# Unsteady flow of chemically reactive Oldroyd-B fluid over oscillatory moving surface with thermo-diffusion and heat absorption/generation effects

- 46 Downloads
- 2 Citations

## Abstract

This theoretical investigation deals with the magnetohydrodynamic mass and heat transportation of oscillatory Oldroyd-B fluid flow under thermo-diffusion effects. Flow is produced due to periodic motion of sheet. Some interesting effects like heat absorption/generation and chemical reaction are superposed in the energy and mass species equations, respectively. By utilizing apposite variables, independent variables in the model equations are reduced. The set of these equations is solved with help of homotopy analysis method. Reliable results for different physical flow constraints are prepared for the velocity, temperature and concentration profiles. It is noted that the Deborah number in terms of relaxation time resists the motion of fluid particles at various time instants. Both temperature and concentration profiles increase by increasing Deborah number in terms of relaxation time. The presence of Dufour number may enhance the thermal boundary layer effectively. It is also concluded that the species concentration profile is promoted by increasing Hartmann number and Soret number. An excellent accuracy of obtained solution is observed with already reported numerical values as a special case.

## Keywords

Magnetohydrodynamic Oldroyd-B fluid Thermo-diffusion effects Homotopy analysis method## List of symbols

- (
*u*,*v*) Velocity component (m/s)

*ω*Frequency (s

^{−1})*λ*_{1}Relaxation time (t)

*σ*Electrical conductivity (s/m)

*α*Thermal diffusivity of fluid (m

^{2}/s)- \(\nu\)
Kinematic viscosity (m

^{2}/s)*Q*Volumetric rate of heat generation/absorption (KW/m

^{3})*C*Concentration

*k*_{T}Thermal diffusion ratio (W/m)

- (
*β*_{1},*β*_{1}) Material parameters

*Pr*Prandtl number

*Du*Dufour number

*δ*Heat source/sink parameter and

*Re*_{x}Local Reynolds number

*Sh*Local Sherwood number

- \(j_{\text{s}}\)
Surface mass flux (W/m

^{2})*b*Stretching rate (s

^{−1})*λ*_{2}Retardation time (t)

- \(f_{y}\)
Dimensionless velocity

*T*Temperature (K)

*ρ*Fluid density (kg/m

^{3})*T*_{m}Mean fluid temperature (K)

- \(c_{\text{p}}\)
Specific heat (J/KgK)

*D*_{m}Molecular diffusivity (m

^{2}/s)*k*_{a}Reaction rate (s)

^{−1}*M*Hartmann number

*Sc*Schmidt number

*Sr*Soret number

*Kr*Chemical reaction parameter

*Nu*_{x}Local Nusselt number

- \(q_{s}\)
Surface heat flux (W/m

^{2})*S*Ratio of oscillating frequency to stretching rate

## Notes

### Acknowledgement

Authors are thankful to the editor and reviewers for their valuable comments to improve the earlier version of the paper.

### Funding

There are no funders to report for this submission.

### Compliance with ethical standards

### Conflict of interest

The authors declare that they have no conflict of interest.

