Exact solutions for natural convection flows of generalized Brinkman type fluids: A Prabhakar-like fractional model with generalized thermal transport

Natural convection flow of unsteady Prabhakar-like fractional Brinkman type fluids over an infinite vertical heated wall with uniform heat flux is studied. Exact dimensionless solutions of momentum and energy equations, under Boussinesq approximation, are obtained using Laplace transforms. They sati...

Full description

Saved in:
Bibliographic Details
Published inCase studies in thermal engineering Vol. 26; p. 101126
Main Authors Sun, Yu-Liang, Shah, Nehad Ali, Khan, Zar Ali, Mahrous, Y.M., Ahmad, Bakhtiar, Chung, Jae Dong, Khan, M. Niaz
Format Journal Article
LanguageEnglish
Published Elsevier Ltd 01.08.2021
Elsevier
Subjects
Brinkman type fluids
Exact solutions
Laplace transform
Mittag-Leffler functions
Prabhakar-like fractional derivative
Prabhakar-like fractional derivative
Mittag-Leffler functions
Brinkman type fluids
Laplace transform
Exact solutions
Online AccessGet full text

Cover

Abstract Natural convection flow of unsteady Prabhakar-like fractional Brinkman type fluids over an infinite vertical heated wall with uniform heat flux is studied. Exact dimensionless solutions of momentum and energy equations, under Boussinesq approximation, are obtained using Laplace transforms. They satisfy all imposed conditions and reduce to known solutions from the literature as special cases. The velocity and heat transfer of Prabhakar-like fractional Brinkman type fluids with generalized thermal transport are compared with ordinary Brinkman type fluids with generalized thermal transport and with the ordinary viscoelastic fluids with classical Fourier thermal flux. Finally, the influence of different parameters like fractional parameters, Prandtl number, Grashof number, time and Brinkman parameter on temperature and velocity is shown by graphs.
AbstractList Natural convection flow of unsteady Prabhakar-like fractional Brinkman type fluids over an infinite vertical heated wall with uniform heat flux is studied. Exact dimensionless solutions of momentum and energy equations, under Boussinesq approximation, are obtained using Laplace transforms. They satisfy all imposed conditions and reduce to known solutions from the literature as special cases. The velocity and heat transfer of Prabhakar-like fractional Brinkman type fluids with generalized thermal transport are compared with ordinary Brinkman type fluids with generalized thermal transport and with the ordinary viscoelastic fluids with classical Fourier thermal flux. Finally, the influence of different parameters like fractional parameters, Prandtl number, Grashof number, time and Brinkman parameter on temperature and velocity is shown by graphs.
ArticleNumber 101126
Author Mahrous, Y.M.
Khan, Zar Ali
Ahmad, Bakhtiar
Sun, Yu-Liang
Khan, M. Niaz
Shah, Nehad Ali
Chung, Jae Dong
Author_xml – sequence: 1
  givenname: Yu-Liang
  surname: Sun
  fullname: Sun, Yu-Liang
  organization: School of Science, Huzhou University, Huzhou, 313000, PR China
– sequence: 2
  givenname: Nehad Ali
  surname: Shah
  fullname: Shah, Nehad Ali
  organization: Department of Mechanical Engineering, Sejong University, Seoul, 05006, South Korea
– sequence: 3
  givenname: Zar Ali
  surname: Khan
  fullname: Khan, Zar Ali
  organization: Department of Mathematics, Government Superior Science College Peshawar, Peshawar, 25000, Khyber Pakhtunkhwa, Pakistan
– sequence: 4
  givenname: Y.M.
  surname: Mahrous
  fullname: Mahrous, Y.M.
