# Group Analysis of the Boundary Layer Equations in the Models of Polymer Solutions

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## Abstract

**:**

## 1. Introduction

## 2. Derivation of Boundary Layer Equations

## 3. Group Classification of System (4)

#### 3.1. Case ${P}_{xx}\ne 0$

**Remark**

**1.**

#### 3.2. Case ${P}_{xx}=0$

## 4. One Class of Solutions of System (7)

## 5. Group Foliation with Respect to ${X}_{h}$

#### 5.1. Deriving the Resolving System

#### 5.2. Some Classes of Solutions of (24)

#### 5.2.1. Case $m=1$

#### 5.2.2. Case $m=2$

#### 5.3. Group Properties of Equation (24)

- (a)
- Solutions invariant with respect to the generator ${X}_{1}$. Such solutions have the representation$$\phi =xf(t,z),\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}z=u{x}^{-1}.$$Substituting this representation of a solution into (28), one finds that$$\psi =\frac{\chi f{f}_{z}+{\left(f{f}_{z}\right)}_{t}+f{f}_{z}-z{\left(f{f}_{z}\right)}_{z}-g}{1-{\left(f{f}_{z}\right)}_{z}}.$$The resolving Equation (24) becomes a partial differential equation with two independent variables,$${f}_{t}+z(f-z{f}_{z})={\left(\frac{\psi}{f}\right)}_{z}{f}^{2}.$$
- (b)
- Solutions invariant with respect to the generator $\alpha {X}_{2}+\beta {X}_{3}$, (${\alpha}^{2}+{\beta}^{2}\ne 0$), have the representation$$\phi (x,t,u)=f\left(t,z\right),\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}z=u-x\frac{{\xi}^{\prime}\left(t\right)}{\xi \left(t\right)},$$$${f}_{t}-{\xi}^{\prime}{\xi}^{-1}z{f}_{z}={\left(\frac{\tilde{\psi}}{f}\right)}_{z}{f}^{2},$$$$\tilde{\psi}=\frac{\chi f{f}_{z}+{\left(f{f}_{z}\right)}_{t}-{\xi}^{\prime}{\xi}^{-1}z{\left(f{f}_{z}\right)}_{z}}{1-{\left(f{f}_{z}\right)}_{z}}.$$
- (c)
- Solutions invariant with respect to the subalgebra $\{{X}_{1},\alpha {X}_{2}+\beta {X}_{3}\}$, (${\alpha}^{2}+{\beta}^{2}\ne 0$). These solutions have the representation$$\phi (x,t,u)=f\left(t\right)\left(u-x\frac{{\xi}^{\prime}\left(t\right)}{\xi \left(t\right)}\right),$$

## 6. Group Classification of Stationary System (4)

#### 6.1. Case ${P}_{x}\ne 0$

#### 6.2. Case ${P}_{x}=0$

#### 6.3. Invariant Solutions

#### 6.4. Group Foliation with Respect to ${X}_{h}$

#### 6.5. Equation (38) in Mises Coordinates

**Remark**

**2.**

## 7. Group Classification of the Boundary Layer Equations of Rivlin-Ericksen Fluids

## 8. Discussion

#### 8.1. Voitkunskii-Amfilokhiev-Pavlovskii Model

#### 8.2. Blasius Problem

#### 8.3. Separation

## 9. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Graphs of the function $\dot{q}$ in the solution of problem (41) for different values of the parameter $\delta $.

**Figure 2.**Solution ${U}_{0}\left(z\right)$ of Equation (47) for different values of ${U}_{0}^{\u2033}\left(a\right):\phantom{\rule{4pt}{0ex}}-10;\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}-5;\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}0;\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}5;\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}50$, presented in bottom-up order.

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Meleshko, S.V.; Pukhnachev, V.V.
Group Analysis of the Boundary Layer Equations in the Models of Polymer Solutions. *Symmetry* **2020**, *12*, 1084.
https://doi.org/10.3390/sym12071084

**AMA Style**

Meleshko SV, Pukhnachev VV.
Group Analysis of the Boundary Layer Equations in the Models of Polymer Solutions. *Symmetry*. 2020; 12(7):1084.
https://doi.org/10.3390/sym12071084

**Chicago/Turabian Style**

Meleshko, Sergey V., and Vladislav V. Pukhnachev.
2020. "Group Analysis of the Boundary Layer Equations in the Models of Polymer Solutions" *Symmetry* 12, no. 7: 1084.
https://doi.org/10.3390/sym12071084