Abstract

This contribution proposes a third-order numerical scheme for solving time-dependent partial differential equations (PDEs). This third-order scheme is further modified, and the new scheme is obtained with second-order accuracy in time and is unconditionally stable. The unconditional stability of the new scheme is proved by employing von Neumann stability analysis. For spatial discretization, a compact fourth-order accurate scheme is adopted. Moreover, a mathematical model for heat transfer of Darcy–Forchheimer flow of micropolar fluid is modified with an oscillatory sheet, nonlinear mixed convection, thermal radiation, and viscous dissipation. Later on, the dimensionless model is solved by the proposed second-order scheme. The results show that velocity and angular velocity have dual behaviors by incrementing coupling parameters. The proposed second-order accurate in-time scheme is compared with an existing Crank–Nicolson scheme and backward in-time and central in space (BTCS) scheme. The proposed scheme is shown to have faster convergence than the existing Crank–Nicolson scheme with the same order of accuracy in time and space. Also, the proposed scheme produces better order of convergence than an existing Crank–Nicolson scheme.

References

1.
Eringen
,
A. C.
,
1966
, “
Theory of Micropolar Fluids
,”
J. Math. Mech.
,
16
(
1
), pp.
1
18
.10.2307/24901466
2.
Rees
,
D. A. S.
, and
Pop
,
I.
,
1998
, “
Free Convection Boundary Layer Flow of a Micropolar Fluid From a Vertical Flat Plate
,”
IMA J. Appl. Math.
,
61
(
2
), pp.
179
197
.10.1093/imamat/61.2.179
3.
Partha
,
M.
,
2010
, “
Nonlinear Convection in a non-Darcy Porous Medium
,”
Appl. Math. Mech.
,
31
(
5
), pp.
565
574
.10.1007/s10483-010-0504-6
4.
Datta
,
N.
, and
Jana
,
R. N.
,
1976
, “
Oscillatory Magneto Hydrodynamic Flow Past a Flat Plate With Hall Effects
,”
J. Phys. Soc. Jpn.
,
40
(
5
), pp.
1469
1474
.10.1143/JPSJ.40.1469
5.
Aboeldahab
,
E. M.
, and
Elbarbary
,
E. M.
,
2001
, “
Hall Current Effect on Magneto Hydrodynamic Freeconvection Flow Past a Semi-Infinite Vertical Plate With Mass Transfer
,”
Int. J. Eng. Sci.
,
39
(
14
), pp.
1641
1652
.10.1016/S0020-7225(01)00020-9
6.
Buongiorno
,
J.
,
2006
, “
Convective Transport in Nanofluids
,”
ASME J. Heat Mass Transfer-Trans. ASME
,
128
(
3
), pp.
240
250
.10.1115/1.2150834
7.
Hayat
,
T.
,
Khan
,
M. I.
,
Waqas
,
M.
,
Alsaedi
,
A.
, and
Khan
,
M. I.
,
2017
, “
Radiative Flow of Micropolar Nanofluid Accounting Thermophoresis and Brownian Moment
,”
Int. J. Hydrogen Energy
,
42
(
26
), pp.
16821
16833
.10.1016/j.ijhydene.2017.05.006
8.
Patel
,
H. R.
, and
Singh
,
R.
,
2019
, “
Thermophoresis, Brownian Motion and Nonlinear Thermal Radiation Effects on Mixed Convection MHD Micropolar Fluid Due to Nonlinear Stretched Sheet in Porous Medium With Viscous Dissipation, Joule Heating and Convective Boundary Condition
,”
Int. Commun. Heat Mass Transfer
,
107
, pp.
68
92
.10.1016/j.icheatmasstransfer.2019.05.007
9.
Sabir
,
Z.
,
Ayub
,
A.
,
Guirao
,
J. L. G.
,
Bhatti
,
S.
, and
Shah
,
S. Z. H.
,
2020
, “
The Effect of Activation Energy and Thermophoretic Diffusion of Nanoparticles on Steady Micropolar Fluid Along With Brownian Motion
,”
Adv. Mat. Sci. Eng.
,
2020
, pp.
1
12
.10.1155/2020/2010568
10.
Chamber
,
P. L.
, and
Young
,
J. D.
,
1958
, “
The Effects of Homogeneous 1st Order Chemical Reactions in the Neighborhood of a Flat Plate for Destructive and Generative Reactions
,”
Phys. Fluids
,
1
, pp.
