# A study of heat and mass transfer in a fractional MHD flow over an infinite oscillating plate

- N. Shahid
^{1}Email author

**Received: **27 April 2015

**Accepted: **12 October 2015

**Published: **24 October 2015

## Abstract

Exact expressions of velocity, temperature and mass concentration have been calculated for free convective flow of fractional MHD viscous fluid over an oscillating plate. Expressions of velocity have been obtained both for sine and cosine oscillations of plate. Corresponding fractional differential equations have been solved by using Laplace transform and inverse Laplace transform. The expression of temperature and mass concentration have been presented in the form of Fox-H function and in the form of general Wright function, respectively and velocity is presented in the form of integral solutions using Generalized function. Some limiting cases of fluid and fractional parameters have been discussed to retrieve some solutions present in literature. The influence of thermal radiation, mass diffusion and fractional parameters on fluid flow has been analyzed through graphical illustrations.

## Keywords

## Background

A great deal of efforts have been put to study the phenomena of mass transfer and radiative heat flux in free convective flows because of its applications in many industrial and chemical processes. In principal, many physical and biological configurations and working owe their appearance and existence to mass and heat transfer. In manufacturing and chemical industry, many processes include radiative heat flows such as construction of satellites, oil and other chemicals’ filtration, construction of accessorizes using solar power, nuclear power processes, fuel combustion and drying of porous mediums’, etc. The process of mass transfer is very similar to heat flow in fluids and this similarity has prompted researchers to study both processes simultaneously. In particular, for the cases of low mass transfer and for low concentration in the fluid, the process of mass transfer and heat flow behave in almost the same way. The study of combined effects of mass transfer and radiation of heat in fluid flow has become a central topic of investigation in recent times. Furthermore, the study of motion of magnetohydrodynamic (MHD) fluid bears its own importance in the context of its vast use in many industrial and nuclear processes.

Exact solutions for various Newtonian and non Newtonian fluids over oscillating bodies for various physical settings have been calculated. Penton (1968) seems to be first to have calculated transient solutions for the flow of Newtonian fluid due to an oscillating plate. An interesting problem of fluid flow with effects of radiative heat on fluid motion was presented by Puri and Kythe (1998). A valuable contribution on this line was made by Erdogan (2000) when he obtained the exact solutions for the motion of viscous fluid due to sine and cosine oscillations of a vertical plate. Rajagopal (1983), Rajagopal and Bhatnagar (1995), Hayat et. al. (2001) and Fetecau et. al. (2005, 2006), etc further extended the study of motion of fluids in various geometrical scenarios for sine oscillations, cosine oscillations, longitudinal and torsional oscillations, etc.

Similarly many authors have worked on determining exact solutions for free convection flows. Soundalgekar (1979) was first to have calculated the exact solution of free convective flow over an infinite oscillating plate and this problem was extended by Soundalgekar and Akolkar (1983) by including the effects of mass transfer on fluid motion. Soundalgekar et. al. (1994) and Das et. al. (1994) further studied the effects of mass transfer on flow past an impulsively started and vertical oscillating plate. Thermal radiation effects on a laminar flow was studied by England and Emry (1969) and Gupta and Gupta (1974) presented the exact solutions of motion of electrically conducting fluid with radiation effects in the presence of uniform magnetic field. Mazumdar and Deka (2007) and Muthucumaraswamy (2006) studied the effects of thermal radiation in an MHD flow and on flow past impulsively started plate, respectively.

But the exact solutions of constitutive models of fluid motion with fractional derivatives are rare in present literature. The study of effects of radiation and mass transfer on free convective flows using fractional calculus tools is even more rare and is a motivation for present investigation. Fractional constitutive relationship model has an advantage over customary constitutive relationship model as it readily assesses the properties of viscous and molecular mediums that are sometimes overlooked by ordinary derivative models. Fractional calculus approach is useful when it comes to generalization of complex dynamics of fluid motion. Many authors (Wang and Xu 2009; Fetecau et al. 2010; Tripathi et al. 2010; Hyat et al. 2010a; Hayat et al. 2010b; Liu and Zheng 2011; Fetecau et al. 2011; Jamil et al. 2011a, b; Tripathi et al. 2011a, b; Tripathi 2011a, b; Zheng et al. 2010; Liu et al. 2011; Zheng et al. 2011a, b, 2012) have recently ventured out on studying the motion of viscous MHD fluids using fractional derivatives and made valuable contributions.