  organization: Department of Studies and Basic Sciences, Faculty of Community, University of Tabuk, P.O. Box 741, Tabuk, 71491, Saudi Arabia
– sequence: 5
  givenname: Bakhtiar
  surname: Ahmad
  fullname: Ahmad, Bakhtiar
  organization: Department of Mathematics, Government College Peshawar, Peshawar, 25000, Khyber Pakhtunkhwa, Pakistan
– sequence: 6
  givenname: Jae Dong
  surname: Chung
  fullname: Chung, Jae Dong
  email: jdchung@sejong.ac.kr
  organization: Department of Mechanical Engineering, Sejong University, Seoul, 05006, South Korea
– sequence: 7
  givenname: M. Niaz
  surname: Khan
  fullname: Khan, M. Niaz
  organization: Department of Mathematics, Lahore Leads University, Lahore, Pakistan
BookMark eNp9kc1uEzEUhS3USpTSJ-jGLzDBfzOTQWJRqgKVKpUFlbqz7tjXjZOJHdlOf3gQnhcnQRVsWNk6555P9j3vyFGIAQk552zGGe8-LGcm-4IzwQTfKVx0b8iJEFw1vO3vj_66vyVnOS8ZY7yXc67UCfl19Qym0BynbfExZOpiogHKNsFETQyPaHY6dVN8yjQ6-oABq-d_oqWfkw-rNQRaXjZYR7be5o_0gn5PMC5gBamZ_KoaCfaQSlxHixN98mXxD6gsMK2rXRKEvImpvCfHDqaMZ3_OU3L35erH5bfm5vbr9eXFTWMUV6XpZN8b24qRS4HzYWStAWYlDr2TzvRymDvTOdeLURrHLJMISjEDCBbHsevkKbk-cG2Epd4kv4b0oiN4vRdietCQijcTaobYGi4quZNqRAZO8RYGpwbeDgPwypIHlkkx54TulceZ3jWll3rflN41pQ9N1dSnQwrrNx89Jp2Nx2DQ-lR3X9_h_5v_DU61o6o
CitedBy_id crossref_primary_10_1142_S0217984923502561
crossref_primary_10_3390_fractalfract6120712
crossref_primary_10_1016_j_csite_2023_103077
crossref_primary_10_1142_S0217979224500577
crossref_primary_10_1007_s00419_022_02188_0
crossref_primary_10_1038_s41598_023_36436_2
crossref_primary_10_1080_17455030_2022_2067379
crossref_primary_10_1016_j_csite_2023_103091
Cites_doi 10.1007/BF01094898
10.1007/s00231-009-0552-3
10.1115/1.3450729
10.1016/j.cjph.2019.05.006
10.1016/j.cnsns.2017.08.018
10.1016/0735-1933(83)90016-7
10.3390/math7020150
10.1007/BF02120318
10.1023/A:1006593330865
10.1016/0017-9310(85)90190-5
10.7153/fdc-06-05
10.1016/j.cnsns.2017.08.002
10.1007/BF00651403
10.1016/j.chaos.2019.109489
10.1016/j.cjph.2018.02.007
10.1007/s11012-009-9257-4
10.1016/j.molliq.2019.110964
10.1142/S0218202507001899
10.1063/1.5032165
10.1016/j.rinp.2018.04.034
10.1007/BF02120313
ContentType Journal Article
Copyright 2021 The Authors
Copyright_xml – notice: 2021 The Authors
DBID 6I.
AAFTH
AAYXX
CITATION
DOA
DOI 10.1016/j.csite.2021.101126
DatabaseName ScienceDirect Open Access Titles
Elsevier:ScienceDirect:Open Access
CrossRef
Open Access: DOAJ - Directory of Open Access Journals
DatabaseTitle CrossRef
DatabaseTitleList

Database_xml – sequence: 1
  dbid: DOA
  name: DOAJ Directory of Open Access Journals
  url: https://www.doaj.org/
  sourceTypes: Open Website
DeliveryMethod fulltext_linktorsrc
Discipline Engineering
EISSN 2214-157X
ExternalDocumentID oai_doaj_org_article_0ee5c127f3634be0af415a9f491599a1
10_1016_j_csite_2021_101126
S2214157X21002896
GroupedDBID 0R~
0SF
457
5VS
6I.
AACTN
AAEDT
AAEDW
AAFTH
AAIKJ
AALRI
AAXUO
ABMAC
ACGFS
ADBBV
ADEZE
AEXQZ
AFTJW
AGHFR
AITUG
ALMA_UNASSIGNED_HOLDINGS
AMRAJ
BCNDV
EBS
EJD
FDB
GROUPED_DOAJ
HZ~
IPNFZ
IXB
KQ8
M41
M~E
NCXOZ
O9-
OK1
RIG
ROL
SSZ
AAYXX
ADVLN
AFJKZ
AKRWK
CITATION
ID FETCH-LOGICAL-c414t-6377cd52b132e89b05ca0d3e97f3fc7398fc6ff72b3cf0d03ea440caeadebb663
IEDL.DBID DOA
ISSN 2214-157X
IngestDate Fri Oct 04 13:12:54 EDT 2024
Thu Sep 26 19:03:03 EDT 2024
Tue Jul 25 20:58:37 EDT 2023
IsDoiOpenAccess true
IsOpenAccess true
IsPeerReviewed true
IsScholarly true
Keywords Prabhakar-like fractional derivative
Mittag-Leffler functions
Brinkman type fluids
Laplace transform
Exact solutions
Language English
License This is an open access article under the CC BY license.
