# Nonlinear instability of an Oldroyd elastico–viscous magnetic nanofluid saturated in a porous medium

## Abstract

Through viscoelastic potential theory, a Kelvin-Helmholtz instability of two semi-infinite fluid layers, of Oldroydian viscoelastic magnetic nanofluids (MNF), is investigated. The system is saturated by porous medium through two semi-infinite fluid layers. The Oldroyd B model is utilized to describe the rheological behavior of viscoelastic MNF. The system is influenced by uniform oblique magnetic field that acts at the surface of separation. The model is used for the MNF incorporated the effects of uniform basic streaming and viscoelasticity. Therefore, a mathematical simplification must be considered. A linear stability analysis, based upon the normal modes analysis, is utilized to find out the solutions of the equations of motion. The onset criterion of stability is derived; analytically and graphs have been plotted by giving numerical values to the various parameters. These graphs depict the stability characteristics. Regions of stability and instability are identified and discussed in some depth. Some previous studies are recovered upon appropriate data choices. The stability criterion in case of ignoring the relaxation stress times is also derived. To relax the mathematical manipulation of the nonlinear approach, the linearity of the equations of motion is taken into account in correspondence with the nonlinear boundary conditions. Taylor's theory is adoptedmore »

- Authors:

- Department of Mathematics, Faculty of Education, Ain Shams University, Roxy (Egypt)
- Department of Mathematics, Faculty of Science (Girls Branch), University of Tabuk, Tabuk, P.O. Box 741 (Saudi Arabia)

- Publication Date:

- OSTI Identifier:
- 22303623

- Resource Type:
- Journal Article

- Journal Name:
- Physics of Plasmas

- Additional Journal Information:
- Journal Volume: 21; Journal Issue: 9; Other Information: (c) 2014 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 1070-664X

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 70 PLASMA PHYSICS AND FUSION TECHNOLOGY; BOUNDARY CONDITIONS; DIAGRAMS; EQUATIONS OF MOTION; GINZBURG-LANDAU THEORY; GRAPH THEORY; HELMHOLTZ INSTABILITY; MAGNETIC FIELDS; MATHEMATICAL SOLUTIONS; NONLINEAR PROBLEMS; NORMAL-MODE ANALYSIS; POROUS MATERIALS; STABILITY; STRESS RELAXATION

### Citation Formats

```
Moatimid, Galal M., E-mail: gal-moa@hotmail.com, Alali, Elham M. M., E-mail: dr-elham-alali@hotmail.com, and Ali, Hoda S. M., E-mail: hoda-ali-1@hotmail.com.
```*Nonlinear instability of an Oldroyd elastico–viscous magnetic nanofluid saturated in a porous medium*. United States: N. p., 2014.
Web. doi:10.1063/1.4894690.

```
Moatimid, Galal M., E-mail: gal-moa@hotmail.com, Alali, Elham M. M., E-mail: dr-elham-alali@hotmail.com, & Ali, Hoda S. M., E-mail: hoda-ali-1@hotmail.com.
```*Nonlinear instability of an Oldroyd elastico–viscous magnetic nanofluid saturated in a porous medium*. United States. doi:10.1063/1.4894690.

```
Moatimid, Galal M., E-mail: gal-moa@hotmail.com, Alali, Elham M. M., E-mail: dr-elham-alali@hotmail.com, and Ali, Hoda S. M., E-mail: hoda-ali-1@hotmail.com. Mon .
"Nonlinear instability of an Oldroyd elastico–viscous magnetic nanofluid saturated in a porous medium". United States. doi:10.1063/1.4894690.
```

```
@article{osti_22303623,
```

title = {Nonlinear instability of an Oldroyd elastico–viscous magnetic nanofluid saturated in a porous medium},

author = {Moatimid, Galal M., E-mail: gal-moa@hotmail.com and Alali, Elham M. M., E-mail: dr-elham-alali@hotmail.com and Ali, Hoda S. M., E-mail: hoda-ali-1@hotmail.com},

abstractNote = {Through viscoelastic potential theory, a Kelvin-Helmholtz instability of two semi-infinite fluid layers, of Oldroydian viscoelastic magnetic nanofluids (MNF), is investigated. The system is saturated by porous medium through two semi-infinite fluid layers. The Oldroyd B model is utilized to describe the rheological behavior of viscoelastic MNF. The system is influenced by uniform oblique magnetic field that acts at the surface of separation. The model is used for the MNF incorporated the effects of uniform basic streaming and viscoelasticity. Therefore, a mathematical simplification must be considered. A linear stability analysis, based upon the normal modes analysis, is utilized to find out the solutions of the equations of motion. The onset criterion of stability is derived; analytically and graphs have been plotted by giving numerical values to the various parameters. These graphs depict the stability characteristics. Regions of stability and instability are identified and discussed in some depth. Some previous studies are recovered upon appropriate data choices. The stability criterion in case of ignoring the relaxation stress times is also derived. To relax the mathematical manipulation of the nonlinear approach, the linearity of the equations of motion is taken into account in correspondence with the nonlinear boundary conditions. Taylor's theory is adopted to expand the governing nonlinear characteristic equation according to of the multiple time scales technique. This analysis leads to the well-known Ginzburg–Landau equation, which governs the stability criteria. The stability criteria are achieved theoretically. To simplify the mathematical manipulation, a special case is considered to achieve the numerical estimations. The influence of orientation of the magnetic fields on the stability configuration, in linear as well as nonlinear approaches, makes a dual role for the magnetic field strength in the stability graphs. Stability diagram is plotted for different sets of physical parameters. A new stability as well as instability region, in the parameter space, appears due to the nonlinear effects.},

doi = {10.1063/1.4894690},

journal = {Physics of Plasmas},

issn = {1070-664X},

number = 9,

volume = 21,

place = {United States},

year = {2014},

month = {9}

}