The following study is undertaken to investigate the thermal radiation and diffusion effects on a free convective MHD fractional fluid flow over a vertical oscillating plate. Exact expressions of velocity field, temperature and concentration of fluid have been calculated and presented in interesting forms. Also, the expressions of velocity have been obtained both for sine and cosine oscillations of plate. Limiting cases of fractional and fluid parameters have also been taken into account to retrieve new and some existing expressions of velocity, temperature and concentration of fluid. The influence of fluid and fractional parameters on fluid motion have also been depicted through graphs and some similarities and differences of velocity profiles for sine and cosine oscillations have also been highlighted through graphs.

## Mathematical formulation

A case of fractional incompressible viscous MHD fluid over an infinite vertical oscillating plate is considered. We take x-axis of the Cartesian coordinate system along the vertical direction of the infinite plate and y-axis will be normal to the plate. \(T_\infty \) and \(C_\infty \) are considered to be initial temperature and concentration of plate and fluid, respectively. At time \(t=0^+\), the plate is given an oscillating motion in its own plane with the velocity \(f_1\cos \omega _1 t\) or \(f_1\sin \omega _1 t\). At the same time the temperature and concentration of the plate are raised to \(T_W\) and \(C_W\), respectively, and magnetic field of uniform strength \(B_0\) is applied to the plate in normal direction. It is assumed that magnetic Reynold’s number is very small and the induced magnetic field is negligible in comparison to transverse magnetic field. The viscous dissipation and Soret and Duoffer effects due to lower level of concentration are assumed to be negligible.

*u*(

*y*,

*t*),

*T*(

*y*,

*t*),

*C*(

*y*,

*t*),\(\nu \),

*g*, \(\beta \), \(\beta ^*\), \(\kappa \), \(q_r\), \(C_P\), \(\rho \) and

*D*are velocity of the fluid, its temperature, species concentration in the fluid, kinematic viscosity, gravitational acceleration, coefficient of thermal expansion, coefficient of expansion with concentration, thermal conductivity of the fluid, radiative heat flux, specific heat at constant pressure, density of fluid and mass diffusion coefficient, respectively.

Also in Eq. (5), \(f_1\) and \(U_0\) are constants, and \(\omega _1\) is the frequency of oscillation.

*M*and

*F*are Prandtl number, Schmidth number, thermal Grashof number, mass Grashof number, Hartmann number and dimensionless thermal radiation parameter, respectively.

## Analytical solutions

Analytical solutions will be obtained by means of Laplace transform and inverse Laplace transform.

*C*(

*y*,

*t*).

*C*(

*y*,

*t*), we write Eq. (18) in the form

*u*(

*y*,

*t*), we apply discrete Laplace transform to Eq. (15) and obtain

*u*(

*y*,

*t*). Also, \(\bar{u}(y,q)\) has to satisfy the condition

## Limiting cases

For \(\alpha ,\,\,\,\beta ,\,\,\,\gamma \rightarrow 1\) in Eqs. (18), (22) and (28), we can retrieve (*y*, *t*) solutions of governing equations in ordinary differential operator. Some significant limiting cases have been discussed below.

### Solution in the absence of magnetic field

### Solution in the case of constant radiative heat flux and \(\beta \rightarrow 1\)

*erfc*(.) represents complementary error function.

### Solution in the absence of magnetic field and constant radiative heat flux

*f*is assumed to be 1.

### Mass concentration corresponding to \(\mathbf {\gamma \rightarrow 1}\)

## Results and discussion

Many interesting physical aspects of radiative heat flow have been brought into light through graphs. These graphs also represent the influence of physical parameters \(G_r\), \(G_m\), \(S_c\), \(P_r\), *M*, *F*, \(\omega \) and fractional parameters \(\alpha \), \(\beta \), \(\gamma \) on motion of MHD fluid over a vertical oscillating plate.

*t*ad for fixed values of \(G_r\), \(G_m\), \(S_c\), \(P_r\),

*M*,

*F*, \(\alpha \), \(\beta \), \(\gamma \),

*f*and \(\omega \) for sine and cosine oscillations. It can be observed that near the plate for starting time, the velocity profiles override each other but the velocity increases eventually for increasing values of time

*t*. Also, it can be seen in both graphs that velocity is vanishing for higher values of

*y*as was expected because the impact of oscillations on fluid gets weaker as fluid gets farther away from the plate.