48
54
.10.1063/1.1724336
11.
Khan
,
M. I.
,
Waqas
,
M.
,
Hayat
,
T.
, and
Alsaedi
,
A.
,
2017
, “
A Comparative Study of Casson Fluid With Homogeneous-Heterogenous Reactions
,”
J. Colloid Interface Sci.
,
498
, pp.
85
90
.10.1016/j.jcis.2017.03.024
12.
Khan
,
M. I.
, and
Alzahrani
,
F.
,
2020
, “
Activation Energy and Binary Chemical Reaction Effect in Nonlinear Thermal Radiative Stagnation Point Flow of Walter-B Nanofluid: Numerical Computations
,”
Int. J. Mod. Phys. B
,
34
(
13
), p.
2050132
.10.1142/S0217979220501325
13.
Abbas
,
S. Z.
,
Khan
,
M. I.
,
Kadry
,
S.
,
Khan
,
W. A.
,
Israr-Ur-Rehman
,
M.
, and
Waqas
,
M.
,
2020
, “
Fully Developed Entropy Optimized Second Order Velocity Slip MHD Nanofluid Flow With Activation Energy
,”
Comput. Methods Programs Biomed.
,
190
, p.
105362
.10.1016/j.cmpb.2020.105362
14.
Chaudhary
,
R. C.
, and
Jha
,
A. K.
,
2008
, “
Effects of Chemical Reactions on MHD Micropolar Fluid Flow Past a Vertical Plate in Slip-Flow Regime
,”
Appl. Math. Mech.
,
29
(
9
), pp.
1179
1194
.10.1007/s10483-008-0907-x
15.
Sheri
,
S. R.
, and
Shamshuddin
,
M. D.
,
2015
, “
Heat and Mass Transfer on the MHD Flow of Micropolar Fluid in the Presence of Viscous Dissipation and Chemical Reaction
,”
Proc. Eng.
,
127
, pp.
885
895
.10.1016/j.proeng.2015.11.426
16.
Siva Reddy
,
S.
, and
Shamshuddin
,
M. D.
,
2018
, “
Finite Element Analysis on Transient Magnetohydro-Dynamic (MHD) Free Convective Chemically Reacting Micropolar Fluid Flow Past a Vertical Porous Plate With Hall Current and Viscous Dissipation
,”
Prop. Power Res.
,
7
(
4
), pp.
353
365
.10.1016/j.jppr.2018.11.003
17.
Das
,
K.
,
2011
, “
Effect of Chemical Reaction and Thermal Radiation on Heat and Mass Transfer Flow of MHD Micropolar Fluid in a Rotating Frame of Reference
,”
Int. J. Heat Mass Transfer
,
54
(
15–16
), pp.
3505
3513
.10.1016/j.ijheatmasstransfer.2011.03.035
18.
Shamshuddin
,
M. D.
,
Siva Reddy
,
S.
, and
Anwar Beg
,
O.
,
2017
, “
Oscillatory Dissipative Conjugate Heat and Mass Transfer in Chemically-Reacting Micropolar Flow With Wall Couple Stress: Finite Element Numerical Study
,”
Proc. Inst. Mech. Eng. E
,
233
(
1
), pp.
48
64
.10.1177/0954408917743372
19.
Anjanna
,
M.
, and
Nagaraju
,
G.
,
2018
, “
Order of Chemical Reaction and Convective Boundary Condition Effects on Micropolar Fluid Flow Over a Stretching Sheet
,”
AIP Adv.
,
8
(
11
), p.
115212
.10.1063/1.5053445
20.
Fatunmbi
,
E. O.
, and
Agbolade
,
A. O.
,
2019
, “
Heat and Mass Transfer of Thermophoretic Magneto-Micropolar Fluid Passing an Inclined Plate With Chemical Reaction in Porous Medium
,”
School Pure Appl. Sci. J.
,
1
(
1
), pp.
15
22
.
21.
Arif
,
M. S.
,
Raza
,
A.
,
Rafiq
,
M.
,
Bibi
,
M.
,
Abbasi
,
J. N.
,
Nazeer
,
A.
, and
Javed
,
U.
,
2020
, “
Numerical Simulations for Stochastic Computer Virus Propagation Model
,”
Comput. Mater. Continua
,
62
, pp.
