Computational analysis of unsteady and steady magnetohydrodynamic radiating nano fluid flows past a slippery stretching sheet immersed in a permeable medium

Document Type : Research Note


1 Department of Mathematics, School of Natural Sciences (SNS), National University of Sciences and Technology (NUST), Sector H-12, 44000, Islamabad, Pakistan

2 Lahore University of Management Sciences, bDepartment of Mathematics, Lahore University of Management Sciences (LUMS), 54792, Lahore, PakistanLahore, Pakistan

3 Chah Ranjhay Wala, Post Office Bangla Morr, Tehsil Shujaabad, Multan, Pakistan

4 Seksjon for Matematikk, Nord Universitet, 8026 Bodo, Norway


This work gives an account of unsteady and steady radiating magnetohydrodynamics (MHD) nanofluid flow over a slippery stretching sheet embedded in a porous medium. A widely used similarity variables of such flows reduces a governing partial differential equations (PDEs) into a new set of partial differential equations (PDEs) in which a dependent variable is a function of two independent variables rather than three. At the same time, third order PDEs are converted into a second order PDEs by defining a new variable wherever it is deemed necessary. For the time integration, we perform first order explicit Euler method and spatial derivatives are approximated by the finite difference formulas in the context of node-centered finite volume method. It is revealed that for the unsteady flow the temperature of the nanofluid is higher near the surface without thermophoresis parameter Nt and reduced significantly when Nt is present. We also show that the concentration boundary layer thickness decreases with an increase of Darcy number Da.


1. Khan, Z., Rasheed, H.U., Alkanhal, T.A., Ullah, M.,
Khan, I., and Tlili, I. E ect of magnetic eld and heat
source on upper-convected-Maxwell
uid in a porous
channel", Open Physics, 16, pp. 917{928 (2018).
2. Kumar, P.S., Gireesha, B.J., Mahanthesh, B., and
Chamkha, A.J. Thermal analysis of nano
containing gyrotactic microorganisms in bioconvection
and second-order slip with convective condition", J.
Therm. Anal., 136, pp. 1947{1957 (2019).
3. Choi, S.U. and Eastman, J.A. Enhancing thermal
conductivity of
uids with nanoparticles", No.
ANL/MSD/CP-84938; CONF-951135-29; Argonne
National Lab., IL (United States) (1995).
4. Philip, J., Shima, P.D., and Raj, B. Nano
uid with
tunable thermal properties", Appl. Phys. Lett., 92, p.
043108 (2008).
5. Kao, M.J., Lo, C.H., Tsung, T.T., Wu, Y.Y., Jwo,
C.S., and Lin, H.M. Copper-oxide brake nano
manufactured using arc-submerged nanoparticle synthesis
system", J. Alloys Compd., 434, pp. 672{674
6. Yu, W. and Xie, H. A review on nano
uids: preparation,
stability mechanisms, and applications", J.
Nanomater., 2012, pp. 1{17 (2012).
7. Mahian, O., Kianifar, A., Kalogirou, S.A., Pop, I.,
and Wongwises, S. A review of the applications of
uids in solar energy", Int. J. Heat Mass Transfer,
57, pp. 582{594 (2013).
8. Angayarkanni, S.A. and Philip, J. Review on thermal
properties of nano
uids: Recent developments",
Advances in Colloid and Interface Science, 225, pp.
146{176 (2015).
9. Hung Thang, B., Le Quang, D., Manh Hong, N., Khoi,
P.H., and Minh, P.N. Application of multiwalled
carbon nanotube nano
uid for 450 W LED
light", J. Nanomater., 2014, pp. 1{6 (2014).
10. Mahendran, V. and Philip, J. Naked eye visualization
of defects in ferromagnetic materials and components",
NDT & E. Int, 60, pp. 100{109 (2013).
11. Liakos, I., Grumezescu, A.M., and Holban, A.M.
Magnetite nanostructures as novel strategies for antiinfectious
therapy", Molecules, 19, pp. 12710{12726
12. Buongiorno, J. Convective transport in nano
J. Heat Transfer, 128, pp. 240{250 (2006).
13. Das, K., Duari, P.R., and Kundu, P.K. Nano