*M*and other fluid and fractional parameters are taken to be fixed. Besides the different shape of velocity profiles for sine oscillations and cosine oscillations, it is observed that velocity increases with increase in \(G_r\) and \(G_m\) and decreases with increase in

*M*. The influence of parameters

*M*,

*F*and \(S_c\) on free convective fluid motion has been depicted through Figs. 5, 6, 7 and 8. All these graphs point to the fact that velocity has inverse relation with Schmidth number, \(S_c\), Harmann number,

*M*and thermal radiation parameter, F even if the pattern of velocity profiles is different for sine and cosine oscillations. It is also noted that velocity responds to the changes of

*M*faster than the changes in \(S_c\) and

*F*.

*F*i.e. increasing values of \(P_r\) and

*F*decrease the temperature of fluid.

It can be seen from Figs. 14 and 15 that concentration of fluid increases with increasing time but increasing values of Schmidth number, \(S_c\) have negative impact on concentration of fluid. Figures 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 and 12 validate the boundary condition \(u(y,t)\rightarrow 0\) as \(y\rightarrow \infty \). Influence of fractional parameters on fluid motion is studied through Figs. 16, 17 and 18. These graphs clearly show that velocity, temperature and concentration of fluid decrease for increasing values of fractional parameters \(\alpha \), \(\beta \) and \(\gamma \), respectively. In Figs. 17 and 18, we have also retrieved profiles of temperature and concentration for ordinary MHD convective flow over an oscillating plate by assuming \(\beta \rightarrow 1\) and \(\gamma \rightarrow 1\).

*p*is a positive integer,

*r*] denotes the integer part of the real number r. According to Table 1, the absolute error being of order \(10^{-6}\), there exists a good agreement of the numerical results. Similar comparisons can be made for temperature and velocity.

y | C(y,t), Eq. (20) | C(y,t) Eq. (18) | Absolute error |
---|---|---|---|

0 | 5 | 5.00001 | \(6.031\times 10^{-6}\) |

0.1 | 4.669 | 4.669 | \(3.15\times 10^{-6}\) |

0.2 | 4.35855 | 4.35856 | \(1.008\times 10^{-5}\) |

0.3 | 4.06747 | 4.06747 | \(8.068\times 10^{-6}\) |

0.4 | 3.79465 | 3.79465 | \(6.578\times 10^{-6}\) |

0.5 | 3.53903 | 3.53903 | \(7.199\times 10^{-6}\) |

0.6 | 3.29961 | 3.29961 | \(4.718\times 10^{-6}\) |

0.7 | 3.07544 | 3.07544 | \(4.313\times 10^{-7}\) |

0.8 | 2.86562 | 2.86562 | \(6.646\times 10^{-6}\) |

0.9 | 2.66929 | 2.6693 | \(7.367\times 10^{-6}\) |

1 | 2.48566 | 2.48566 | \(2.036\times 10^{-6}\) |

## Conclusion

Exact solutions have been calculated for fractional MHD free convective viscous fluid over a vertical oscillating plate and influence of thermal radiation and mass diffusion on fluid motion have been analyzed. Expressions of velocity field, temperature and mass concentration of fluid have been obtained by applying Laplace transform to fractional differential equations governing present fluid flow problem. In particular, Caputo fractional differential operator is favored and motivation to employ fractional calculus tool is generalization of dynamics of such fluid flow problems.

Expressions of velocity field have been obtained for both sine and cosine oscillations of plate and are presented in series form and in the form of integral solutions. The part of velocity corresponding to oscillations of plate is nicely presented in the form of Fox- H function and the part of velocity corresponding to thermal radiation, mass diffusion and magnetic field effects has been presented in integral solutions form, employing the concept of Generalized function. The expression of mass concentration of fractional MHD fluid has been presented in the form of general Wright function and the exact expression of temperature is written in the form of Fox- H function form.

All solutions satisfy initial and boundary conditions.

Some significant limiting cases of fractional and fluid parameters have also been taken into account and expressions of mass concentration and temperature, present in literature, have been retrieved for \(\gamma \rightarrow 1\) and, \(\beta \rightarrow 1\) and \(F=0\), respectively. Also, velocity field expression has been separately calculated for the case when magnetic field is absent as well as for the case of absence of thermal radiation.

- 1.
The velocity of fluid for both sine and cosine oscillations increase with increasing

*t*, eventually. - 2.
The velocity has inverse relation with fluid parameters Hartmann number,

*M*, thermal radiation parameter,*F*and Schmidth number, \(S_c\) and has direct relation with thermal Grashof number, \(G_r\) and mass Grashof number, \(G_m\). - 3.
Temperature of fluid increases for decreasing values of Prandtl number, \(P_r\) and thermal radiation parameter,

*F*. - 4.
Mass concentration of fluid is negatively influenced by Schmidth number, \(S_c\) but it increases with increasing time.

