# Viscoelastic Thermovibrational Flow Driven by Sinusoidal and Pulse (Square) Waves

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Mathematical Model

## 3. Numerical Method

^{®}is employed here to solve numerically the set of balance equations previously defined. In particular, the related segregated time-marching procedure directly relies on the PISO (Pressure Implicit with Splitting of Operator) algorithm to couple velocity and pressure fields, and the Rie and Chow scheme [36] to avoid checkerboarding of the latter one.

## 4. Results

#### 4.1. Forcing and Related System Response

#### 4.2. Patterning Behaviour and 3D Evolution

## 5. Discussion and Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Time evolution of the Axial velocity (probe being located in the center of the layer), and Nusselt number. Numerical simulation with $AR=15$, $P{r}_{g}=8$, $\Omega =25$, $\xi =0.5$, $\vartheta =0.1$, and $R{a}_{\omega}=4000$. (

**a**) Sinusoidal and (

**b**) pulse waves. In black (

**—**) is represented the axial velocity, in blue (

**—**) $Nu\left(t\right)$, and in red (

**—**) the qualitative evolution of $h\left(t\right)$.

**Figure 3.**Angular frequency of the Axial velocity signal (probe being located in the center of the layer). Numerical simulation with $AR=15$, $P{r}_{g}=8$, $\Omega =25$, $\xi =0.5$, $\vartheta =0.1$, and $R{a}_{\omega}=4000$. (

**a**) Sinusoidal and (

**b**) Pulse waves.

**Figure 4.**Thermovibrational convection in a layer of FENE-CR fluid delimited by differentially heated solid walls (snapshots of the isosurfaces of the vertical component of velocity evenly distributed over the interval ${\mathcal{I}}_{1}$). $P{r}_{g}=8$, $\Omega =25$, $R{a}_{\omega}=4000$, $\xi =0.5$, $\vartheta =0.1$, sinusoidal acceleration.

**Figure 5.**Thermovibrational convection in a layer of FENE-CR fluid delimited by differentially heated solid walls (snapshots of the isosurfaces of the vertical component of velocity evenly distributed over the interval ${\mathcal{I}}_{2}$). $P{r}_{g}=8$, $\Omega =25$, $R{a}_{\omega}=4000$, $\xi =0.5$, $\vartheta =0.1$, sinusoidal acceleration.

**Figure 6.**Thermovibrational convection in a layer of FENE-CR fluid delimited by differentially heated solid walls (snapshots of the isosurfaces of the vertical component of velocity evenly distributed over the interval ${\mathcal{I}}_{1}$). $P{r}_{g}=8$, $\Omega =25$, $R{a}_{\omega}=4000$, $\xi =0.5$, $\vartheta =0.1$, pulse wave acceleration.

**Figure 7.**Thermovibrational convection in a layer of FENE-CR fluid delimited by differentially heated solid walls (snapshots of the isosurfaces of the vertical component of velocity evenly distributed over the interval ${\mathcal{I}}_{2}$). $P{r}_{g}=8$, $\Omega =25$, $R{a}_{\omega}=4000$, $\xi =0.5$, $\vartheta =0.1$, pulse wave acceleration.

**Figure 8.**Color field of $tr\left(\tilde{\tau}\right)$ as a function of the time and the x coordinate in the center of the cavity. $0<x<15$, $y=0.5$, $z=7.5$. Numerical simulation with $AR=15$, $P{r}_{g}=8$, $\Omega =25$, $\xi =0.5$, $\vartheta =0.1$, and $R{a}_{\omega}=4000$. (

**a**) Sinusoidal and (

**b**) pulse waves.

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**MDPI and ACS Style**

Lappa, M.; Boaro, A.
Viscoelastic Thermovibrational Flow Driven by Sinusoidal and Pulse (Square) Waves. *Fluids* **2021**, *6*, 311.
https://doi.org/10.3390/fluids6090311

**AMA Style**

Lappa M, Boaro A.
Viscoelastic Thermovibrational Flow Driven by Sinusoidal and Pulse (Square) Waves. *Fluids*. 2021; 6(9):311.
https://doi.org/10.3390/fluids6090311

**Chicago/Turabian Style**

Lappa, Marcello, and Alessio Boaro.
2021. "Viscoelastic Thermovibrational Flow Driven by Sinusoidal and Pulse (Square) Waves" *Fluids* 6, no. 9: 311.
https://doi.org/10.3390/fluids6090311