LinkModel DirectLink
MergedId FETCHMERGED-LOGICAL-c414t-6377cd52b132e89b05ca0d3e97f3fc7398fc6ff72b3cf0d03ea440caeadebb663
OpenAccessLink https://doaj.org/article/0ee5c127f3634be0af415a9f491599a1
ParticipantIDs doaj_primary_oai_doaj_org_article_0ee5c127f3634be0af415a9f491599a1
crossref_primary_10_1016_j_csite_2021_101126
elsevier_sciencedirect_doi_10_1016_j_csite_2021_101126
PublicationCentury 2000
PublicationDate August 2021
2021-08-00
2021-08-01
PublicationDateYYYYMMDD 2021-08-01
PublicationDate_xml – month: 08
  year: 2021
  text: August 2021
PublicationDecade 2020
PublicationTitle Case studies in thermal engineering
PublicationYear 2021
Publisher Elsevier Ltd
Elsevier
Publisher_xml – name: Elsevier Ltd
– name: Elsevier
References Awan, Shah, Ahmed, Ali, Riaz (bib23) 2019; 60
Singh, Kumar (bib3) 1984; 98
Ishak (bib6) 2009; 46
Elnaqeeb, Shah, Mirza (bib33) 2020
Gorla, Mansour, Sahar (bib15) 1999; 36
Garra, Garrappa (bib27) 2018; 56
Ali, Farman, Sheikh, Khan, Nisar (bib17) 2020; 131
Ahmed, Shah, Ahmad, Shah, Haq, Gorji (bib26) 2019; 5
Baleanu, Diethelm, Scalas, Trujillo (bib19) 2011
Ahmed, Shah, Vieru (bib22) 2018; 56
Hajizadeh, Shah, Shah, Animasaun, Gorji, Alarifi (bib24) 2019
Hsu, Cheng (bib11) 1985; 28
Ishak (bib7) 2010; 45
Ali, Khan, Haq, Shafie (bib12) 2012; 67a
Shah, Fetecau, Vieru (bib30) 2020
Khan, Haq, Khan, Khan, Tlili (bib16) 2018; 9
Ahmed, Vieru, Fetecau, Shah (bib21) 2018; 30
Soundalgekar (bib1) 1997; 99
Giusti, Colombaro (bib29) 2018; 56
Brinkman (bib9) 1949; 1
Raptis, Singh (bib2) 1983; 10
Haq, Fetecau, Khan, Ali, Shafie (bib8) 2012; 67a
Polito, Tomovski (bib32) 2016; 6
Mansour (bib5) 1990; 166
Varma, Babu (bib13) 1985; 16
Ortigueira, Machado (bib20) 2019; 7
Rajagopal (bib14) 2007; 17
Shah, Ahmed, Elnaqeeb, Rashidi (bib25) 2019; 5
Soundalgekar (bib4) 1973; 333
Yang (bib28) 2019
Jacquot, Steadman, Rhodine (bib34) 1983; 5
Brinkman (bib10) 1949; 1
Sabatier, Agrawal, Machado (bib18) 2007
Giusti, Colombaro (bib31) 2018; 56
Shah (10.1016/j.csite.2021.101126_bib25) 2019; 5
Khan (10.1016/j.csite.2021.101126_bib16) 2018; 9
Rajagopal (10.1016/j.csite.2021.101126_bib14) 2007; 17
Hajizadeh (10.1016/j.csite.2021.101126_bib24) 2019
Ahmed (10.1016/j.csite.2021.101126_bib21) 2018; 30
Haq (10.1016/j.csite.2021.101126_bib8) 2012; 67a
Garra (10.1016/j.csite.2021.101126_bib27) 2018; 56
Polito (10.1016/j.csite.2021.101126_bib32) 2016; 6
Ishak (10.1016/j.csite.2021.101126_bib6) 2009; 46
Awan (10.1016/j.csite.2021.101126_bib23) 2019; 60
Singh (10.1016/j.csite.2021.101126_bib3) 1984; 98
Elnaqeeb (10.1016/j.csite.2021.101126_bib33) 2020
Jacquot (10.1016/j.csite.2021.101126_bib34) 1983; 5
Raptis (10.1016/j.csite.2021.101126_bib2) 1983; 10
Ahmed (10.1016/j.csite.2021.101126_bib22) 2018; 56
Giusti (10.1016/j.csite.2021.101126_bib31) 2018; 56
Ortigueira (10.1016/j.csite.2021.101126_bib20) 2019; 7
Soundalgekar (10.1016/j.csite.2021.101126_bib1) 1997; 99
Baleanu (10.1016/j.csite.2021.101126_bib19) 2011
Ishak (10.1016/j.csite.2021.101126_bib7) 2010; 45
Varma (10.1016/j.csite.2021.101126_bib13) 1985; 16
Soundalgekar (10.1016/j.csite.2021.101126_bib4) 1973; 333
Shah (10.1016/j.csite.2021.101126_bib30) 2020
Sabatier (10.1016/j.csite.2021.101126_bib18) 2007
Ali (10.1016/j.csite.2021.101126_bib12) 2012; 67a
Ali (10.1016/j.csite.2021.101126_bib17) 2020; 131
Ahmed (10.1016/j.csite.2021.101126_bib26) 2019; 5
Mansour (10.1016/j.csite.2021.101126_bib5) 1990; 166
Brinkman (10.1016/j.csite.2021.101126_bib10) 1949; 1
Gorla (10.1016/j.csite.2021.101126_bib15) 1999; 36
Giusti (10.1016/j.csite.2021.101126_bib29) 2018; 56
Brinkman (10.1016/j.csite.2021.101126_bib9) 1949; 1
Yang (10.1016/j.csite.2021.101126_bib28) 2019
Hsu (10.1016/j.csite.2021.101126_bib11) 1985; 28
References_xml – year: 2007
  ident: bib18
  article-title: Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering
  contributor:
    fullname: Machado
– volume: 1
  start-page: 27
  year: 1949
  ident: bib9
  article-title: A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles
  publication-title: Flow, Turbul. Combust.
