# Unsteady Magnetohydrodynamic Convective Fluid Flow of Oldroyd-B Model Considering Ramped Wall Temperature and Ramped Wall Velocity

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## Abstract

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## 1. Introduction

## 2. Model Formulation

## 3. Analytical Solution of the Problem

#### Laplace and Inverse Laplace Transforms

## 4. Limiting Models

#### 4.1. Case 1

#### 4.2. Case 2

## 5. Results and Discussion

## 6. Conclusions

- An increase in the magnetic parameter (M) on velocity causes decrease in the thickness of the momentum boundary layer.Momentum boundary layer increases as parameters values such as, ${\lambda}_{1},{\lambda}_{2}$, K, Gr and $t<1$ increase.
- An increase in relaxation time ${\lambda}_{1}$ results in a decrease in velocity (related to skin friction) on the plate.

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

B | Total magnetic field |

J | Current density |

E | Electric field |

T | Cauchy Stress Tensor |

r | Darcy resistance vector |

S | Extra Stress tensor |

A_{1} | Rivlin-Erickson tensor |

$u,v,w$ | Velocity components |

$x,y,z$ | Space variables |

T | Temperature |

p | Pressure |

g | Acceleration due to gravity |

$\rho $ | Fluid density |

${\mathbf{\mu}}_{m}$ | Magnetic permeability |

$\sigma $ | Electrical conductivity of the fluid |

$\varphi $ | Porosity parameter |

$\mu $ | Viscosity of the fluid |

$\lambda ,{\lambda}_{r}$ | Relaxation, retardation times |

${u}^{\ast}$ | Non-dimensional velocity |

${t}^{\ast}$ | Non-dimensional time |

${T}^{\ast}$ | Non-dimensional temperature |

M | Hartmann Number |

K | Non-dimensional porosity parameter |

${\lambda}_{1},{\lambda}_{2}$ | Non-dimensional relaxation & retardation times |

Gr | Grashof’s number |

Pr | Prandtl’s Number |

$\mathcal{L}$ | Laplace transform operator |

s | Laplace transform parameter |

## References

- Khan, M. Partial slip effects on the oscillatory flows of a fractional Jeffery fluid in a porous medium. J. Porous Media
**2007**, 10, 473–487. [Google Scholar] [CrossRef] - Ghosh, A.K.; Sana, P. On hydromagnetic flow of an Oldroyd-B fluid near a pulsating plate. Acta Astronaut.
**2009**, 64, 272–280. [Google Scholar] [CrossRef] - Nadeem, S.; Mehmood, R.; Akbar, N.S. Non-orthogonal stagnation point flow of a nano non-Newtonian fluid towards a stretching surface with heat transfer. Int. J. Heat Mass Transf.
**2013**, 57, 679–689. [Google Scholar] [CrossRef] - Nadeem, S.; Saleem, S. Analytical study of third grade fluid over a rotating vertical cone in the presence of nanoparticles. Int. J. Heat Mass Transf.
**2015**, 85, 1041–1048. [Google Scholar] [CrossRef] - Sharmilaa, K.; Kaleeswari, S. Dufour effects on unsteady free convection and mass transfer through a porous medium in a slip regime with heat source/sink. Int. J. Sci. Eng. Appl. Sci. (IJSEAS)
**2015**, 1, 307–320. [Google Scholar] - Geindreau, C.; Auriault, J.-L. Magnetohydrodynamic flows in porous media. J. Fluid Mech.
**2002**, 466, 343–363. [Google Scholar] [CrossRef] - Sobral Filho, D.C. A new proposal to guide velocity and inclination in the ramp protocol for the Treadmill Ergometer. Arq. Bras. Cardiol.
**2003**, 81, 48–53. [Google Scholar] - Bruce, R.A. Evaluation of functional capacity and exercise tolerance of cardiac patients. Mod. Concepts Cardiovasc. Dis.
**1956**, 25, 321–326. [Google Scholar] - Ästrand, P.O.; Rodahl, K. Avaliação da capacidade de trabalho físico na base dos testes. In Tratado de Fisiologia do Exercício, 2nd ed.; Interamericana: Rio de Janeiro, Brazil, 1977; pp. 304–336. [Google Scholar]
- Myers, J.; Bellin, D. Ramp exercise protocol for clinical and cardiopulmonary exercise testing. Sports Med.
**2000**, 30, 23–29. [Google Scholar] [CrossRef] [PubMed] - Hayday, A.A.; Bowlus, D.A.; McGraw, R.A. Free convection from a vertical flat plate with step discontinuities in surface temperature. J. Heat Transf.
**1967**, 89, 244–249. [Google Scholar] [CrossRef] - Schetz, J.A. On the approximate solution of viscous flow problems. ASME J. Appl. Mech.
**1963**, 30, 263–268. [Google Scholar] [CrossRef] - Malhotra, C.P.; Mahajan, R.L.; Sampath, W.S.; Barth, K.L.; Enzenroth, R.A. Control of temperature uniformity during the manufacture of stable thin-film photovoltaic devices. Int. J. Heat Mass Transf.
**2006**, 49, 2840–2850. [Google Scholar] [CrossRef] - Kundu, B. Exact analysis for propagation of heat in a biological tissue subject to different surface conditions for therapeutic applications. Appl. Math. Comput.
**2016**, 285, 204–216. [Google Scholar] [CrossRef] - Seth, G.S.; Hussain, S.M.; Sarkar, S. Hydromagnetic natural convection flow with heat and mass transfer of a chemically reacting and heat absorbing fluid past an accelerated moving vertical plate with ramped temperature and ramped surface concentration through a porous medium. J. Egypt. Math. Soc.
