# Porous and Magnetic Effects on Modified Stokes’ Problems for Generalized Burgers’ Fluids

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## Abstract

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## 1. Introduction

**T**,

**S**,

**A**and $-\tilde{p}\mathit{I}$ have well known significations, $\mu $ is the dynamic viscosity, ${\tilde{\lambda}}_{i}(i=1,2,3,4)$ are material constants and $\delta /\delta \tau $ denotes the upper-convected time derivative. As compared with the incompressible Burgers’ model [2], a novel material parameter represented by ${\tilde{\lambda}}_{4}$ emerges within the constitutive equations. The first precise solutions for unsteady motions of IGBFs appear to be those proposed by Fetecau et al. [1] within rectangular regions. Additional noteworthy solutions describing different unsteady motions of the same fluids in such a domain have been identified by Zheng et al. [3], Jamil [4], Fetecau et al. [5] and Khan et al. [6]. Starting solutions for oscillatory motions of these fluids have been established by Tong [7] in cylindrical domains.

## 2. Governing Equations

**j**is the unit vector along the y-axis. For such motions, the continuity equation is identically satisfied. We also assume that the extra-stress tensor

**S**, as well as the velocity vector $\tilde{\mathit{w}}$, is a function of z and $\tau $ only.

**S**are zero. In addition, the tangential shear stress $\tilde{\vartheta}(z,\tau )={S}_{yz}(z,\tau )$ has to satisfy the partial differential equation

## 3. Analytical Expressions for the Dimensionless Steady-State Solutions

#### 3.1. Analytical Expressions for ${\tilde{w}}_{cp}(z,\tau )$ and ${\tilde{w}}_{sp}(z,\tau )$

#### 3.2. Exact Expressions for ${\tilde{\vartheta}}_{cp}(z,\tau ),{\tilde{\vartheta}}_{sp}(z,\tau )$ and ${\tilde{R}}_{cp}(z,\tau ),{\tilde{R}}_{sp}(z,\tau )$

#### 3.3. Limiting Case $\omega \to 0$ (Modified Stokes’ First Problem)

#### 3.4. A Simple but Useful Observation Regarding Governing Equations for Velocity and Shear Stress

**S**is a function of y and $\tau $ only, the fluid motion is governed by Equation (3) and

## 4. Some Numerical Results and Discussion

## 5. Conclusions

- Closed-form expressions are provided for the dimensionless steady-state velocity, shear stress and Darcy’s resistance of MHD modified Stokes’s problems for IGBFs through a porous medium. For validation, the fluid velocities are presented in equivalent forms.
- The obtained expressions can be immediately particularized to find similar solutions for Burgers, Oldroyd-B, Maxwell, second-grade and Newtonian fluids subject to same motions.
- Convergence of the starting velocities (numerical solutions) to their steady components is graphically proven, and the necessary time to reach a steady state is found.
- This time proportionally increases with the augmentation of the porous and magnetic parameters K and M, respectively. Consequently, the establishment of a steady state is more expeditiously achieved in the absence of a porous medium or a magnetic field.
- The flow resistance of fluid exhibits a propensity to escalate in the presence of a porous medium or magnetic field. This results in a decelerated flow rate of the fluid within a porous medium or in the presence of a magnetic field.
- The governing equations for the fluid velocity and the non-trivial shear stress corresponding to the MHD motions of IGBFs between parallel plates are identical in form. Consequently, MHD motion problems with shear stress on the boundary can be solved for incompressible rate-type fluids.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

T | Cauchy stress tensor |

S | Extra-stress tensor |

A | First Rivlin–Ericksen tensor |

I | Identity tensor |

$\tilde{p}$ | Hydrostatic pressure |

$\tilde{W}$ | Amplitude of the oscillations |

${\lambda}_{1},{\lambda}_{2},{\lambda}_{3},{\lambda}_{4}$ | Material constants |

$\mu $ | Dynamic viscosity |

$\nu $ | Kinematic viscosity |

$\tau $ | Time |

$x,y,z$ | Cartesian coordinates |

$\omega $ | Frequency of oscillations |

$\tilde{w}$ | Velocity vector |

$\tilde{w}(z,\tau )$ | Fluid velocity |

$\tilde{\vartheta}(z,\tau )$ | Tangential shear stress |

$\tilde{R}(z,\tau )$ | Darcy’s resistance |

d | Distance between plates |

$\rho $ | Fluid density |

$\sigma $ | Electrical conductivity |

$\phi $ | Porosity |

k | Permeability of porous medium |

Q | Volume flux |

$\alpha ,\beta ,\gamma ,\delta $ | Dimensionless constants |

M | Magnetic parameter |

K | Porous parameter |

B | Magnitude of magnetic field |

${K}_{eff}$ | Effective permeability |

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**Figure 1.**Diagrams of the velocities ${\tilde{w}}_{cp}(z,\tau )$ and ${\tilde{w}}_{sp}(z,\tau )$ given by Equations (25) and (32) and (26) and (34), respectively, for $\alpha =0.7,\beta =0.6,\gamma =0.5,$ $\delta =0.4,\omega =\pi /6,M=0.8$ and $\tau =10$.