  contributor:
    fullname: Brinkman
– volume: 56
  start-page: 138
  year: 2018
  end-page: 143
  ident: bib31
  article-title: Prabhakar-like fractional viscoelasticity
  publication-title: Commun. Nonlinear Sci. Numer. Simulat
  contributor:
    fullname: Colombaro
– volume: 67a
  start-page: 572
  year: 2012
  end-page: 580
  ident: bib8
  article-title: Radiation and porosity effects on the magnetohydrodynamic flow past an oscillating vertical plate with uniform heat flux
  publication-title: Z. Naturforsch.
  contributor:
    fullname: Shafie
– volume: 56
  start-page: 314
  year: 2018
  end-page: 329
  ident: bib27
  article-title: The Prabhakar or three parameter Mittag–Leffler function: theory and application
  publication-title: Commun. Nonlinear Sci. Numer. Simulat.
  contributor:
    fullname: Garrappa
– volume: 5
  start-page: 4
  year: 1983
  end-page: 8
  ident: bib34
  article-title: The Gaver–Stehfest algorithm for approximate inversion of Laplace transforms
  publication-title: CircuitsSyst Mag
  contributor:
    fullname: Rhodine
– volume: 16
  start-page: 796
  year: 1985
  end-page: 806
  ident: bib13
  article-title: A Brinkman model for MHD viscous incompressible flow through a porous channel
  publication-title: Indian J. Pure Appl. Math.
  contributor:
    fullname: Babu
– volume: 46
  start-page: 147
  year: 2009
  end-page: 151
  ident: bib6
  article-title: Mixed convection boundary layer flow over a horizontal plate with thermal radiation
  publication-title: Heat Mass Tran.
  contributor:
    fullname: Ishak
– volume: 131
  start-page: 109489
  year: 2020
  ident: bib17
  article-title: Caputo-Fabrizio fractional derivatives modeling of transient MHD Brinkman nano liquid: applications in food technology, Chaos
  publication-title: Solitons \& Fractals
  contributor:
    fullname: Nisar
– volume: 36
  start-page: 357
  year: 1999
  end-page: 371
  ident: bib15
  article-title: Natural convection from a vertical plate in a porous medium using Brinkman's model
  publication-title: Transport Porous Media
  contributor:
    fullname: Sahar
– year: 2011
  ident: bib19
  article-title: Fractional Calculus: Models and Numerical Methods
  contributor:
    fullname: Trujillo
– volume: 56
  start-page: 138
  year: 2018
  end-page: 143
  ident: bib29
  article-title: Prabhakar-like fractional viscoelasticity
  publication-title: Commun. Nonlinear Sci. Numer. Simulat
  contributor:
    fullname: Colombaro
– volume: 99
  start-page: 499
  year: 1997
  ident: bib1
  article-title: Free convection effects on the Stokes problem for an infinite vertical plate
  publication-title: J. Heat Tran.
  contributor:
    fullname: Soundalgekar
– volume: 45
  start-page: 367
  year: 2010
  end-page: 373
  ident: bib7
  article-title: Thermal boundary layer flow over a stretching sheet in a micropolar fluid with radiation effect
  publication-title: Meccanica
  contributor:
    fullname: Ishak
– volume: 60
  start-page: 98
  year: 2019
  end-page: 106
  ident: bib23
  article-title: Analysis of free convection flow of viscous fluid with damped thermal and mass fluxes
  publication-title: Chin. J. Phys
  contributor:
    fullname: Riaz
– year: 2019
  ident: bib24
  article-title: Transient free convection flow of nanofluids between two vertical parallel plates with damped thermal flux
  publication-title: J. Mol. Liq
  contributor:
    fullname: Alarifi
– start-page: 1
  year: 2020
  end-page: 14
  ident: bib30
  article-title: Natural convection flows of Prabhakar-like fractional Maxwell fluids with generalized thermal transport
  publication-title: J.Therm. Anal. Calorim
  contributor:
    fullname: Vieru
– volume: 7
  start-page: 150
  year: 2019
  ident: bib20
  article-title: Fractional Derivatives: the perspective of system theory
  publication-title: Mathematics
  contributor:
    fullname: Machado
– volume: 9
  start-page: 1602
  year: 2018
  end-page: 1608
  ident: bib16
  article-title: Unsteady MHD flow of a Brinkman type fluid between two side walls perpendicular to an infinite plate
  publication-title: Results Phys
  contributor:
    fullname: Tlili
– volume: 1
  start-page: 81
  year: 1949
  ident: bib10
  article-title: On the permeability of media consisting of closely packed porous particles
  publication-title: Flow, Turbul. Combust.
  contributor:
    fullname: Brinkman
– volume: 333
  start-page: 25
  year: 1973
  end-page: 36
  ident: bib4
  article-title: Free convection effects on the oscillatory flow past an infinite, vertical, porous plate with constant suction
  publication-title: Proc. Roy. Soc. A.
  contributor:
    fullname: Soundalgekar
– volume: 28
  start-page: 683
  year: 1985
  end-page: 697
  ident: bib11
  article-title: The Brinkman model for natural convection about a semi-infinite vertical flat plate in a porous medium
  publication-title: Int. J. Heat Mass Tran.