**2015**, 23, 197–207. [Google Scholar] [CrossRef] [Green Version] - Seth, G.S.; Sharma, R.; Sarkar, S. Natural convection heat and mass transfer flow with Hall current, rotation, radiation and heat absorption past an accelerated moving vertical plate with ramped temperature. J. Appl. Fluid Mech.
**2015**, 8, 7–20. [Google Scholar] - Seth, G.S.; Sarkar, S.; Hussain, S.M.; Mahato, G.K. Effects of hall current and rotation on hydromagnetic natural convection flow with heat and mass transfer of a heat absorbing fluid past an impulsively moving vertical plate with ramped temperature. J. Appl. Fluid Mech.
**2015**, 8, 159–171. [Google Scholar] - Seth, G.S.; Sarkar, S. MHD natural convection heat and mass transfer flow past a time dependent moving vertical plate with ramped temperature in a rotating medium with hall effects, radiation and chemical reaction. J. Mech.
**2015**, 31, 91–104. [Google Scholar] [CrossRef] - Seth, G.S.; Sarkar, S. Hydromagnetic natural convection flow with induced magnetic field and nth order chemical reaction of a heat absorbing fluid past an impulsively moving vertical plate with ramped temperature. Bulg. Chem. Commun.
**2015**, 47, 66–79. [Google Scholar] - Chandran, P.; Sacheti, N.C.; Singh, A.K. Natural convection near a vertical plate with ramped wall temperature. Heat Mass Transf.
**2005**, 41, 459–464. [Google Scholar] [CrossRef] - Seth, G.S.; Ansari, M.S. MHD natural convection flow past an impulsively moving vertical plate with ramped wall temperature in the presence of thermal diffusion with heat absorption. Int. J. Appl. Mech. Eng.
**2010**, 15, 199–215. [Google Scholar] - Das, S.; Jana, M.; Jana, R.N. Radiation effect on natural convection near a vertical plate embedded in a porous medium with ramped wall temperature. Open J. Fl. Dyn.
**2011**, 1, 1. [Google Scholar] [CrossRef] - Nandkeolyar, R.; Das, M.; Sibanda, P. Exact solutions of unsteady MHD free convection in a heat absorbing fluid flow past a flat plate with ramped wall temperature. Bound. Value Probl.
**2013**, 2013, 247. [Google Scholar] [CrossRef] - Zin, N.A.B.M.; Khan, I.; Shafie, S. Influence of thermal radiation on unsteady MHD free convection flow of Jeffrey fluid over a vertical plate with ramped wall temperature. Math. Probl. Eng.
**2016**. [Google Scholar] [CrossRef] - Ahmed, N.; Dutta, M. Transient mass transfer flow past an impulsively started infinite vertical plate with ramped plate velocity and ramped temperature. Int. J. Phys. Sci.
**2013**, 8, 254–263. [Google Scholar] - Maqbool, K.; Mann, A.B.; Tiwana, M.H. Unsteady MHD convective flow of a Jeffery fluid embedded in a porous medium with ramped wall velocity and temperature. Alex. Eng. J.
**2018**, 57, 1071–1078. [Google Scholar] [CrossRef] - Hayat, T.; Siddiqui, A.M.; Asghar, S. Some simple flows of an Oldroyd-B fluid. Int. J. Eng. Sci.
**2001**, 39, 135–147. [Google Scholar] [CrossRef] - Asghar, S.; Parveen, S.; Hanif, S.; Siddiqui, A.M.; Hayat, T. Hall effects on the unsteady hydromagnetic flows of an Oldroyd-B fluid. Int. J. Eng. Sci.
**2003**, 41, 609–619. [Google Scholar] [CrossRef] - Rajagopal, K.R.; Ruzicka, M.; Srinivasa, A.R. On the Oberbeck-Boussinesq approximation. Math. Mod. Methods Appl. Sci.
**1996**, 6, 1157–1167. [Google Scholar] [CrossRef] - Wilbur, R. Lepage. In Complex Variables and the Laplace Transform for Engineers; McGraw Hill Book Company: New York, NY, USA, 1961. [Google Scholar]
- Knight, J.H.; Raiche, A.P. Transient electromagnetic calculations using the Gaver-Stehfest inverse Laplace transform method. Geophysics
**1982**, 47, 47–50. [Google Scholar] [CrossRef] - Zakian, V. Numerical inversion of Laplace transform. Electr. Lett.
**1969**, 5, 120–121. [Google Scholar] [CrossRef] - Kenny, S. Crump, Numerical inversion of Laplace transforms using a Fourier series approximation. J. Assoc. Comput. Mach.
**1976**, 23, 89–96. [Google Scholar] - Khan, A.; Khan, I.; Ali, F. Sami ulhaq and S. Shafie, Effects of wall shear stress on unsteady MHD conjugate flow in a porous medium with ramped wall temperature. PLoS ONE
**2014**, 9, e90280. [Google Scholar] - Seth, G.S.; Nandkeolyar, R.; Ansari, M.S. Effect of rotation on unsteady hydromagnetic natural convection flow past an impulsively moving vertical plate with ramped temperature in a porous medium with thermal diffusion and heat absorption. Int. J. Appl. Math. Mech.
**2011**, 7, 52–69. [Google Scholar] - Gargano, F.; Ponetti, G.; Sammartino, M.; Sciacca, V. Route to chaos in the weakly stratified Kolmogorov flow. Phys. Fluids
**2019**, 31, 024106. [Google Scholar] [CrossRef] - Balmforth, N.J.; Young, Y.N. Stratified Kolmogorov flow. J. Fluid Mech.
**2002**, 450, 131–167. [Google Scholar] [CrossRef] [Green Version] - Vadasz, P.; Olek, S. Route to Chaos for Moderate Prandtl Number Convection in a Porous Layer Heated from Below. Transp. Porous Med.
**2000**, 41, 211–239. [Google Scholar] [CrossRef]