- 5.
A contrasting behavior of velocity profiles for different values of oscillating frequency, \(\omega \) for both cases of sine and cosine oscillations has been noted through graphs. These graphs show that the velocity is decreasing for increasing values of oscillating frequency for cosine oscillations and decreases for decreasing frequency for sine oscillations.

- 6.
The influence of fractional parameters on fluid motion is also studied through graphs. These graphs show that for decreasing values of \(\alpha \), \(\beta \) and \(\gamma \), velocity, temperature and concentration increase, respectively.

- 7.
For concentration of fluid

*C*(*y*,*t*), the accuracy of obtained analytical solutions has been checked by making a comparison between the numerical and analytical results. Numerical data is in good agreement with analytical results. Same can be done for temperature and velocity field.

## Declarations

### Acknowledgements

This whole work has been completed by me (Author: Dr. Nazish Shahid) and I thank the administration of Forman Christian College, A Chartered University, Lahore for providing me the right atmosphere that helped me in completing my research article.

### Competing interests

The author declares that they have no competing interests.

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## Authors’ Affiliations

## References

- Das UN, Deka RK, Soundalgekar VM (1994) Efects of mass transfer on flow past an impulsively started vertical infinite plate with constant heat flux and chemical reaction. Forschung in Ingenieurwesen 60:284–287View ArticleGoogle Scholar
- England WG, Emery AF (1969) Thermal radiation Effects on the laminar free convection boundary layer of an absorbing gas. J Heat Trans 91:37–44View ArticleGoogle Scholar
- Erdogan ME (2000) A note on an unsteady flow of a viscous fluid due to an oscillating plane wall. Internat J Non-Linear Mech 35:1–6View ArticleGoogle Scholar
- Fetecau C (2005) Corina Fetecau, Starting solutions for some unsteady unidirectional flows of a Second grade fluid. Internat J Engrg Sci 43:781–789View ArticleGoogle Scholar
- Fetecau C (2006) Corina Fetecau, Starting solutions for the motion of a second grade fluid due to longitudinal and torsional oscillations of a circular cylinder. Internat J Engrg Sci 44:788–796View ArticleGoogle Scholar
- Fetecau C, Fetecau C, Jamil M, Mahmood A (2011) Flow of fractional Maxwell fluid between coaxial cylinders. Arch App Mech 81:1153–1163View ArticleGoogle Scholar
- Fetecau C, Mahmood A, Jamil M (2010) Exact solutions for the flow of a viscoelastic fluid induced by a circular cylinder subject to a time dependent shear stress. Commun Nonlinear Sci Numer Simulat 15:3931–3938View ArticleGoogle Scholar
- Gupta PS, Gupta AS (1974) Radiation effect on hydromagnetic convection in a vertical channel. Int J Heat Mass Transfer 17:1437–1442View ArticleGoogle Scholar
- Hayat T, Siddiqui AM, Asghar S (2001) Some simple flows of an Oldroyd-B fluid. Internat J Engrg Sci 39:135–147View ArticleGoogle Scholar
- Hayat T, Zaib S, Fetecau C (2010) Flows in a fractional generalized Burgers’ fluid. J Porus Media 13:725–739View ArticleGoogle Scholar
- Hyat T, Najam S, Sjid M, Ayub M, Mesloub S (2010) On exact solutions for oscillatory flows in a generalized Burgers’ fluid with slip condition. Z Naturforsch 65:381–391Google Scholar
- Jamil M, Rauf A, Zafar AA, Khan NA (2011) New exact analytical solutions for Stokes’ first problem of Maxwell fluid with fractional derivative approach. Comput Math Appl 62:1013–1023View ArticleGoogle Scholar
- Jamil M, Zafar AA, Khan NA (2011) Translational flows of an Oldroyd-B fluid with fractional derivatives. Comput Math Appl 62:1540–1553View ArticleGoogle Scholar
- Liu Y, Zheng L (2011) Unsteady MHD Couette flow of a generalized Oldroyd-B fluid with fractional derivative. Comput Math Appl 61:443–450View ArticleGoogle Scholar
- Liu Y, Zheng L, Zhang X (2011) MHD flow and heat transfer of ageneralized Burgers’ fluid due to an exponential accelerating plate with the effect of radiation. Appl Math Comput 62(8):3123–3131View ArticleGoogle Scholar
