The Cattaneo-Christov heat flux model with nonlinear radiation, exponential heat generation, Joule heating, and homo-heterogenic reactions in association with the melting heat peripheral condition has been envisioned as a mathematical representation of the unsteady flow of Prandtl nanofluid carried over a curvy sheet that allows stretching and shrinking geometry enabled by a magnetic dipole. Solution graphs were generated using the Runge-Kutta-Fehlberg 4–5th order tool. Other parameters are simultaneously set to their default values while showing the solution graphs for all flow defining profiles with the appropriate parameters. Each produced graph has been the subject of a thorough discussion. This study investigates the effects of various parameters on velocity, thermal, and concentration distributions in stretching and shrinking sheets. The velocity curves for the magnetic and stretching/shrinking parameters exhibit rising or decreasing trends during sheet stretching, reversing when the sheet shrinks. Significant differences in thermal distribution are observed between stretching and shrinking sheets for each parameter. The velocity distribution decreases with increasing unsteadiness parameter values due to the time factor, while thermal distribution increases and concentration decreases with rising unsteadiness. The melting heat parameter enhances temperature distribution in both stretching and contracting cases. Additionally, an increase in the homogeneous reaction parameter decreases concentration, while the heterogeneous reaction parameter reduces the mass transfer profile in both cases. The thermal relaxation parameter negatively impacts thermal panels when the sheet is stretched and positively when contracting, with both the Nusselt and Eckert numbers increasing regardless of the sheet’s motion. Streamlines and isotherms are provided to further illustrate the flow and heat patterns.
RoşcaNCPopI.Unsteady boundary layer flow over a permeable curved stretching/shrinking surface. Eur J Mecj B/Fluids2015; 51: 61–67.
2.
NaveedMAbbasZSajidM.Hydromagnetic flow over an unsteady curved stretching surface. Eng Sci Technol Int J2016; 19(2): 841–845.
3.
AmjadMZehraINadeemS, et al. Influence of Lorentz force and induced magnetic field effects on Casson micropolar nanofluid flow over a permeable curved stretching/shrinking surface under the stagnation region. Surf Interf2020; 21: 100766.
4.
ZehraIAbbasNAmjadM, et al. Casson nanoliquid flow with Cattaneo-Christov flux analysis over a curved stretching/shrinking channel. Case Stud Thermal Eng2021; 27: 101146.
5.
AbbasNNadeemSKhanMN.Numerical analysis of unsteady magnetized micropolar fluid flow over a curved surface. J Thermal Anal Calorimet2022; 147(11): 6449–6459.
6.
SaranyaSRagupathiPAl-MdallalQ.Analysis of bio-convective heat transfer over an unsteady curved stretching sheet using the shifted Legendre collocation method. Case Stud Thermal Eng2022; 39: 102433.
7.
AlmeidaFBasavarajappaNKumarP, et al. Irreversibility analysis of nonlinear mixed convective time-based flow analysis of casson-williamson nanofluid accelerated by curved stretching surface. Comput Thermal Sci Int J2024; 16(5): 15–41.
8.
VidhyaKGNagarajaBAlmeidaF, et al. Numerical exploration of micropolar fluid stratified flow over curved stretching sheet under buoyancy effect. ZAMM J Appl Math Mech Zeitschrift für Angewandte Mathematik und Mechanik2024; 104(12): e202300965.
9.
KumarPAlmeidaFAl-MdallalQ.Entropy optimization of inverse Darcy-Forchheimer model of Jeffrey fluid flow over a curved stretching surface using ANOVA-Taguchi technique. Part Diff Equ Appl Math2025; 14: 101183.
10.
VidhyaKGAlmeidaFKumarP, et al. Intervention of the Koo-Kleinstreuer and Li model in nanofluid flow over magnetic dipole centered curved sheet and optimizing entropy using response surface methodology. J Appl Comput Mech2025; 11(2): 382–398.
11.
KumarPVidhyaKGAlmeidaF, et al. Optimization using response surface methodology for Eyring-powell fluid flow with Cattaneo-Christov heat flux and cross diffusion effects. Int J Thermofluids2025; 25: 100981.
12.
NadeemSIjazSAkbarNS.Nanoparticle analysis for blood flow of Prandtl fluid model with stenosis. Int Nano Lett2013; 3: 1–3.
13.