  contributor:
    fullname: Cheng
– volume: 17
  start-page: 215
  year: 2007
  end-page: 252
  ident: bib14
  article-title: On a hierarchy of approximate models for flows of incompressible fluids through porous solids
  publication-title: Math. Model Methods Appl. Sci.
  contributor:
    fullname: Rajagopal
– volume: 5
  start-page: 150
  year: 2019
  end-page: 161
  ident: bib25
  article-title: Magnetohydrodynamic free convection flows with thermal memory over a moving vertical plate in porous medium
  publication-title: J. Appl. Comput. Mech.
  contributor:
    fullname: Rashidi
– start-page: 1
  year: 2020
  end-page: 14
  ident: bib33
  article-title: Natural convection flows of carbon nanotubes nanofluids with Prabhakar-like thermal transport
  publication-title: Math Method ApplSci
  contributor:
    fullname: Mirza
– volume: 30
  year: 2018
  ident: bib21
  article-title: Convective flows of generalized time-nonlocal nanofluids through a vertical rectangular channel
  publication-title: Phys. Fluids
  contributor:
    fullname: Shah
– volume: 10
  start-page: 313
  year: 1983
  end-page: 321
  ident: bib2
  article-title: MHD free convection flow past an accelerated vertical plate
  publication-title: Int. Commun. Heat Mass Tran.
  contributor:
    fullname: Singh
– volume: 56
  start-page: 630
  year: 2018
  end-page: 644
  ident: bib22
  article-title: Natural convection with damped thermal flux in a vertical circular cylinder
  publication-title: Chin. J. Phys
  contributor:
    fullname: Vieru
– volume: 166
  start-page: 269
  year: 1990
  end-page: 275
  ident: bib5
  article-title: Radiative and free-convection effects on the oscillatory flow past a vertical plate
  publication-title: Astrophys. Space Sci.
  contributor:
    fullname: Mansour
– year: 2019
  ident: bib28
  article-title: General Fractional Derivatives: Theory, Methods and Applications
  contributor:
    fullname: Yang
– volume: 6
  start-page: 73
  year: 2016
  end-page: 94
  ident: bib32
  article-title: Some properties of Prabhakar-type fractional calculus operators
  publication-title: Fractional Differential Calculus
  contributor:
    fullname: Tomovski
– volume: 98
  start-page: 245
  year: 1984
  end-page: 248
  ident: bib3
  article-title: Free-convection flow past an exponentially accelerated vertical plate
  publication-title: Astrophys. Space Sci.
  contributor:
    fullname: Kumar
– volume: 67a
  start-page: 377
  year: 2012
  end-page: 380
  ident: bib12
  article-title: A note on new exact solutions for some unsteady flows of Brinkman-type fluids over a plane wall
  publication-title: ZeitschriftfürNaturforschung
  contributor:
    fullname: Shafie
– volume: 5
  start-page: 592
  year: 2019
  end-page: 602
  ident: bib26
  article-title: Transient MHD convective flow of fractional nanofluid between vertical plates
  publication-title: J. Appl. Comput. Mech.
  contributor:
    fullname: Gorji
– volume: 166
  start-page: 269
  issue: 2
  year: 1990
  ident: 10.1016/j.csite.2021.101126_bib5
  article-title: Radiative and free-convection effects on the oscillatory flow past a vertical plate
  publication-title: Astrophys. Space Sci.
  doi: 10.1007/BF01094898
  contributor:
    fullname: Mansour
– volume: 46
  start-page: 147
  issue: 2
  year: 2009
  ident: 10.1016/j.csite.2021.101126_bib6
  article-title: Mixed convection boundary layer flow over a horizontal plate with thermal radiation
  publication-title: Heat Mass Tran.
  doi: 10.1007/s00231-009-0552-3
  contributor:
    fullname: Ishak
– year: 2011
  ident: 10.1016/j.csite.2021.101126_bib19
  contributor:
    fullname: Baleanu
– year: 2007
  ident: 10.1016/j.csite.2021.101126_bib18
  contributor:
    fullname: Sabatier
– volume: 99
  start-page: 499
  year: 1997
  ident: 10.1016/j.csite.2021.101126_bib1
  article-title: Free convection effects on the Stokes problem for an infinite vertical plate
  publication-title: J. Heat Tran.
  doi: 10.1115/1.3450729
  contributor:
    fullname: Soundalgekar
– volume: 60
  start-page: 98
  year: 2019
  ident: 10.1016/j.csite.2021.101126_bib23
  article-title: Analysis of free convection flow of viscous fluid with damped thermal and mass fluxes
  publication-title: Chin. J. Phys
  doi: 10.1016/j.cjph.2019.05.006
  contributor:
    fullname: Awan
– volume: 56
  start-page: 314
  year: 2018
  ident: 10.1016/j.csite.2021.101126_bib27
  article-title: The Prabhakar or three parameter Mittag–Leffler function: theory and application
  publication-title: Commun. Nonlinear Sci. Numer. Simulat.
