<sup>ε</sup>* and *ε*�2*e*�*x=<sup>ε</sup>* are also transcendentally small. Within the boundary layer when *<sup>x</sup>* <sup>¼</sup> *<sup>O</sup>*ð Þ*<sup>ε</sup>* , we have *<sup>e</sup>*�*x=<sup>ε</sup>* <sup>¼</sup> *<sup>O</sup>*ð Þ<sup>1</sup> . The order of *<sup>u</sup>*ex <sup>0</sup> and its derivatives are given below:


This indicates that *x* is the appropriate independent variable outside the boundary layer where *u*ex <sup>0</sup> and its derivatives are of *O*ð Þ1 quantities. However, within the boundary layer the appropriately scaled independent variable is *s* ¼ *x=ε*, then

$$
\frac{du}{d\mathfrak{x}} = \varepsilon^{-1} \frac{dv}{ds}, \quad \frac{d^2u}{d\mathfrak{x}^2} = \varepsilon^{-2} \frac{d^2v}{ds^2},
$$

so that within the boundary layer

$$\frac{du}{d\mathfrak{x}} = O(\mathfrak{1}), \quad \text{and} \quad \frac{d^2u}{d\mathfrak{x}^2} = O(\mathfrak{1}).$$

The variable *s* ¼ *x=ε* is called a *stretched variable*. The differential equations becomes

$$\frac{d^2v}{ds^2} + \frac{dv}{ds} = \varepsilon + 2\varepsilon^2 \mathfrak{s}.\tag{48}$$

We assume a boundary layer expansion, called the *inner expansion* of the form

$$
\upsilon(\mathfrak{s};\mathfrak{e}) \sim \upsilon\_0(\mathfrak{s}) + \epsilon \upsilon\_1(\mathfrak{s}) + \cdots. \tag{49}
$$

The inner expansion will satisfy the boundary condition at *x* ¼ *s* ¼ 0 namely *v*0ð Þ¼ *s* ¼ 0 1 giving *v*0ð Þ¼ 0 1, and *vn*ð Þ¼ 0 0, *n* ¼ 1, 2, ⋯. Substituting (49) into the DE (48), we obtain the following set of equations:

#### *A Collection of Papers on Chaos Theory and Its Applications*

$$\begin{cases} O(1): \quad \frac{d^2 v\_0}{ds^2} + \frac{dv\_0}{ds} = 0, \quad v\_0(0) = 1 \\ O(\epsilon): \quad \frac{d^2 v\_1}{ds^2} + \frac{dv\_1}{ds} = 1, \quad v\_1(0) = 0 \\\\ O(\epsilon^2): \quad \frac{d^2 v\_2}{ds^2} + \frac{dv\_2}{ds} = 2\epsilon, \quad v\_1(0) = 0 \\\\ O(\epsilon^n): \quad \frac{d^2 v\_n}{ds^2} + \frac{dv\_n}{ds} = 0, \quad v\_n(0) = 0, \quad n = 3, 4, \cdots \end{cases} \tag{50}$$

with solutions

$$\begin{cases} v\_0 = A + (1 - A)e^{-s} \\ v\_1 = B - Be^{-s} + s \\ v\_2 = C - Ce^{-s} + s^2 - 2s \\ v\_n = D\_n - D\_n e^{-s}, \quad n = 3, 4, \dots \end{cases} \tag{51}$$

The boundary condition at *x* ¼ 1 cannot be used to determine the constants appearing in these solutions because the DEs (50) are only valid in the boundary layer. The constants in (51) are determined by matching the inner and outer expansions. We shall first restrict our attention to matching the leading order expansions *u*<sup>0</sup> and *v*0. The method which we shall apply is *Prandtl's matching condition*.