61
77
.10.32604/cmc.2020.08595
22.
Bibi
,
M.
,
Nawaz
,
Y.
,
Arif
,
M. S.
,
Abbasi
,
J. N.
,
Javed
,
U.
, and
Nazeer
,
A.
,
2020
, “
A Finite Difference Method and Effective Modification of Gradient Descent Optimization Algorithm for MHD Fluid Flow Over a Linearly Stretching Surface
,”
Comput., Mater. Continua
,
62
(
2
), pp.
657
677
.10.32604/cmc.2020.08584
24.
Shamshuddin
,
M. D.
, and
Mabood
,
F.
,
2021
, “
A Numerical Model for Analysis of Binary Chemical Reaction and Activation Energy of Thermo Solutal Micropolar Nanofluid Flow Through Permeable Stretching Sheet: Nanoparticle Study
,”
Phys. Scr.
,
96
, p.
075206
.10.1088/1402-4896/abf794
25.
Turkyilmazoglu
,
M.
,
2010
, “
Unsteady Mhd Flow With Variable Viscosity: Applications of Spectral Scheme
,”
Int. J. Therm. Sci.
,
49
(
3
), pp.
563
570
.10.1016/j.ijthermalsci.2009.10.007
26.
Shamshuddin
,
M. D.
,
Ferdows
,
M.
,
Anwar Beg
,
O.
,
Bég
,
T. A.
, and
Leonard
,
H. J.
,
2022
, “
Computation of Reactive Thermosolutal Micropolar Nanofluid Sakiadis Convection Flow With Gold/Silver Metallic Nanoparticles
,”
Waves Random Complex Media
, pp.
1
37
.
27.
Shamshuddin
,
M. D.
,
Mabood
,
F.
,
Rajput
,
G. R.
,
Bég
,
O. A.
, and
Badruddin
,
I. A.
,
2022
, “
Thermo-Solutal Dual Stratified Convective Magnetized Fluid Flow From an Exponentially Stretching Riga Plate Sensor Surface With Thermophoresis
,”
Int. Commun. Heat Mass Transfer
,
134
, p.
105997
.10.1016/j.icheatmasstransfer.2022.105997
28.
Mabood
,
F.
,
Shamshuddin
,
M. D.
, and
Mishra
,
S. R.
,
2022
, “
Characteristics of Thermophoresis and Brownian Motion on Radiative Reactive Micropolar Fluid Flow Towards Continuously Moving Flat Plate: HAM Solution
,”
Math. Comput. Simul.
,
191
, pp.
187
202
.10.1016/j.matcom.2021.08.004
29.
Shamshuddin
,
M. D.
,
Mabood
,
F.
, and
Bég
,
O. A.
,
2022
, “
Thermomagnetic Reactive Ethylene Glycol-Metallic Nanofluid Transport From a Convectively Heated Porous Surface With Ohmic Dissipation, Heat Source, Thermophoresis and Brownian Motion Effects
,”
Int. J. Modell. Simul.
,
42
(
5
), pp.
782
796
.10.1080/02286203.2021.1977531
30.
Shamshuddin
,
M. D.
,
Mabood
,
F.
,
Khan
,
W. A.
, and
Rajput
,
G. R.
,
2022
, “
Exploration of Thermal Péclet Number, Vortex Viscosity, and Reynolds Number on Two‐Dimensional Flow of Micropolar Fluid Through a Channel Due to Mixed Convection
,”
Heat Transfer
, 52(1), pp.
854
873
.
31.
Salawu
,
S. O.
,
Obalalu
,
A. M.
, and
Shamshuddin
,
M. D.
,
2022
, “
Nonlinear Solar Thermal Radiation Efficiency and Energy Optimization for Magnetized Hybrid Prandtl–Eyring Nanoliquid in Aircraft
,”
Arabian J. Sci. Eng.
, pp.
1
12
.
32.
Turkyilmazoglu
,
M.
,
2011
, “
An Optimal Analytic Approximate Solution for the Limit Cycle of Duffing–Van Der Pol Equation
,”
ASME J. Appl. Mech
, 78(2), p.
021005
.10.1115/1.4002567
33.
Turkyilmazoglu
,
M.