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**MDPI and ACS Style**

Tiwana, M.H.; Mann, A.B.; Rizwan, M.; Maqbool, K.; Javeed, S.; Raza, S.; Khan, M.S.
Unsteady Magnetohydrodynamic Convective Fluid Flow of Oldroyd-B Model Considering Ramped Wall Temperature and Ramped Wall Velocity. *Mathematics* **2019**, *7*, 676.
https://doi.org/10.3390/math7080676

**AMA Style**

Tiwana MH, Mann AB, Rizwan M, Maqbool K, Javeed S, Raza S, Khan MS.
Unsteady Magnetohydrodynamic Convective Fluid Flow of Oldroyd-B Model Considering Ramped Wall Temperature and Ramped Wall Velocity. *Mathematics*. 2019; 7(8):676.
https://doi.org/10.3390/math7080676

**Chicago/Turabian Style**

Tiwana, Mazhar Hussain, Amer Bilal Mann, Muhammad Rizwan, Khadija Maqbool, Shumaila Javeed, Saqlain Raza, and Mansoor Shaukat Khan.
2019. "Unsteady Magnetohydrodynamic Convective Fluid Flow of Oldroyd-B Model Considering Ramped Wall Temperature and Ramped Wall Velocity" *Mathematics* 7, no. 8: 676.
https://doi.org/10.3390/math7080676