  doi: 10.1016/j.cnsns.2017.08.018
  contributor:
    fullname: Garra
– start-page: 1
  year: 2020
  ident: 10.1016/j.csite.2021.101126_bib30
  article-title: Natural convection flows of Prabhakar-like fractional Maxwell fluids with generalized thermal transport
  publication-title: J.Therm. Anal. Calorim
  contributor:
    fullname: Shah
– volume: 10
  start-page: 313
  issue: 4
  year: 1983
  ident: 10.1016/j.csite.2021.101126_bib2
  article-title: MHD free convection flow past an accelerated vertical plate
  publication-title: Int. Commun. Heat Mass Tran.
  doi: 10.1016/0735-1933(83)90016-7
  contributor:
    fullname: Raptis
– volume: 7
  start-page: 150
  year: 2019
  ident: 10.1016/j.csite.2021.101126_bib20
  article-title: Fractional Derivatives: the perspective of system theory
  publication-title: Mathematics
  doi: 10.3390/math7020150
  contributor:
    fullname: Ortigueira
– volume: 5
  start-page: 4
  issue: 1
  year: 1983
  ident: 10.1016/j.csite.2021.101126_bib34
  article-title: The Gaver–Stehfest algorithm for approximate inversion of Laplace transforms
  publication-title: CircuitsSyst Mag
  contributor:
    fullname: Jacquot
– volume: 1
  start-page: 81
  issue: 1
  year: 1949
  ident: 10.1016/j.csite.2021.101126_bib10
  article-title: On the permeability of media consisting of closely packed porous particles
  publication-title: Flow, Turbul. Combust.
  doi: 10.1007/BF02120318
  contributor:
    fullname: Brinkman
– volume: 36
  start-page: 357
  issue: 3
  year: 1999
  ident: 10.1016/j.csite.2021.101126_bib15
  article-title: Natural convection from a vertical plate in a porous medium using Brinkman's model
  publication-title: Transport Porous Media
  doi: 10.1023/A:1006593330865
  contributor:
    fullname: Gorla
– volume: 28
  start-page: 683
  issue: 3
  year: 1985
  ident: 10.1016/j.csite.2021.101126_bib11
  article-title: The Brinkman model for natural convection about a semi-infinite vertical flat plate in a porous medium
  publication-title: Int. J. Heat Mass Tran.
  doi: 10.1016/0017-9310(85)90190-5
  contributor:
    fullname: Hsu
– volume: 67a
  start-page: 377
  year: 2012
  ident: 10.1016/j.csite.2021.101126_bib12
  article-title: A note on new exact solutions for some unsteady flows of Brinkman-type fluids over a plane wall
  publication-title: ZeitschriftfürNaturforschung
  contributor:
    fullname: Ali
– volume: 6
  start-page: 73
  issue: 1
  year: 2016
  ident: 10.1016/j.csite.2021.101126_bib32
  article-title: Some properties of Prabhakar-type fractional calculus operators
  publication-title: Fractional Differential Calculus
  doi: 10.7153/fdc-06-05
  contributor:
    fullname: Polito
– volume: 56
  start-page: 138
  year: 2018
  ident: 10.1016/j.csite.2021.101126_bib31
  article-title: Prabhakar-like fractional viscoelasticity
  publication-title: Commun. Nonlinear Sci. Numer. Simulat
  doi: 10.1016/j.cnsns.2017.08.002
  contributor:
    fullname: Giusti
– volume: 56
  start-page: 138
  year: 2018
  ident: 10.1016/j.csite.2021.101126_bib29
  article-title: Prabhakar-like fractional viscoelasticity
  publication-title: Commun. Nonlinear Sci. Numer. Simulat
  doi: 10.1016/j.cnsns.2017.08.002
  contributor:
    fullname: Giusti
– volume: 98
  start-page: 245
  issue: 2
  year: 1984
  ident: 10.1016/j.csite.2021.101126_bib3
  article-title: Free-convection flow past an exponentially accelerated vertical plate
  publication-title: Astrophys. Space Sci.
  doi: 10.1007/BF00651403
  contributor:
    fullname: Singh
– volume: 5
  start-page: 150
  issue: 1
  year: 2019
  ident: 10.1016/j.csite.2021.101126_bib25
  article-title: Magnetohydrodynamic free convection flows with thermal memory over a moving vertical plate in porous medium
  publication-title: J. Appl. Comput. Mech.
  contributor:
    fullname: Shah
– volume: 131
  start-page: 109489
  year: 2020
  ident: 10.1016/j.csite.2021.101126_bib17
  article-title: Caputo-Fabrizio fractional derivatives modeling of transient MHD Brinkman nano liquid: applications in food technology, Chaos
  publication-title: Solitons \& Fractals
  doi: 10.1016/j.chaos.2019.109489
  contributor:
    fullname: Ali
– volume: 56
  start-page: 630
  year: 2018
  ident: 10.1016/j.csite.2021.101126_bib22
  article-title: Natural convection with damped thermal flux in a vertical circular cylinder
  publication-title: Chin. J. Phys
  doi: 10.1016/j.cjph.2018.02.007
  contributor:
    fullname: Ahmed
– start-page: 1
  year: 2020
  ident: 10.1016/j.csite.2021.101126_bib33
  article-title: Natural convection flows of carbon nanotubes nanofluids with Prabhakar-like thermal transport
  publication-title: Math Method ApplSci
  contributor:
    fullname: Elnaqeeb
– volume: 45
  start-page: 367
  year: 2010
  ident: 10.1016/j.csite.2021.101126_bib7
  article-title: Thermal boundary layer flow over a stretching sheet in a micropolar fluid with radiation effect
  publication-title: Meccanica
  doi: 10.1007/s11012-009-9257-4
  contributor:
    fullname: Ishak
– year: 2019
  ident: 10.1016/j.csite.2021.101126_bib24
  article-title: Transient free convection flow of nanofluids between two vertical parallel plates with damped thermal flux
  publication-title: J. Mol. Liq
  doi: 10.1016/j.molliq.2019.110964
  contributor:
    fullname: Hajizadeh
– volume: 17
  start-page: 215
  issue: 2
  year: 2007
  ident: 10.1016/j.csite.2021.101126_bib14
  article-title: On a hierarchy of approximate models for flows of incompressible fluids through porous solids
  publication-title: Math. Model Methods Appl. Sci.