KhanIMalikMYHussainA, et al. Effect of homogenous-heterogeneous reactions on MHD Prandtl fluid flow over a stretching sheet. Res Phys2017; 7: 4226–4231.
14.
HamidMZubairTUsmanM, et al. Natural convection effects on heat and mass transfer of slip flow of time-dependent Prandtl fluid. J Comput Des Eng2019; 6(4): 584–592.
15.
SalmiAMadkhaliHAAliB, et al. Numerical study of heat and mass transfer enhancement in Prandtl fluid MHD flow using Cattaneo-Christov heat flux theory. Case Stud Thermal Eng2022; 33: 101949.
16.
PatilABPatilNSPatilVS, et al. Unsteady thermally radiative Prandtl fluid flow past a magnetized inclined porous stretching device with double-diffusion, viscous dissipation, and Joule heating. Proceed Instit Mech Eng Part E J Proc Mech Eng2023; 237(3): 762–770.
17.
UddinMJKhanWAIsmailAM, et al. Computational study of three-dimensional stagnation point nanofluid bioconvection flow on a moving surface with anisotropic slip and thermal jump effect. J Heat Transfer2016; 138(10): 104502.
18.
RazaqAHayatTKhanSA, et al. ATSS model based upon applications of Cattaneo-Christov thermal analysis for entropy optimized ternary nanomaterial flow with homogeneous-heterogeneous chemical reactions. Alexandria Eng J2023; 79: 390–401.
19.
DasMKUddinMJBégTA, et al. Blowing and multiple slip effects on bio-nano-convection flow in porous media within the gap of a rotating cone-disc system. Chin J Physics2025; 95: 685–713.
20.
BégOAKumarDUddinMJ, et al. Simulation of magneto-nano-bioconvective coating flow with blowing and multiple slip effects. Proceed Instit Mech Eng Part N J Nanomat Nanoeng Nanosystems. Epub ahead of print 25 July 2024. DOI: 10.1177/23977914241259815.
21.
KhanWAUddinMJ.Nano-bioconvective anisotropic slip flow in anisotropic porous medium with Coriolis force effects. Heat Transfer2024; 53(2): 558–583.
22.
KumarPGuruprasadMNAlmeidaF.Thermal analysis of magnetized ZnO-blood nanofluid anticipating couple stresses in vertical microchannel using differential transform method. J Mol Liq2025; 424: 127051.
23.
KumarPAlmeidaFAl-MdallalQ.Advancement of nanoparticles in blood flow with non-linear radiation and optimisation of irreversibility within the microchannel using analysis of variance and Taguchi approach. Int J Thermofluids2024; 24: 100975.
24.
FelicitaAVenkateshPGireeshaBJ, et al. Entropy scrutinization of magnetized-hyperbolic tangent nanofluid in the microchannel stuffed by porous media. ZAMM J Appl Math Mech/Zeitschrift für Angewandte Mathematik und Mechanik2024; 104(8): e202300444.
25.
OdenbachS.Magnetic fluids-suspensions of magnetic dipoles and their magneticcontrol. J Phys Cond Matt2003; 15(15): S1497.
26.
YasmeenTHayatTKhanMI, et al. Ferrofluid flow by a stretched surface in the presence of magnetic dipole and homogeneous-heterogeneous reactions. J Mol Liq2016; 223: 1000–1005.
27.
HayatTAhmadSKhanMI, et al. Exploring magnetic dipole contribution on radiative flow of ferromagnetic Williamson fluid. Res Phys2018; 8: 545–551.
28.
KumarRNJyothiAMAlhumadeH, et al. Impact of magnetic dipole on thermophoretic particle deposition in the flow of Maxwell fluid over a stretching sheet. J Mol Liq2021; 334: 116494.
29.
MajeedAZeeshanAEllahiR.Unsteady ferromagnetic liquid flow and heat transfer analysis over a stretching sheet with the effect of dipole and prescribed heat flux. J Mol Liq2016; 223: 528–533.
30.
NandiSKumbhakarB.Viscous dissipation and chemical reaction effects on tangent hyperbolic nanofluid flow past a stretching wedge with convective heating and Navier’s slip conditions. Iran J Sci Technol Transact Mech Eng2022; 46(2): 379–397.
31.