ow over an unsteady stretching surface in presence
of thermal radiation", Alex. Eng. J., 53, pp. 737{745
14. Atif, S.M., Hussain, S., and Sagheer, M. Heat and
mass transfer analysis of time-dependent tangent hyperbolic
ow past a wedge", Physics Letters
A, 383(11), pp. 1187{1198 (2019).
15. Azam, M., Shakoor, A., Rasool, H.F., and Khan,
M. Numerical simulation for solar energy aspects on
unsteady convective
ow of MHD cross nano
uid: A
revised approach", Int. J. Heat Mass Transf., 131, pp.
495{505 (2019).
16. Khan, M., Huda, N.U., and Hamid, A. Non-linear
radiative heat transfer analysis during the
ow of
Carreau nano
uid due to wedge-geometry: A revised
model", Int. J. Heat Mass Transf., 131, pp. 1022{1031
17. Hayat, T., Aziz, A., Muhammad, T., and Alsaedi, A.
Numerical simulation for three-dimensional
ow of
Carreau nano
uid over a nonlinear stretching surface
with convective heat and mass conditions", J. Braz.
Soc. Mech. Sci., 41, pp. 41{55 (2019).
18. Ghadikolaei, S.S., Hosseinzadeh, K., Ganji, D.D., and
Hatami, M. Fe3O4(CH2OH)2 nano
uid analysis in a
porous medium under MHD radiative boundary layer
and dusty
uid", J. Mol. Liq, 258, pp. 172{185 (2018).
19. Rossow, V.J. On
ow of electrically conducting
over a
at plate in the presence of a transverse
magnetic eld", NASA, Report No. 1358, 489 (1958).
20. Raptis, A., Perdikis, C., and Takhar, H.S. E ect
of thermal radiation on MHD
ow", Appl. Math.
Comput., 153, pp. 645{649 (2004).
21. Fang, T., Zhang, J., and Yao, S. Slip MHD viscous

ow over a stretching sheet. An exact solution", Commun.
Nonlinear Sci. Numer. Simul., 14, pp. 3731{3737
22. Makinde, O.D., Mabood, F., Khan, W.A., and
Tshehla, M.S. MHD
ow of a variable viscosity
uid over a radially stretching convective surface
with radiative heat", J. Mol. Liq, 219, pp. 624{630
23. Nield, D.A. and Kuznetsov, A.V. The Cheng-
Minkowycz problem for natural convective boundarylayer