The leading order terms in the 'inner' and 'outer' expansions are to be matched at the 'edge of the boundary layer'. Of course there is no precise edge of the boundary layer, we simply know that it has thickness of order *O*ð Þ*ε* . A plausible matching procedure would be to equate *u*<sup>0</sup> and *v*<sup>0</sup> at a value of *x* such that the region of rapid change has passed. We might choose to equate the terms at the point *x* ¼ 5*ε*. The leading order expansions are

$$
\mu\_0 = \mathfrak{x}^2 + \mathfrak{x} + \mathfrak{z} \qquad \nu\_0 = A + (\mathfrak{1} - A)e^{-s}.
$$

Equating at *x* ¼ 5*ε* gives the following:

$$A = \frac{2 + \mathbf{5e} + 2\mathbf{5e}^2 - e^{-\mathbf{5}}}{1 - e^{-\mathbf{5}}}.$$

If, instead we choose to match at *x* ¼ 6*ε*, then we obtain

$$A = \frac{2 + \mathfrak{G}\varepsilon + \mathfrak{G}\mathfrak{G}^2 - e^{-6}}{1 - e^{-6}}.$$

These two expressions differ in the argument of the exponential and differ algebraically with 5*ε* replaced by 6*ε*. The exponential functions are approaching transcendentally small values so that their contribution can be neglected. The algebraic difference is of *O*ð Þ*ε* . Thus, the arbitrariness in the decision of the point at which we choose to equate the expansions leads to a difference of *O*ð Þ*ε* . But we are only dealing with leading order expansions anyway. The difference between the exact solution and the leading order expansions will of *O*ð Þ*ε* so that an arbitrariness in *v*<sup>0</sup> and *u*<sup>0</sup> of *O*ð Þ*ε* is immaterial. Rather than choose between, for example, 5*ε* and 6*ε* as the value of *x* to evaluate *u*<sup>0</sup> we may take the value at *x* ¼ 0, since

*Perturbation Expansion to the Solution of Differential Equations DOI: http://dx.doi.org/10.5772/intechopen.94173*

$$
\mathfrak{u}\_0[\mathfrak{x} = O(\mathfrak{e})] = \mathfrak{u}\_0(\mathfrak{0}) + O(\mathfrak{e}),
$$

where the remainder is uniformly *O*ð Þ*ε* since the gradient of *u*<sup>0</sup> is *O*ð Þ1 . For the inner expansion we are to ensure that the rapidly varying function has achieved its asymptotic value at the edge of the boundary layer. This means that the term *e*�*x=<sup>ε</sup>* should be replaced by zero. This can be achieved by taking the limit *s* ! ∞. Thus, rather than choosing a specific point to equate the inner and outer terms er are led to the following *Prandtl's matching condition*:

$$\lim\_{\mathfrak{x}\to\mathbf{0}}\omega\_{0}(\mathfrak{x})=\lim\_{\mathfrak{s}\to\mathbf{s}}\upsilon\_{0}(\mathfrak{s}).\tag{52}$$

The limit *s* ! ∞ may appear rather dangerous since although it certainly removes the exponential term it could lead to an algebraically unbounded term. For example, if *<sup>v</sup>*<sup>0</sup> <sup>¼</sup> *As* <sup>þ</sup> ð Þ <sup>1</sup> � *<sup>A</sup> <sup>e</sup>*�*<sup>s</sup>* , then the first member would be unbounded as *s* ! ∞. This possibility can be eliminated since the inner expansion must be of a form which varies rapidly for *x* ¼ *O*ð Þ*ε* but not for *x* ¼ *O*ð Þ1 , i.e., not for *s* ! ∞. In practice, if the boundary layer has been properly located and the correct inner variable is used then Prandtl's matching condition is valid and elegantly avoids the need to choose an arbitrary 'edge' of the boundary layer.

Applying these conditions to the current example leads to

$$\lim\_{\mathfrak{x}\to 0} \left(\mathfrak{x}^2 + \mathfrak{x} + \mathfrak{z}\right) = \lim\_{\mathfrak{x}\to \infty} [A + (\mathfrak{1} - A)e^{-\mathfrak{z}}],$$

which yields *A* ¼ 2. Thus the leading order terms in the expansion solutions are

Outer region : *<sup>u</sup>*<sup>0</sup> <sup>¼</sup> *<sup>x</sup>*<sup>2</sup> <sup>þ</sup> *<sup>x</sup>* <sup>þ</sup> 2, for *<sup>x</sup>* <sup>¼</sup> *<sup>O</sup>*ð Þ<sup>1</sup> Inner region : *v*<sup>0</sup> ¼ 2 � *e* �*x=ε* , for *x* ¼ *O*ð Þ*ε*

To prove that these are valid leading terms we consider *u*ex:

$$\text{If } \mathfrak{x} = \mathcal{O}(\mathbf{1}), \quad \text{then } u\_0^{\text{ex}} = \mathfrak{x}^2 + \mathfrak{x} + \mathfrak{Z} + \text{T.S.T.}$$

$$\text{If } \mathfrak{x} = \mathcal{O}(\varepsilon), \quad \text{then } u\_0^{\text{ex}} = \mathfrak{Z} - e^{-\mathfrak{x}/\varepsilon} + \mathcal{O}(\varepsilon)$$

We conclude that the matching condition has correctly predicted the leading order terms.