**Figure 2.**Equivalence of expressions of ${\tilde{w}}_{Cp}(z)$ given by Equations (51) and (54) for two values of the effective permeability ${K}_{eff}$.

**Figure 3.**Convergence of ${\tilde{w}}_{c}(z,\tau )$ (numerical solutions) with ${\tilde{w}}_{cp}(z,\tau )$ for $\alpha =0.7,\beta =0.6,$ $\gamma =0.5,$$\delta =0.4,\omega =\pi /6,M=0.8,$ two values of K and increasing values of time $\tau $.

**Figure 4.**Convergence of ${\tilde{w}}_{c}(z,\tau )$ (numerical solutions) with ${\tilde{w}}_{cp}(z,\tau )$ for $\alpha =0.7,\beta =0.6,$ $\gamma =0.5,$$\delta =0.4,\omega =\pi /4,K=0.1,$ two values of M and increasing values of time $\tau $.

**Figure 5.**Convergence of ${\tilde{w}}_{s}(z,\tau )$ (numerical solutions) with ${\tilde{w}}_{sp}(z,\tau )$ for $\alpha =0.7,\beta =0.6,$ $\gamma =0.5,$$\delta =0.4,\omega =\pi /6,M=0.8,$ two values of K and increasing values of time $\tau $.

**Figure 6.**Convergence of ${\tilde{w}}_{s}(z,\tau )$ (numerical solutions) with ${\tilde{w}}_{sp}(z,\tau )$ for $\alpha =0.7,\beta =0.6,$ $\gamma =0.5,$$\delta =0.4,\omega =\pi /4,K=0.1,$ two values of M and increasing values of time $\tau $.

**Figure 7.**The time variations of the mid plane velocities ${\tilde{w}}_{cp}(1/2,\tau )$ and ${\tilde{w}}_{sp}(1/2,\tau )$ for $\alpha =0.7,\beta =0.6,\gamma =0.5,$$\delta =0.4,\omega =\pi /12,M=0.8$ and increasing values of K.

**Figure 8.**The time variations of the mid plane velocities ${\tilde{w}}_{cp}(1/2,\tau )$ and ${\tilde{w}}_{sp}(1/2,\tau )$ for $\alpha =0.7,\beta =0.6,\gamma =0.5,$$\delta =0.4,\omega =\pi /12,K=0.5$ and increasing values of M.

**Figure 9.**Variations in Darcy’s resistance ${\tilde{R}}_{Cp}(z)$ from Equation (53) for $\alpha =0.7,\beta =0.6,$ $\gamma =0.5,$$\delta =0.4,M=0.8$ with three values of K and $K=0.5$ with three values of M.

**Figure 10.**Influence of the effective permeability ${K}_{eff}$ on the fluid velocity ${\tilde{w}}_{Cp}(z)$ given by Equation (51) and the steady volume flux ${Q}_{Cp}$ given by Equation (56).

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**MDPI and ACS Style**

Fetecau, C.; Akhtar, S.; Moroşanu, C.
Porous and Magnetic Effects on Modified Stokes’ Problems for Generalized Burgers’ Fluids. *Dynamics* **2023**, *3*, 803-819.
https://doi.org/10.3390/dynamics3040044

**AMA Style**

Fetecau C, Akhtar S, Moroşanu C.
Porous and Magnetic Effects on Modified Stokes’ Problems for Generalized Burgers’ Fluids. *Dynamics*. 2023; 3(4):803-819.
https://doi.org/10.3390/dynamics3040044

**Chicago/Turabian Style**

Fetecau, Constantin, Shehraz Akhtar, and Costică Moroşanu.
2023. "Porous and Magnetic Effects on Modified Stokes’ Problems for Generalized Burgers’ Fluids" *Dynamics* 3, no. 4: 803-819.
https://doi.org/10.3390/dynamics3040044