MishraSSwainKDalaiR.Joule heating and viscous dissipation effects on heat transfer of hybrid nanofluids with thermal slip. Iran J Sci Technol Transact Mech Eng2024; 48(2): 531–539.
32.
HayatTQayyumSAlsaediA, et al. 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 Liq2016; 224: 811–817.
33.
IrfanMFarooqMAMushtaqA, et al. Unsteady MHD bionanofluid flow in a porous medium with thermal radiation near a stretching/shrinking sheet. Math Prob Eng2020; 2020(1): 8822999.
34.
IbrahimWNegeraM.Viscous dissipation effect on Williamson nanofluid over stretching/shrinking wedge with thermal radiation and chemical reaction. J Phys Commun2020; 4(4): 045015.
35.
DasMKumbhakarBSinghJ.Analysis of unsteady MHD Williamson nanouid flow past a stretching sheet with nonlinear mixed convection, thermal radiation and velocity slips. J Comput Anal Appl2022; 30(1): 176–195.
36.
HayatTRazzaqAKhanSA, et al. TASA formulation for nonlinear radiative flow of Walter-B nanoliquid invoking microorganism and entropy generation. Res Eng2024; 24: 103346.
37.
Shankar GoudBDharmendar ReddyYMishraS. Joule heating and thermal radiation impact on MHD boundary layer Nanofluid flow along an exponentially stretching surface with thermal stratified medium. Proceed Instit Mech Eng Part N J Nanomat Nanoeng Nanosys2023; 237(3–4): 107–19. DOI: 10.1177/239779 14221100961.
38.
BeheraSDashAKMishraSR.Impact of partial slip on the radiative conducting nanofluid flow through an expanding sheet for the interaction of heat source/sink. Proceed Instit Mech Eng Part N J Nanomat Nanoeng Nanosys2023; 237(1–2): 31–43.
39.
MehmoodRAliIIjazS, et al. Radiative slip transport of magnetized gyrotactic micro-organisms submerged with nano fluid along a vertical stretching surface with suction/injection effects. Proceed Instit Mech Eng Part N J Nanomat Nanoeng Nanosys2023: 23977914231176866. DOI: 10.1177/23977914231176866.
40.
KumarPAlmeidaFAl-MdallalQ.Artificial neural network model using Levenberg Marquardt algorithm to analyse transient flow and thermal characteristics of micropolar nanofluid in a microchannel. Partial Diff Eqn Appl Math2025; 13: 101061.
41.
KhanSARazaqAAlsaediA, et al. Modified thermal and solutal fluxes through convective flow of Reiner-Rivlin material. Energy2023; 283: 128516.
42.
KumarPGuruprasadMNAlmeidaF, et al. Optimising the thermal characteristics of Williamson fluid flow through a microchannel influenced by the Hall effect using response surface methodology. Case Stud Thermal Eng2025; 70: 106139.
43.
KumarPAlmeidaFAl-MdallalQ.Artificial neural network algorithm for time dependent radiative Casson fluid flow with couple stresses through a microchannel. Alexand Eng J2025; 125: 167–184.
44.
HayatTAlsaediARazaqA, et al. HARK formulation for entropy optimized convective flow beyond constant thermophysical properties. Case Stud Thermal Eng2024; 54: 103983.
45.
ImtiazMHayatTAlsaediA, et al. Homogeneous-heterogeneous reactions in MHD flow due to an unsteady curved stretching surface. J Mol Liq2016; 221: 245–253.
46.
MahdyA.Impacts of homogeneous-heterogeneous chemical reactions and inclined magnetic field on unsteady nanofluids flow. AIP Adv2018; 8(11): 115109.
47.
MahatoRDasMSenSS, et al. Entropy generation on unsteady stagnation-point Casson nanofluid flow past a stretching sheet in a porous medium under the influence of an inclined magnetic field with homogeneous and heterogeneous reactions. Heat Transfer2022; 51(6): 5723–5747.
48.
NagarajaBAjaykumarARFelicitaR, et al. Non-Darcy-Forchheimer flow of Casson-Williamson nanofluid on melting curved stretching sheet influenced by magnetic dipole. ZAMM J Appl Math Mech/Zeitschrift für Angewandte Mathematik und Mechanik2023; 103(10): e202300134. DOI: 10.1002/zamm.202300134.
49.