### **4.1 Composite expansion**

As single composite expression for these leading order terms can be constructed using the combination

$$
\mu\_0^{\text{comp}} = \mu\_0 + v\_0 - \mu\_0^{\text{match}},\tag{53}
$$

where *u*match <sup>0</sup> is given by (52). Then,

$$\text{for } \varkappa = O(1), \quad v\_0 = u\_0^{\text{match}} + \text{T.S.T.}, \quad \text{so that } u\_0^{\text{comp}} = u\_0^{\text{match}} + \text{T.S.T.}$$

$$\text{for } \varkappa = O(\varepsilon), \quad u\_0 = u\_0^{\text{match}} + O(\varepsilon), \qquad \text{so that } u\_0^{\text{comp}} = v\_0 + O(\varepsilon)$$

For the current example, *u*match <sup>0</sup> ¼ 2, so the composite expansion is *A Collection of Papers on Chaos Theory and Its Applications*

$$
\mu\_0^{\text{match}} = \mathbf{x}^2 + \mathbf{x} + \mathbf{2} - e^{-\mathbf{x}/\mathbf{r}}.\tag{54}
$$

Prandtl's matching condition can only be used for the leading order terms in the asymptotic expansions.

The outer, inner and composite expansions of the BVP (38) are presented in **Figures 2** and **3** for different values of *ε*. From these figures, one can easily identify the need and efficiency of the composite expansion.

#### **4.2 Boundary layer location**

Consider the following linear DE

$$a\frac{d^2u}{d\mathfrak{x}^2} + a(\mathfrak{x})\frac{du}{d\mathfrak{x}} + b(\mathfrak{x})u = c(\mathfrak{x}), \quad \mathfrak{x} \in (\mathfrak{x}\_1, \mathfrak{x}\_2). \tag{55}$$

The following general statements can be made about the boundary layer location and the nature of the inner expansion.

**Case I.** If *a x*ð Þ>0 throughout ð Þ *x*1, *x*<sup>2</sup> , then the boundary layer will occur at *x* ¼ *x*1. The stretching transformation will be *s* ¼ ð Þ *x* � *x*<sup>1</sup> *=ε*, and the one-term inner expansion will satisfy

#### **Figure 2.**

*Outer, inner and composite expansions. (a) For ε = 0.2; (b) For ε = 0.1.*

**Figure 3.** *Outer, inner and composite expansions. (a) For ε = 0.05; (b) For ε = 0.025.*

*Perturbation Expansion to the Solution of Differential Equations DOI: http://dx.doi.org/10.5772/intechopen.94173*

$$\frac{d^2v\_0}{ds^2} + a(\mathbf{x}\_1)\frac{dv\_0}{ds} = \mathbf{0}.$$

The solution of this equation is

$$v\_0 = A + B e^{a(\mathbf{x}\_1)(\mathbf{x} - \mathbf{x}\_1)/\sigma} \mathbf{s}$$

where *A* þ *B* ¼ *u x*ð Þ ¼ *x*<sup>1</sup> . The other condition to determine the constants *A* and *B* is obtained by matching with the value of the outer expansion at *x* ¼ *x*1.

**Case II.** If *a x*ð Þ<0 throughout ð Þ *x*1, *x*<sup>2</sup> , then the boundary layer will occur at *x* ¼ *x*2. The stretching transformation will be *s* ¼ ð Þ *x*<sup>2</sup> � *x =ε*, and the one-term inner expansion will involve the rapidly decaying function *ea x*ð Þ<sup>2</sup> ð Þ *<sup>x</sup>*2�*<sup>x</sup> <sup>=</sup><sup>ε</sup>* .

**Case III.** If *a x*ð Þ changes sign in the interval *x*<sup>1</sup> <*x*<*x*2, then a boundary layer occurs at an interior point *x*0, where *a x*ð Þ¼ <sup>0</sup> 0 and boundary layers may also occur at both ends *x*<sup>1</sup> and *x*2.