- Mainardi F (2010) Fractional calculus and waves in viscoelasticity: an introduction to mathematical models. Imperial College Press, LondonView ArticleGoogle Scholar
- Mainardi F, Mura A, Pagnini G (2010) The M-Wright function in time-fractional diffusion processes: a tutorial survey. Int J Diff Eqs. doi:10.1155/2010/104505 Google Scholar
- Mathai AM, Saxena RK, Haubold HJ (2010) The H-functions: theory and applications. Springer, New YorkView ArticleGoogle Scholar
- Mazumdar MK, Deka RK (2007) MHD flow past an impulsively started infinite vertical plate in presence of thermal radiation. Romanian J Phys 52(5–6):529–535Google Scholar
- Muthucumaraswamy R (2006) The interaction of thermal radiation on vertical oscillating plate with variable temperature and mass diffusion. Theoret Appl Mech 33(2):107–121View ArticleGoogle Scholar
- Penton R (1968) The transient for stokes’ oscillating plane: a solution in terms of tabulated functions. J Fluid Mech 31:810–825Google Scholar
- Podlubny I (1999) Fractional Differential Equations. Academic press, San DiegoGoogle Scholar
- Puri P, Kythe PK (1998) Thermal effect in Stokes’ second problem. Acta Mech 112:40–44Google Scholar
- Rajagopal KR (1983) Longitudinal and torsional oscillations of a rod in a non-Newtonian fluid. Acta Mech 49:281–285View ArticleGoogle Scholar
- Rajagopal KR, Bhatnagar RK (1995) Exact solutions for some simple flows of an Oldroyd-B fluid. Acta Mech 113:233–239View ArticleGoogle Scholar
- Soundalgekar VM (1979) Free convection effects on the flow past a vertical oscillating plate. Astrophys Space Sci 66:165–172View ArticleGoogle Scholar
- Soundalgekar VM, Akolkar SP (1983) Effects of free convection currents and mass transfer on flow past a vertical oscillating plate. Astrophys Space Sci 89:241–254View ArticleGoogle Scholar
- Soundalgekar VM, Lahurikar RM, Pohanerkar SG, Birajdar NS (1994) Effects of mass transfer on the flow past an oscillating infinite vertical plate with constant heat flux. Thermophys AeroMech 1:119–124Google Scholar
- Stehfest H (1970) Algorithm 368: numerical inversion of Laplace transform. Commun ACM 13:47–49View ArticleGoogle Scholar
- Tripathi D (2011) Peristaltic transport of fractional Maxwell fluids in uniform tubes: application of an endoscope. Comp Math Appl 62:1116–1126View ArticleGoogle Scholar
- Tripathi D (2011) Peristaltic transport of a viscoelastic fluid in a channel. Acta Astronautica 68:1379–1385View ArticleGoogle Scholar
- Tripathi D, Gupta PK, Das S (2011) Influence of slip condition on peristaltic transport of a viscoelastic fluid with model. Thermal Sci 15:501–515View ArticleGoogle Scholar
- Tripathi D, Pandey SK, Das S (2010) Peristaltic flow of viscoelastic fluid with fractional Maxwell model through a channel. Appl Math Comput 215:3645–3654View ArticleGoogle Scholar
- Tripathi D, Pandey SK, Das S (2011) Peristaltic transport of a generalized Burgers fluid: Application to the movement of chyme in small intestine. Acta Astronautica 69:30–38View ArticleGoogle Scholar
- Wang S, Xu M (2009) Axial Coutte flow of two kinds of fractional viscoelastic fluids in an annulus. Nonlinear Anal Real World Appl 10:1087–1096View ArticleGoogle Scholar
- Zheng L, Liu Y, Zhang X (2011) Exact solutions forMHD flow of generalized Oldroyd-B fluid MHD flow of generalized Oldroyd-B fluid due to an infinite accelerating plate. Math Comp Model 54(1–2):780–788View ArticleGoogle Scholar
- Zheng L, Liu Y, Zhang X (2012) Slip effects on MHD flow of a generalized Oldroyd-B fluid with fractional derivative. Nonlinear Anal Real World Appl 13(2):513–523View ArticleGoogle Scholar
- Zheng L, Wang KN, Gao YT (2011) Unsteady flow and heat transfer of a generalized Maxwell fluid due to a hyperbolic sine accelerating plate. Comp Math Appl 61(8):2209–2212View ArticleGoogle Scholar
- Zheng L, Zhao F, Zhang X (2010) Exact solutions for generalized Maxwell fluid flow due to oscillatory and constantly accelerating plate. Nonlinear Anal Real World Appl 11(5):3744–3751View ArticleGoogle Scholar