ow in a porous medium saturated by a
uid", Int. J. Heat Mass Transfer., 52, pp. 5792{
5795 (2009).
24. Hayat, T., Saif, R.S., Ellahi, R., Muhammad, T.,
and Alsaedi, A. Simultaneous e ects of melting heat
and internal heat generation in stagnation point
of Je rey
uid towards a nonlinear stretching surface
with variable thickness", Int. J. Therm. Sci., 132, pp.
344{354 (2018).
M.A. Farooq et al./Scientia Iranica, Transactions F: Nanotechnology 27 (2020) 3454{3466 3465
25. Makinde, O.D., Khan, Z.H., Ahmad, R., and Khan,
W.A. Numerical study of unsteady hydromagnetic
ow past a slippery stretching sheet
embedded in a porous medium", Phys. Fluids, 30, p.
083601 (2018).
26. Malik, M.Y., Naseer, M., Nadeem, S., and Rehman,
A. The boundary layer
ow of Casson nano
uid over
a vertical exponentially stretching cylinder", Appl.
Nanosci., 4, pp. 869{873 (2014).
27. Jusoh, R., Nazar, R., and Pop, I. Flow and
heat transfer of magnetohydrodynamic threedimensional
Maxwell nano
uid over a permeable
stretching/shrinking surface with convective boundary
conditions", J. Comput. Theor. Nanosci., 14, pp.
1644{1652 (2017).
28. Bhaskar Reddy, N., Poornima, T., and Sreenivasulu, P.
uence of variable thermal conductivity on MHD
boundary layer slip
ow of ethylene-glycol based Cu
uids over a stretching sheet with convective
boundary condition", Int. J. of Eng. Math., 2014, pp.
1{10 (2014).
29. Hakeem, A.A., Kalaivanan, R., Ganesh, N.V., and
Ganga, B. E ect of partial slip on hydromagnetic
over a porous stretching sheet with non-uniform heat
source/sink, thermal radiation and wall mass transfer",
Eng. Phys. and Math., 5, pp. 913{922 (2014).
30. Cortell, R. Flow and heat transfer of a
uid through
a porous medium over a stretching surface with internal
heat generation/absorption andsuction/blowing",
Fluid Dyn. Res., 37, p. 231 (2005).
31. Cai, W., Su, N., and Liu, X. Unsteady convection
and heat transfer over a vertical stretching surface",
PLoS One, 9, p. 111 (2014).
32. Wang, C.Y. Flow due to stretching boundary with
partial slip- an exact solution of the navier-stokes
equation", Chem. Eng. Sci., 57, pp. 3745{3747 (2002).
33. Noghrehabadi, A., Pourrajab, R., and Ghalambaz, M.
E ect of partial slip boundary condition on the
and heat transfer of nano
uids past stretching sheet
prescribed constant wall temperature", Int. J. Therm.
Sci., 54, pp. 253{261 (2012).
34. Ariel, P.D., Hayat, T., and Asghar, S. The
ow of
an elastico-viscous
uid past a stretching sheet with
partial slip", Acta Mechanica, 187(1{4), pp. 29{35
35. Fatunmbi, E.O. and Fenuga, O.J. Heat and mass
transfer of a chemically reacting MHD micropolar

ow over an exponentially stretching sheet with slip
e ects", Physical Science International Journal, 18(3),
pp. 1{15 (2018).
36. Khan, N.S., Zuhra, S., Shah, Z., Bonyah, E., Khan,
W., and Islam, S. Slip
ow of Eyring-Powell nanoliquid
lm containing graphene nanoparticles", AIP
Advances, 8(11), p. 115302 (2018).
37. LeVeque, R.J., Finite Di erence Methods for Ordinary
and Partial Di erential Equations: Steady-State and
Time-Dependent Problems, Siam (2007).
38. LeVeque, R.J., Finite Volume Methods for Hyperbolic
Problems, Cambridge University Press (2002).
39. Sheikholeslami, M. CuO-water nano
ow due
to magnetic eld inside a porous media considering
Brownian motion", J. Mol. Liq., 249, pp. 921{929
40. Reza-E-Rabbi, Sk, Arifuzzaman, S.M., Tanmoy
Sarkar, Md Shakhaoath Khan, and Sarder Firoz
Ahmmed. Explicit nite di erence analysis of an
unsteady MHD
ow of a chemically reacting Casson

uid past a stretching sheet with Brownian motion
and thermophoresis e ects", Journal of King Saud
University-Science, 32(1), pp. 690-701 (2020).
41. Shampine, L.F., Kierzenka, J., and Reichelt, M.W.
Solving boundary value problems for ordinary di erential
equations in MATLAB with bvp4c", Tutorial
Notes, pp. 1{17 (2000).
42. Mabood, F. and Das, K. Melting heat transfer on
ow of a nano
uid over a stretching
sheet with radiation and second-order slip", Eur. Phys.
J. Pl., 131(1), pp. 1{12 (2016).
43. Mabood, F. and Shateyi, S. Multiple slip e ects on
MHD unsteady
ow heat and mass transfer impinging
on permeable stretching sheet with radiation", Model.
Sim. Engi., 2019 (2019).