## References

- 1.Davies AR, Devlin J (1993) On corner flows of Oldroyd-B fluids. J Non Newton Fluid Mech 50:173–191CrossRefGoogle Scholar
- 2.Hayat T, Siddiqui AM, Asghar S (2001) Some simple flows of an Oldroyd-B fluid. Int J Eng Sci 39:135–147CrossRefGoogle Scholar
- 3.Ghosh AK, Sana P (2009) On hydromagnetic channel flow of an Oldroyd-B fluid induced by rectified sine pulses. J Adv Math 28:365–395MathSciNetzbMATHGoogle Scholar
- 4.Jamil M, Khan NA, Zafar AA (2011) Translational flows of an Oldroyd-B fluid with fractional derivative. Comput Math Appl 62:1540–1553MathSciNetCrossRefGoogle Scholar
- 5.Fetecau C, Prasad SC, Rajagopal KR (2007) A note on the flow induced by a constantly accelerating plate in an Oldroyd B fluid. Appl Math Model 31:647654CrossRefGoogle Scholar
- 6.Sakiadis BC (1961) Boundary-layer behavior on continuous solid surface: I. Boundary layer equations for two-dimensional and axisymmetric flow. AIChE J 7:26–28CrossRefGoogle Scholar
- 7.Sakiadis BC (1961) Boundary-layer behavior on continuous solid surface: II. Boundary layer on a continuous flat surface. AIChE J 7:221–225CrossRefGoogle Scholar
- 8.Wang CY (1991) Exact solutions of the steady state Navier–Stokes equations. Annu Rev Fluid Mech 23:159MathSciNetCrossRefGoogle Scholar
- 9.Ariel PD (2007) The three-dimensional flow past a stretching sheet and the homotopy perturbation method. Comput Math Appl 54:920–925MathSciNetCrossRefGoogle Scholar
- 10.Fang T (2007) Flow over a stretchable disk. Phys Fluids 19:128105CrossRefGoogle Scholar
- 11.Sajid M, Abbas Z, Javed T, Ali N (2010) Boundary layer flow of an Oldroyd-B fluid in the region of stagnation point over a stretching sheet. Can J Phys 88:635–640CrossRefGoogle Scholar
- 12.Javed T, Ali N, Abbas Z, Sajid M (2013) Flow of an Eyring–Powell non-Newtonian fluid over a stretching sheet. Chem Eng Commun 200:327–336CrossRefGoogle Scholar
- 13.Hayat T, Qayyum S, Alsaedi A, Waqas M (2016) Simultaneous influences of mixed convection and nonlinear thermal radiation in stagnation point flow of Oldroyd-B fluid towards an unsteady convectively heated stretched surface. J Mol Liq 224:811–817CrossRefGoogle Scholar
- 14.Ghadikolaei SS, Hosseinzadeh K, Yassari M, Sadeghi H, Ganji DD (2018) Analytical and numerical solution of non-Newtonian second-grade fluid flow on a stretching sheet. Therm Sci Eng Prog 5:309–316CrossRefGoogle Scholar
- 15.Rahimi J, Ganji DD, Khaki M, Hosseinzadeh K (2017) Solution of the boundary layer flow of an Eyring–Powell non-Newtonian fluid over a linear stretching sheet by collocation method. Alex Eng J 56:621–627CrossRefGoogle Scholar
- 16.Andersson HI, Bech KH (1992) Magnetohydrodynamic flow of a Power-law fluid over a stretching sheet. Int J Nonlinear Mech 27:929–936CrossRefGoogle Scholar
- 17.Chauhana DS, Agrawal R (2011) MHD flow through a porous medium adjacent to a stretching sheet: Numerical and an approximate solution. Eur Phys J Plus 126:47CrossRefGoogle Scholar
- 18.Abbasi M, Khaki M, Rahbari A, Ganji DD, Rahimipetroudi I (2016) Analysis of MHD flow characteristics of an UCM viscoelastic flow in a permeable channel under slip conditions. J Braz Soc Mech Sci Eng 38:977–988CrossRefGoogle Scholar
- 19.Turkyilmazoglu M (2013) The analytical solution of mixed convection heat transfer and fluid flow of a MHD viscoelastic fluid over a permeable stretching surface. Int J Mech Sci 77:263–268CrossRefGoogle Scholar
- 20.Ghadikolaei SS, Hosseinzadeh K, Ganji DD (2017) Analysis of unsteady MHD Eyring–Powell squeezing flow in stretching channel with considering thermal radiation and Joule heating effect using AGM. Case Stud Therm Eng 10:579–594CrossRefGoogle Scholar
- 21.Ghadikolaei SS, Hosseinzadeh K, Ganji DD, Jafari B (2018) Nonlinear thermal radiation effect on magneto Casson nanofluid flow with Joule heating effect over an inclined porous stretching sheet. Case Stud Therm Eng 12:176–187CrossRefGoogle Scholar
- 22.Hosseinzadeh K, Amiri AJ, Ardahaie SS, Ganji DD (2017) Effect of variable Lorentz forces on nanofluid flow in movable parallel plates utilizing analytical method. Case Stud Therm Eng 10:595–610CrossRefGoogle Scholar