JahangirHEsfahaniMR.Bond behavior investigation between steel reinforced grout composites and masonry substrate. Iran J Sci Technol Transact Civil Eng2022; 46(5): 3519–3535.
50.
VasuBRayAKGorlaRS.Free convective heat transfer in Jeffrey fluid with suspended nanoparticles and Cattaneo–Christov heat flux. Proceed Instit Mech Eng Part N J Nanomat Nanoeng Nanosys2020; 234(3–4): 99–114.
51.
UmavathiJC.Micropolar nanofluid overlying a porous layer: Thermosolutal convection. Proceed Instit Mech Eng Part N J Nanomat Nanoeng Nanosys2024; 238(1–2): 47–64.
52.
ChamkhaAJRashadAMAl-MeshaieiE.Melting effect on unsteady hydromagnetic flow of a nanofluid past a stretching sheet. Int J Chem Reactor Eng. Epub ahead of print 1 December 2011. DOI: 10.2202/1542-6580.2613.
53.
BaagSMishraSMathurP.Characteristics of variable plate conditions on the time-dependent flow of polar fluid over an expanding/contracting surface. Proceed Instit Mech Eng Part N J Nanomat Nanoeng Nanosys2023; 237(3-4): 97–106.
54.
FelicitaAVenkateshPGireeshaBJ, et al. Slip and convective flow of Williamson nanofluid influenced by Brownian motion and thermophoresis mechanism in a horizontal microchannel. Proceed Instit Mech Eng Part N J Nanomat Nanoeng Nanosys2023: 23977914231177340.
55.
AnimasaunILAdebileEAFagbadeAI.Casson fluid flow with variable thermo-physical property along exponentially stretching sheet with suction and exponentially decaying internal heat generation using the homotopy analysis method. J Nigerian Math Soc2016; 35(1): 1–7.
56.
ZiaQZUllahIWaqasMA, et al. Cross diffusion and exponential space dependent heat source impacts in radiated three-dimensional (3D) flow of Casson fluid by heated surface. Res Phys2018; 8: 1275–1282.
57.
MishraSMahantheshBMackolilJ, et al. Nonlinear radiation and cross-diffusion effects on the micropolar nanoliquid flow past a stretching sheet with an exponential heat source. Heat Transfer2021; 50(4): 3530–3546.
58.
NagarajaBGireeshaBJSoumyaDO, et al. Characterization of MHD convective flow of Jeffrey nanofluid driven by a curved stretching surface by employing Darcy–Forchheimer law of porosity. Waves Random Complex Med2025; 35(1): 73–92.
59.
NagarajaBGireeshaBJ.Exponential space-dependent heat generation impact on MHD convective flow of Casson fluid over a curved stretching sheet with chemical reaction. J Thermal Anal Calori2021; 143: 4071–4079.
60.
FourierJBJDarbouxG. Théorie analytique de la chaleur. Paris: Didot. 1822: 504.
61.
CattaneoC.Sulla conduzione del calore. Atti Sem Mat Fis Univ Modena1948; 3: 83–101.
62.
ChristovCI.On frame indifferent formulation of the Maxwell–Cattaneo model of finite-speed heat conduction. Mech Res Commun2009; 36(4): 481–486.
63.
HayatTFarooqMAlsaediA, et al. Impact of Cattaneo-Christov heat flux in the flow over a stretching sheet with variable thickness. Aip Adv2015; 5(8): 087159.
64.
GireeshaBJShankaralingappaBMPrasannakumarBC, et al. MHD flow and melting heat transfer of dusty Casson fluid over a stretching sheet with Cattaneo–Christov heat flux model. Int J Amb Energy2022; 43(1): 2931–2939.
65.
KhanSAHayatTAlsaediA.KHA model comprising MoS 4 and CoFe 2 O 3 in engine oil invoking non-similar Darcy–Forchheimer flow with entropy and Cattaneo–Christov heat flux. Nanoscale Adv2023; 5(22): 6135–6147.
66.
KhanSARazaqAHayatT.SAT formulation for entropy generated hybrid nanomaterial flow: modified Cattaneo-Christov analysis. Res Eng2024; 24: 102895.
67.
ZhangXHAbidiAAhmedAE, et al. MHD stagnation point flow of nanofluid over a curved stretching/shrinking surface subject to the influence of Joule heating and convective condition. Case Stud Thermal Eng2021; 26: 101184.