  doi: 10.1142/S0218202507001899
  contributor:
    fullname: Rajagopal
– volume: 16
  start-page: 796
  issue: 7
  year: 1985
  ident: 10.1016/j.csite.2021.101126_bib13
  article-title: A Brinkman model for MHD viscous incompressible flow through a porous channel
  publication-title: Indian J. Pure Appl. Math.
  contributor:
    fullname: Varma
– volume: 30
  issue: 5
  year: 2018
  ident: 10.1016/j.csite.2021.101126_bib21
  article-title: Convective flows of generalized time-nonlocal nanofluids through a vertical rectangular channel
  publication-title: Phys. Fluids
  doi: 10.1063/1.5032165
  contributor:
    fullname: Ahmed
– volume: 9
  start-page: 1602
  year: 2018
  ident: 10.1016/j.csite.2021.101126_bib16
  article-title: Unsteady MHD flow of a Brinkman type fluid between two side walls perpendicular to an infinite plate
  publication-title: Results Phys
  doi: 10.1016/j.rinp.2018.04.034
  contributor:
    fullname: Khan
– volume: 67a
  start-page: 572
  year: 2012
  ident: 10.1016/j.csite.2021.101126_bib8
  article-title: Radiation and porosity effects on the magnetohydrodynamic flow past an oscillating vertical plate with uniform heat flux
  publication-title: Z. Naturforsch.
  contributor:
    fullname: Haq
– volume: 5
  start-page: 592
  issue: 4
  year: 2019
  ident: 10.1016/j.csite.2021.101126_bib26
  article-title: Transient MHD convective flow of fractional nanofluid between vertical plates
  publication-title: J. Appl. Comput. Mech.
  contributor:
    fullname: Ahmed
– volume: 1
  start-page: 27
  issue: 1
  year: 1949
  ident: 10.1016/j.csite.2021.101126_bib9
  article-title: A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles
  publication-title: Flow, Turbul. Combust.
  doi: 10.1007/BF02120313
  contributor:
    fullname: Brinkman
– volume: 333
  start-page: 25
  issue: 1592
  year: 1973
  ident: 10.1016/j.csite.2021.101126_bib4
  article-title: Free convection effects on the oscillatory flow past an infinite, vertical, porous plate with constant suction
  publication-title: Proc. Roy. Soc. A.
  contributor:
    fullname: Soundalgekar
– year: 2019
  ident: 10.1016/j.csite.2021.101126_bib28
  contributor:
    fullname: Yang
SSID ssj0001738144
Score 2.2943695
Snippet Natural convection flow of unsteady Prabhakar-like fractional Brinkman type fluids over an infinite vertical heated wall with uniform heat flux is studied....