- 23.Hatami M, Hosseinzadeh K, Domairry G, Behnamfar MT (2014) Numerical study of MHD two-phase Couette flow analysis for fluid-particle suspension between moving parallel plates. J Taiwan Inst Chem Eng 45:2238–2245CrossRefGoogle Scholar
- 24.Balazadeh N, Sheikholeslami M, Ganji DD, Li Z (2018) Semi analytical analysis for transient Eyring–Powell squeezing flow in a stretching channel due to magnetic field using DTM. J Mol Liq 260:30–36CrossRefGoogle Scholar
- 25.Ganga B, Ansari SMY, Ganesh NV, Hakeem AKA (2015) MHD radiative boundary layer flow of nanofluid past a vertical plate with internal heat generation/absorption, viscous and Ohmic dissipation effects. J Niger Math Soc 34:181–194MathSciNetCrossRefGoogle Scholar
- 26.Hakeem AKA, Govindaraju M, Ganga B, Kayalvizhi M (2016) Second law analysis for radiative MHD slip flow of a nanofluid over a stretching sheet with non-uniform heat source effect. Sci Iran F 23:1524–1538Google Scholar
- 27.Hakeem AKA, Ganesh NV, Ganga B (2015) Magnetic field effect on second order slip flow of nanofluid over a stretching/shrinking sheet with thermal radiation effect. J Magn Magn Mater 381:243–257CrossRefGoogle Scholar
- 28.Hosseinzadeh K, Afsharpanah F, Zamani S, Gholinia M, Ganji DD (2018) A numerical investigation on ethylene glycol-titanium dioxide nanofluid convective flow over a stretching sheet in presence of heat generation/absorption. Case Stud Therm Eng 12:228–236CrossRefGoogle Scholar
- 29.Hayat T, Safdar A, Awais M, Mesloub S (2012) Soret and Dufour effects for three-dimensional flow in a viscoelastic fluid over a stretching surface. Int J Heat Mass Transf 55:2129–2136CrossRefGoogle Scholar
- 30.Cheng CY (2009) Soret and Dufour effects on natural convection heat and mass transfer from a vertical cone in a porous medium. Int Commun Heat Mass Transf 36:1020–1024CrossRefGoogle Scholar
- 31.Turkyilmazoglu M, Pop I (2012) Soret and heat source effects on the unsteady radiative MHD free convection flow from an impulsively started infinite vertical plate. Int J Heat Mass Transf 55:7635–7644CrossRefGoogle Scholar
- 32.Nawaz M, Alsaedi A, Hayat T, Alhothauli MS (2013) Dufour and Soret effects in an axisymmetric stagnation point flow of second grade fluid with Newtonian heating. J Mech 29:27–34CrossRefGoogle Scholar
- 33.Wang CY (1988) Nonlinear streaming due to the oscillatory stretching of a sheet in a viscous fluid. Acta Mech 72:261–268CrossRefGoogle Scholar
- 34.Abbas Z, Wang Y, Hayat T, Oberlack M (2008) Hydromagnetic flow in a viscoelastic fluid due to the oscillatory stretching surface. Int J Nonlinear Mech 43:783–797CrossRefGoogle Scholar
- 35.Ali N, Khan SU, Abbas Z (2015) Hydromagnetic flow and heat transfer of a Jeffrey fluid over an oscillatory stretching surface. Z Naturforschung A 70:567–576CrossRefGoogle Scholar
- 36.Zheng LC, Jin X, Zhang XX, Zhang JH (2013) Unsteady heat and mass transfer in MHD flow over an oscillatory stretching surface with Soret and Dufour effects. Acta Mech Sin 29:667–675MathSciNetCrossRefGoogle Scholar
- 37.Ali N, Khan SU, Abbas Z, Sajid M (2016) Soret and Dufour effects on hydromagnetic flow of viscoelastic fluid over porous oscillatory stretching sheet with thermal radiation. J Braz Soc Mech Sci Eng 38:2533–2546CrossRefGoogle Scholar
- 38.Khan SU, Ali N (2017) Unsteady hydromagnetic flow of Oldroyd-B fluid over an oscillatory stretching surface: a mathematical model. Tech Sci 20:87–100Google Scholar
- 39.Turkyilmazoglu M (2011) Numerical and analytical solutions for the flow and heat transfer near the equator of an MHD boundary layer over a porous rotating sphere. Int J Therm Sci 50:831–842CrossRefGoogle Scholar
- 40.Liao SJ (2014) Advances in the homotopy analysis method, 5 Toh Tuck Link. World Scientific Publishing, SingaporeCrossRefGoogle Scholar
- 41.Sui J, Zheng L, Zhang X (2016) Boundary layer heat and mass transfer with Cattaneo–Christov double-diffusion in upper-convected Maxwell nanofluid past a stretching sheet with slip velocity. Int J Therm Sci 104:461–468CrossRefGoogle Scholar
- 42.Hayat T, Muhammad T, Shehzad SA, Alsaedi A (2017) Simultaneous effects of magnetic field and convective condition in three-dimensional flow of couple stress nanofluid with heat generation/absorption. J Braz Soc Mech Sci Eng 39:1165–1176CrossRefGoogle Scholar