,
2013
, “
Effective Computation of Exact and Analytic Approximate Solutions to Singular Nonlinear Equations of Lane–Emden–Fowler Type
,”
Appl. Math. Modell.
,
37
(
14–15
), pp.
7539
7548
.10.1016/j.apm.2013.02.014
34.
Turkyilmazoglu
,
M.
,
2017
, “
Solution of Initial and Boundary Value Problems by an Effective Accurate Method
,”
Int. J. Comput. Methods
,
14
(
06
), p.
1750069
.10.1142/S0219876217500694
35.
Shamshuddin
,
M. D.
, and
Ibrahim
,
W.
,
2022
, “
Finite Element Numerical Technique for Magneto-Micropolar Nanofluid Flow Filled With Chemically Reactive Casson Fluid Between Parallel Plates Subjected to Rotatory System With Electrical and Hall Currents
,”
Int. J. Modell. Simul.
, 42(6), pp. 985–1004.10.1080/02286203.2021.2012634
36.
Rao
,
P. S.
, and
Shamshuddin
,
M. D.
,
2021
, “
Second‐Order Slip and Newtonian Cooling Impact on Unsteady Mixed Convective Radiative Chemically Reacting Fluid With Hall Current and Cross‐Diffusion Over a Stretching Sheet
,”
Heat Transfer
,
50
(
7
), pp.
7380
7405
.10.1002/htj.22234
37.
Shamshuddin
,
M. D.
,
Mabood
,
F.
, and
Salawu
,
S. O.
,
2021
, “
Flow of Three‐Dimensional Radiative Williamson Fluid Over an Inclined Stretching Sheet With Hall Current and Nth‐Order Chemical Reaction
,”
Heat Transfer
,
50
(
6
), pp.
5400
5417
.10.1002/htj.22130
38.
Shamshuddin
,
M. D.
,
Khan
,
S. U.
,
Bég
,
O. A.
, and
Beg
,
T. A.
,
2020
, “
Hall Current, Viscous and Joule Heating Effects on Steady Radiative 2-D Magneto-Power-Law Polymer Dynamics From an Exponentially Stretching Sheet With Power-Law Slip Velocity: A Numerical Study
,”
Therm. Sci. Eng. Prog.
,
20
, p.
100732
.10.1016/j.tsep.2020.100732
39.
Raptis
,
A.
,
1998
, “
Flow of a Micropolar Fluid Past a Continuously Moving Plate by the Presence of Radiation: Technical Note
,”
Int. J. Heat Mass Transfer
,
41
(
18
), pp.
2865
2866
.10.1016/S0017-9310(98)00006-4
40.
Hayat
,
T.
,
Momani
,
S.
,
Muhammad
., and
K.
,
Inayatullah
,
2021
, “
FDM Analysis for Nonlinear Mixed Convective Nanofluid Flow With Entropy Generation
,”
Int. Commun. Heat Mass Transfer
,
126
, p.
105389
.10.1016/j.icheatmasstransfer.2021.105389
41.
Nawaz
,
Y.
,
Arif
,
M. S.
, and
Abodayeh
,
K.
,
2022
, “
A Third-Order Two-Stage Numerical Scheme for Fractional Stokes Problems: A Comparative Computational Study
,”
ASME J. Comput. Nonlinear Dyn.
17
(
10
), p.
101004
.10.1115/1.4054800
42.
Arif
,
M. S.
,
Abodayeh
,
K.
, and
Nawaz
,
Y.
,
2022
, “
The Modified Finite Element Method for Heat and Mass Transfer of Unsteady Reacting Flow With Mixed Convection
,”
Front. Phys.
,
10
, p.
802
.10.3389/fphy.2022.952787
43.
Arif
,
M. S.
,
Abodayeh
,
K.
, and
Nawaz
,
Y.
,
2022
, “
An Application of the Finite Element Method for Heat and Mass Transfer of Boundary Layer Flow Using Variable Thermal Conductivity and Mass Diffusivity
,”
J. Math.
2022
, p.
7929161
.10.1155/2022/7929161
44.
Nawaz
,
Y.
,
Arif
,
M. S.
, and
Abodayeh
,
K.
,
2022
, “
A Numerical Scheme for Fractional Mixed Convection Flow Over Flat and Oscillatory Plates
,”
ASME J. Comput. Nonlinear Dyn.
,
17
(
7
), p.
071008
.10.1115/1.4054483
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