# Stokes’ First Problem for Viscoelastic Fluids with a Fractional Maxwell Model

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Fractional Maxwell Model-Material Functions

**Proposition**

**1.**

- (a)
- $0<\alpha \le \beta \le 1$;
- (b)
- $G\left(t\right)$ is monotonically non-increasing for $t>0$;
- (c)
- $J\left(t\right)$ is monotonically non-decreasing for $t>0$;
- (d)
- $G\left(t\right)$ is a completely monotone function;
- (e)
- $J\left(t\right)$ is a complete Bernstein function.

**Proof.**

## 3. Stokes’ First Problem

**Proposition**

**2.**

**Proof.**

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

## 4. Explicit Representation of Solution and Numerical Results

**Theorem**

**3.**

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A

- (A)
- The class $\mathcal{CMF}$ is closed under point-wise multiplication.
- (B)
- If $\phi \in \mathcal{BF}$ then $\phi \left(x\right)/x\in \mathcal{CMF}$.
- (C)
- If $\phi \in \mathcal{CMF}$ and $\psi \in \mathcal{BF}$ then the composite function $\phi \left(\psi \right)\in \mathcal{CMF}$.
- (D)
- $\phi \in \mathcal{CBF}$ if and only if $1/\phi \in \mathcal{SF}$.
- (E)
- If $\phi ,\psi \in \mathcal{CBF}$ then $\sqrt{\phi .\psi}\in \mathcal{CBF}$.
- (F)
- The Mittag-Leffler function ${E}_{\alpha ,\beta}(-x)\in \mathcal{CMF}$ for $0<\alpha \le 1$, $\alpha \le \beta $.
- (G)
- If $\alpha \in [0,1]$ then ${x}^{\alpha}\in \mathcal{CBF}$ and ${x}^{\alpha -1}\in \mathcal{SF}$;
- (H)
- If $\phi \in \mathcal{SF}$ or $\mathcal{CBF}$ then it can be analytically extended to $\mathbb{C}\setminus (-\infty ,0]$ and $\left|\mathrm{arg}\phi \right(z\left)\right|\le \left|\mathrm{arg}z\right|$ for $z\in \mathbb{C}\setminus (-\infty ,0].$

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**Figure 1.**Velocity field $u(x,t)$ for $\beta =0.8$ and different values of $\alpha $: $0.2,0.4,0.6$, and $0.8$; $a={10}^{\alpha}$, $b={10}^{\beta -1}$; (

**a**) $u(x,t)$ as a function of x for $t=4$; (

**b**) $u(x,t)$ as a function of t for $x=1$.

**Figure 2.**Velocity field $u(x,t)$ for $\alpha =0.5$ and three different values of $\beta $: $0.5,0.7,0.9$; $a={10}^{\alpha}$, $b={10}^{\beta -1}$; (

**a**) $u(x,t)$ as a function of x for $t=4$; (

**b**) $u(x,t)$ as a function of t for $x=1$.

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**MDPI and ACS Style**

Bazhlekova, E.; Bazhlekov, I.
Stokes’ First Problem for Viscoelastic Fluids with a Fractional Maxwell Model. *Fractal Fract.* **2017**, *1*, 7.
https://doi.org/10.3390/fractalfract1010007

**AMA Style**

Bazhlekova E, Bazhlekov I.
Stokes’ First Problem for Viscoelastic Fluids with a Fractional Maxwell Model. *Fractal and Fractional*. 2017; 1(1):7.
https://doi.org/10.3390/fractalfract1010007

**Chicago/Turabian Style**

Bazhlekova, Emilia, and Ivan Bazhlekov.
2017. "Stokes’ First Problem for Viscoelastic Fluids with a Fractional Maxwell Model" *Fractal and Fractional* 1, no. 1: 7.
https://doi.org/10.3390/fractalfract1010007