# On the Stability of a Convective Flow with Nonlinear Heat Sources

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## Abstract

**:**

## 1. Introduction

- The analysis of the nonlinear boundary value problem for the temperature distribution is performed using rigorous mathematical tools such as Krasnosel’skiĭ–Guo cone expansion/contraction theorem. It is shown in the paper that, depending on the value of the Frank–Kamenetskii parameter (characterizing the thermal effect of the reaction), the number of solutions is 0, 1 or 2.
- In addition, it is proved that in the region of interest for linear stability analysis, there are two solutions of the nonlinear boundary value problem such that the maximum norm of one solution is smaller than 1 while the maximum norm of the second solution is larger than 1. This gives a simple criterion for the base flow selection—a physically realizable solution is with the smallest maximum norm—and this solution should be chosen for stability analysis.
- Bifurcation analysis is performed to numerically investigate the effect of the parameters of the problem on bifurcation diagrams.
- A linear stability problem is formulated and solved numerically for different values of the parameters characterizing the problem: the Frank–Kamenetskii parameter $\lambda $ and the Reynolds number $Re$ (based on the velocity of the flow through permeable walls). Critical values of the Grasshof number are found for different values of $\lambda $ and $Re$.
- Recommendations for the choice of parameters that result in more intensive mixing are provided.

## 2. Mathematical Formulation of the Problem

## 3. Nonlinear Boundary Value Problem

**Proposition 1.**

- 1.
- The zero function $x\equiv 0$ is not a solution of (11).
- 2.
- A function x solves (11) if and only if $y\left(t\right):=x(-t)$ solves$${y}^{\u2033}+(-\alpha ){y}^{\prime}+\lambda {e}^{y}=0,\phantom{\rule{1.em}{0ex}}y(-1)=0,\phantom{\rule{1.em}{0ex}}y\left(1\right)=0;$$besides, ${y}^{\prime}(-1)=-{x}^{\prime}\left(1\right)$ and ${y}^{\prime}\left(1\right)=-{x}^{\prime}(-1)$.
- 3.

## 4. Some Preliminaries

#### 4.1. Linear Part of the Problem

**Proposition 2.**

- 1.
- $k(t,s)>0$ for every $(t,s)\in \stackrel{\circ}{Q}$ and $k(t,s)=0$ for every $(t,s)\in \partial Q$.
- 2.
- $\mathrm{\Phi}\left(s\right)>0$ for every $s\in (-1,1)$ and $\mathrm{\Phi}(\pm 1)=0$.
- 3.
- $k(t,s)\le \mathrm{\Phi}\left(s\right)$ for every $t,s\in [-1,1]$.
- 4.
- Let a and b be two real numbers such that $-1<a<b<0$. Let$${c}_{1}:=\frac{{e}^{\alpha (1-b)}-1}{{e}^{2\alpha}-1},\phantom{\rule{0.277778em}{0ex}}{c}_{2}:=\frac{{e}^{2\alpha}-{e}^{\alpha (1-a)}}{{e}^{2\alpha}-1},\phantom{\rule{0.277778em}{0ex}}c:=min\{{c}_{1},{c}_{2}\}.$$Then, $c\in (0,1)$ and $c\phantom{\rule{0.166667em}{0ex}}\mathrm{\Phi}\left(s\right)\le k(t,s)$ for every $t\in [a,b]$ and every $s\in [-1,1]$.

**Proof.**

#### 4.2. Integral Operator

**Definition 1**

**.**Let E be a Banach space. A nonempty convex closed subset M of E is called a cone if (a) $\lambda x\in M$ for every $x\in M$ and every $\lambda \ge 0$; (b) $x\in M$, $-x\in M$ implies $x=\theta $, where θ is the zero element of E.

**Definition 2**

**Proposition 3.**

- 1.
- For every $x\in {C}_{[-1,1]}$, $Tx\left(t\right)>0$ for all $t\in (-1,1)$ and $Tx(\pm 1)=0$.
- 2.
- $T\left(\right)open="("\; close=")">{C}_{[-1,1]}$.
- 3.
- $T\left(P\right)\subset P$ and the operator $T:P\to P$ is completely continuous.
- 4.
- $T\left(K\right)\subset K$ and the operator $T:K\to K$ is completely continuous.

**Proof.**

**Corollary 1.**

**Proof.**

#### 4.3. Krasnosel’skiĭ–Guo Cone Expansion/Contraction Theorem

**Theorem 1**

- (H1)
- $\parallel Tx\parallel \le \parallel x\parallel $ for every $x\in K$ with $\parallel x\parallel =r$ and $\parallel Tx\parallel \ge \parallel x\parallel $ for every $x\in K$ with $\parallel x\parallel =R$,
- (H2)
- $\parallel Tx\parallel \ge \parallel x\parallel $ for every $x\in K$ with $\parallel x\parallel =r$ and $\parallel Tx\parallel \le \parallel x\parallel $ for every $x\in K$ with $\parallel x\parallel =R$,

**Lemma 1.**

**Proof.**

**Corollary 2.**

**Proof.**

**Lemma 2.**

**Proof.**

**Lemma 3.**

- 1.
- The function ${\lambda}^{*}:(0,+\infty )\to {\mathbb{R}}_{+}$ defined in (21) has the following properties.
- (1a)
- $\underset{r\to 0+}{lim}{\lambda}^{*}\left(r\right)=0$, $\underset{r\to +\infty}{lim}{\lambda}^{*}\left(r\right)=0$.
- (1b)
- The function ${\lambda}^{*}$ strictly increases in $(0,1]$ and strictly decreases in $[1,+\infty )$; the function ${\lambda}^{*}$ has a unique global maximum point $r=1$.