SourceID doaj
crossref
elsevier
SourceType Open Website
Aggregation Database
Publisher
StartPage 101126
SubjectTerms Brinkman type fluids
Exact solutions
Laplace transform
Mittag-Leffler functions
Prabhakar-like fractional derivative
SummonAdditionalLinks – databaseName: Elsevier Free Content
  dbid: IXB
  link: http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwnV3JTsMwELUQJzggVlE2-cCRqE7ibNzaqlWFBEICpN4irxAKLQqpQHwI38uMk0C5cOAYb7E8I8_zeOaZkFOdGEAdPMWMZe1xbmIvNVJ41s-kLzWLmEbXwOVVPL7jF5NoskIGbS4MhlU2e3-9p7vduinpNqvZfSmK7k0Q-GB9kkngu-sypN1Gbk9M4pv0f_wsCdgk96YrtvewQ0s-5MK8lLukDcDUYYmPJAtLBsrx-C_ZqSXbM9okGw1opL16Xltkxcy2yfoSleAO-Ry-C1XRb02iAEapY-2Eji603CUwUPs0f3ulc0vva77p4sNo2ocxps9iRtEhC00WhX49pz16XQr5IKai9J6KKVSUdRYEjOge0KHoxP01EKLJZ6iuWsr0XXI3Gt4Oxl7z5oKnuM8rLw6TROkokHBKNWkmWaQEiMtkiQ2tSsIstSq2NglkqCzTLDSCc6YExl1LCfBlj6zO5jOzT2ichUangPCkiDi3cLbjPJSpZVIyzmXaIWftQucvNbVG3sacPeZOLjnKJa_l0iF9FMZ3U-TFdgXz8j5vFCNnxkTKD2CyccilYcKCiojM8gxgWyb8DolbUea_1AyGKv76-8F_Ox6SNfyqYwaPyGpVLswx4JhKnjhF_QLp5PPf
  priority: 102
  providerName: Elsevier
Title Exact solutions for natural convection flows of generalized Brinkman type fluids: A Prabhakar-like fractional model with generalized thermal transport
URI https://dx.doi.org/10.1016/j.csite.2021.101126
https://doaj.org/article/0ee5c127f3634be0af415a9f491599a1
Volume 26
hasFullText 1
inHoldings 1
isFullTextHit
isPrint
link http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwrV07T8MwELYQEwyIpygveWAkwk4uLzaKWgESiAGkbpGfUAotaoNA_BB-L2e7QWGBhSWD7Zwj30n33eX8HSGHOjeIOqBwN5Z1BGCyqDBSRJaXkkvNUqZdauDqOju_g8tBOmi1-nI1YYEeOBzcMTMmVTzObZIlIA0TFl2OKC2U6IhLEQIfnraCKZ9dydET-U6uccwh4mk-aCiHfHGX8r9mMfDnboQ7aoWWW_Ls_S3v1PI4_VWyMoeK9DR84hpZMON1stwiENwgn713oWr6bT8UISj1XJ34oi8o99cWqH2avM3oxNL7wDI9_DCadlHG6FmMqUvD4pLXoZ6d0FN6MxXyQYzENHoajnBiGu4-oETfNoe61O0PQQ5DPuN03RClb5K7fu_27Dyad1qIFHCooyzJc6XTWGJsaopSslQJVJIp8ditypOysCqzNo9loizTLDECgCnhqq2lRNCyRRbHk7HZJjQrE6MLxHVSpAAWIzqARBaWSckAZNEhR81BVy-BUKNqKs0eK6-XyumlCnrpkK5TxvdSx4btB9BGqrmNVH_ZSIdkjSqrObAIgAFFDX_bfec_dt8lS05kqBrcI4v19NXsI5Kp5YE3WnxeDLpfJuDz1w
link.rule.ids 315,786,790,870,2115,3525,27955,27956,45907
linkProvider Directory of Open Access Journals
linkToHtml http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwnV1LT9wwEB5Remh7qEppxfaFDz0SrZNMXr2xCLRtASEB0t4sPyEs7KIQ1Ko_pL-3Yyehy6WHXm3PJPKMPJ_tmc8An01hCXVg6SuWTYRo86i0SkYurlSsDM-48UcDR8f59By_zbLZGuwNtTA-rbJf-7s1PazWfcu4n83xbV2PT5MkpuhTzJI4XJflT-ApZrTX81V8s8nfg5aCglJ41NULRF5iYB8KeV463NImFOt8S-xZFlYiVCDyXwlUK8Hn4BW87FEj2-1-bAPW7OI1vFjhEtyE3_s_pW7ZgysxQqMs0HaSYMgtDxUMzF0vf9yxpWMXHeF0_csaNiEd8xu5YP5Elobc1-buC9tlJ41Ul3Ium-i6nlNH05VBkMbwgg7zp7iPFHk4eUPd7cCZ_gbOD_bP9qZR_-hCpDHGNsrTotAmSxRtU21ZKZ5pSfayVeFSp4u0Kp3OnSsSlWrHDU-tRORa-sRrpQi_vIX1xXJht4DlVWpNSRBPyQzR0eYOMVWl40pxRFWOYGeYaHHbcWuIIensSgS7CG8X0dllBBNvjIehnhg7NCybC9F7huDWZjpO6GfzFJXl0pGPyMphRbitkvEI8sGU4pGfkar6X19_97-C2_BsenZ0KA6_Hn9_D899T5dA-AHW2-befiRQ06pPwWn_AD8r9wQ
openUrl ctx_ver=Z39.88-2004&ctx_enc=info%3Aofi%2Fenc%3AUTF-8&rfr_id=info%3Asid%2Fsummon.serialssolutions.com&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Exact+solutions+for+natural+convection+flows+of+generalized+Brinkman+type+fluids%3A+A+Prabhakar-like+fractional+model+with+generalized+thermal+transport&rft.jtitle=Case+studies+in+thermal+engineering&rft.au=Yu-Liang+Sun&rft.au=Nehad+Ali+Shah&rft.au=Zar+Ali+Khan&rft.au=Y.M.+Mahrous&rft.date=2021-08-01&rft.pub=Elsevier&rft.issn=2214-157X&rft.eissn=2214-157X&rft.volume=26&rft.spage=101126&rft_id=info:doi/10.1016%2Fj.csite.2021.101126&rft.externalDBID=DOA&rft.externalDocID=oai_doaj_org_article_0ee5c127f3634be0af415a9f491599a1
thumbnail_l http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/lc.gif&issn=2214-157X&client=summon
thumbnail_m http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/mc.gif&issn=2214-157X&client=summon
thumbnail_s http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/sc.gif&issn=2214-157X&client=summon