# Slip Effects on Heat and Mass Transfer in MHD Visco-Elastic Fluid Flow Through a Porous Channel

## Abstract

The problem of MHD flow of visco-elastic fluid through a horizontal channel has been analysed in the presence of heat and mass transfer. The fluid is subjected to a transverse magnetic field and the slip velocity at the lower wall of the channel taken into consideration. A mathematical model (Walters liquid model B′) has been analyzed using appropriate mathematical techniques. Expressions for velocity, temperature, concentration, wall shear stress and rate of heat and mass transfer have been obtained. Variations of the quantities with different parameters are computed by using MATLAB software. The results are discussed graphically to measure the impact of visco-elasticity.

## Keywords

Visco-elastic MHD Slip velocity Free convection Shear stress## Nomenclature

## .

*L*The mean-free path

*m*_{1}The Maxwell’s reflexion

*B*_{0}Transverse magnetic field

*D*_{m}Co-efficient of mass diffusivity

*D*The mean width of the channel

*p*′Pressure

*T*′Temperature distribution

*x, y*Cartesian co-ordinate

*u, v*Velocity components along x and y axis

*K*Co-efficient of thermal conductivity

*T*_{1}′,*T*_{2}′Wall temperatures

- \( C_{1}^{{\prime }} ,C_{2}^{{\prime }} \)
Wall concentrations

*K*_{1}Visco-elastic parameter \( \left( {\frac{{K_{0} V_{0} }}{{\eta_{0} d}}} \right) \)

## Dimensionless Parameters

*M*Magnetic Parameter \( \left( {\frac{{\sigma B_{0}^{2} \mu {\prime }d^{2} }}{{\mu_{0} }}} \right) \)

*P*_{r}Prandtl number \( \left( {\frac{{\mu C_{p} }}{K}} \right) \)

- S
Suction parameter \( \left( {\frac{{v_{0} d}}{\upsilon }} \right) \)

- S
_{c} Schmidth number \( \left( {\frac{{v_{0} d}}{Dm}} \right) \)

## Greek Symbols

- ∈
The amplitude parameter

*λ*Frequency parameter \( \left( {K_{0} d} \right) \)

- \( \upsilon \)
Kinematic viscosity \( \left( {\frac{{\mu^{\prime}}}{\rho }} \right) \)

- \( \rho \)
Density

- \( \sigma \)
Electric conductivity

- \( \theta \)
Dimensionless temperature

- \( \phi \)
Dimensionless concentration

## References

- 1.Sparrow, E.M., Cess, R.D.: The effect of a magnetic field on free convection heat transfer. Int. J. Heat Mass Transf.
**3**(4), 267–274 (1961)Google Scholar - 2.Drake, D.G.: Flow in a channel due to periodic pressure gradient, quart. J. Mech. Appl. Math.
**18**(1) (1965)Google Scholar - 3.Ram, P.C., Singh, C.B.: Unsteady MHD fluid flow through a channel. J. Sci. Res.
**28**(2) (1978)Google Scholar - 4.Makinde, O.D., Mhone, P.Y.: Heat transfer to MHD oscillatory flow in a channel filled with porous medium. Ramanian J. Phys.
**50**, 931–938 (2005)Google Scholar - 5.Singh, C.B.: MHD steady flow of liquid between two parallel plates. In: Proceeding of the First Conference of Kenya Mathematical Society, pp. 24–26 (1993)Google Scholar
- 6.Al-Hadhrami, A.K., Elliot, L., Ingham, M.D., Wen, X.: Flow through horizontal channels of porous materials. Inter. Energy Res.
**27**, 875–899 (2003)Google Scholar - 7.Adesanya, O., Gbadeyan, J.A.: Adomian decomposition approach to steady visco-elastic fluid flow with slip through a planner channel. Int. J. Nonlinear Sci., 986–94 (2010)Google Scholar
- 8.Ganesh, S., Krishnambal, S.: Unsteady MHD Stokes flow of viscous fluid between two parallel porous plates. J. Appl. Sci.
**7**, 374–379 (2007)Google Scholar - 9.Sanyal, D.C., Sanyal, M.K.: Hydromagnetic slip flow with heat transfer is an inclined channel. Czechoslov. J. Phys.
**39**(5), 529–536 (1989)Google Scholar - 10.Chauhan, D.S., Rastogi, P.: Heat transfer effects on rotating MHD Couette flow in a channel partially filled by a porous medium will hall current. J. Appl. Sci. Eng.
**15**(3), 281–290 (2012)Google Scholar - 11.Falade, J.A., Ukaegbu, J.C., Egere, A.C., Adesanya, S.O.: MHD oscillatory flow through a porous channel saturated with porous medium. Alex. Eng. J.
**56**, 147–152 (2017)Google Scholar - 12.Raju, K.V.S., Venkataramana, S.: MHD convective flow through porous medium in a horizontal channel with insulated and impermeable bottom wall in the presence of viscous dissipation and Joule heating. Ain Shams Eng. J.
**5**(2), 1–8 (2014)Google Scholar - 13.Prakash, O.M., Makinde, O.D., Kumar, D., Dwivedi, Y.K.: Heat transfer to MHD oscillatory dusty fluid flow in a channel filled with a porous medium. Indian Acad. Sci.
**40**, part-4, 1273–1282 (2015)Google Scholar - 14.Mohamed Ismail, A., Ganesh, S., Kirubhashankar, C.K.: Unsteady MHD flow between two parallel plates through porous medium with one plate moving uniformly and the other plate at rest with uniform suction. Int. J. Sci. Eng. Tech. Res.
**3(**1), 6–10 (2014)Google Scholar - 15.Choudhury, R., Dey, D.: Free convective visco-elastic flow with heat and mass transfer through a porous medium with periodic permeability. Int. J. Heat Mass Transf.
**53**, 1666–1672 (2010)Google Scholar - 16.Choudhury, R., Das, S.: Visco-elastic MHD free convectine flow through porous media in presence of radiation and chemical reaction with heat and mass transfer. J. Appl. Fluid Mech.
**7**(4), 603–609 (2014)Google Scholar - 17.Choudhury, R., Dhar, P.: Mixed convective MHD oscillatory flow of a visco-elastic fluid flow with heat source and variable suction through porous media. Int. J. Math. Stast.
**16**(1), 55–70 (2015)MathSciNetzbMATHGoogle Scholar - 18.Sekhar, D.V., Reddy, G.V.: Chemical reaction effects on MHD unsteady free convective Walter’s memory flow with constant suction and heat sink. Adv. Appl. Sci. Res.
**3**(4), 2141–2150 (2012)Google Scholar - 19.Walters, K.: The solutions of flow problems in the case of material with memories. J. Mec.
**1**, 473–478 (1962)Google Scholar - 20.Walters, K.: Non-Newtonian effects in some elastico-viscous liquids whose behaviour at small rates of shear in characterized by a general linear equation of state. Q. J. Mec. Appl. Math.
**15**, 63–76 (1962)CrossRefGoogle Scholar