- 2.
- The function ${\lambda}_{*}:(0,+\infty )\to {\mathbb{R}}_{+}$ defined in (23) has the following properties.
- (2a)
- $\underset{r\to 0+}{lim}{\lambda}_{*}\left(r\right)=0$, $\underset{r\to +\infty}{lim}{\lambda}_{*}\left(r\right)=0$.
- (2b)
- The function ${\lambda}_{*}$ strictly increases in $(0,1/c]$ and strictly decreases in $[1/c,+\infty )$; the function ${\lambda}_{*}$ has a unique global maximum point $r=1/c$.

- 3.
- ${\lambda}^{*}\left(r\right)<{\lambda}_{*}\left(r\right)$ for every positive r.

**Proof.**

## 5. Existence and Multiplicity of Positive Solutions

#### 5.1. Application of the Krasnosel’skiĭ–Guo Cone Expansion/Contraction Theorem

**Theorem 2.**

**Proof.**

**Remark 1.**

#### 5.2. Bifurcation Analysis

#### 5.2.1. Parameter $\alpha $ Is Zero

#### 5.2.2. Parameter $\alpha $ Is Nonzero

#### 5.3. Parameter Analysis

**Theorem 3.**

**Proof.**

## 6. Linear Stability Analysis

## 7. Numerical Results

## 8. Discussion

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

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**Figure 2.**Bifurcation curve ${\mathrm{\Lambda}}_{0}$ for (25) and three positive solutions of (25): (

**a**) bifurcation curve ${\mathrm{\Lambda}}_{0}$ for (25). The straight line $\lambda ={\lambda}_{0}$, where ${\lambda}_{0}=0.5$, crosses the curve ${\mathrm{\Lambda}}_{0}$ at the points $({\lambda}_{0},{\beta}_{1})=(0.5,0.6241)$ and $({\lambda}_{0},{\beta}_{2})=(0.5,4.1344)$; (

**b**) three positive solutions of (25) corresponding to the points $({\lambda}_{0},{\beta}_{1})$, $({\lambda}_{0},{\beta}_{2})$ and $(\overline{\lambda},\overline{\beta})$ on the curve ${\mathrm{\Lambda}}_{0}$ depicted in (

**a**).

**Figure 3.**Bifurcation curve ${\mathrm{\Lambda}}_{\alpha}$ for (11) if $\alpha =2.5$. The straight line $\lambda ={\lambda}_{0}$, where ${\lambda}_{0}=0.7$, crosses the curve ${\mathrm{\Lambda}}_{\alpha}$ at the points $({\lambda}_{0},{\beta}_{1})=(0.7,1.4467)$ and $({\lambda}_{0},{\beta}_{2})=(0.7,16.1354)$.

**Figure 5.**Base flow temperature distribution for three values of $Re=0,2,4$ and $F=0.5$ (solution with smaller norm).

**Figure 6.**Base flow temperature distribution for three values of $Re=0,2,4$ and $F=0.5$ (solution with larger norm).

**Figure 12.**Critical values of the Grashof number $G{r}_{c}$ versus $Re$ for $\lambda =0.3$ (red curve), $\lambda =0.5$ (green curve) and $\lambda =0.7$ (blue curve).

N | $\mathbf{Gr}$ |
---|---|

30 | 11,157.351253 |

40 | 11,157.569064 |

50 | 11,157.811295 |

60 | 11,157.624778 |

70 | 11,157.625483 |

80 | 11,157.625661 |

90 | 11,157.649458 |

100 | 11,157.639824 |

110 | 11,157.650583 |

120 | 11,157.640381 |

130 | 11,157.632748 |

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**MDPI and ACS Style**

Gritsans, A.; Kolyshkin, A.; Sadyrbaev, F.; Yermachenko, I.
On the Stability of a Convective Flow with Nonlinear Heat Sources. *Mathematics* **2023**, *11*, 3895.
https://doi.org/10.3390/math11183895

**AMA Style**

Gritsans A, Kolyshkin A, Sadyrbaev F, Yermachenko I.
On the Stability of a Convective Flow with Nonlinear Heat Sources. *Mathematics*. 2023; 11(18):3895.
https://doi.org/10.3390/math11183895

**Chicago/Turabian Style**

Gritsans, Armands, Andrei Kolyshkin, Felix Sadyrbaev, and Inara Yermachenko.
2023. "On the Stability of a Convective Flow with Nonlinear Heat Sources" *Mathematics* 11, no. 18: 3895.
https://doi.org/10.3390/math11183895