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Cattaneo–Christov heat-mass transfer rheology in third-grade nanoliquid flow confined by stretchable surface subjected to mixed convection

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Abstract

This research addresses buoyancy-driven stretching flow of non-Newtonian (third-grade) rheological liquid confined by a vertically stretchable surface. The nanoliquid considered for modeling encompasses Brownian movement and thermophoresis aspects. Heat-mass transportation characteristics are scrutinized under modern approaches (i.e., Cattaneo–Christov heat-mass fluxes consideration). Such consideration overwhelms the paradoxes of heat conduction and mass diffusion via heat-mass flux relation times. Steady-state and chemically reactive magnetohydrodynamic boundary-layer flow satisfying incompressibility condition is modeled. The governing nonlinear boundary-layer expressions are coupled and highly nonlinear due to mixed convection consideration. The homotopy scheme yielding convergent solutions is implemented. Numerical data along with plots is presented to ensure convergence. The achieved outcomes are exhibited graphically and elaborated thoroughly.

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Abbreviations

\(\nu =\frac{\mu }{\rho _{f}}\) :

Kinematic viscosity

\(\sigma \) :

Electrical conductivity

x, y :

Cartesian coordinates

\(\rho \) :

Fluid density

u,  v :

Velocity components

\(\mu \) :

Dynamic viscosity

\(u_\textrm{w}\) :

Stretching velocity

\(\alpha _{1}^{*},\) \(\alpha _{2}^{*},\) \(\beta _{3}^{*}\) :

material parameters

g :

Gravitational acceleration

\(\alpha \) \(\left( =\frac{k}{\left( \rho c\right) _{f}}\right) \) :

Thermal diffusivity

k :

Thermal conductivity

c :

Dimensional constants

\(B_{0}\) :

Magnetic field potency

\(\Lambda ^{*}\) :

Thermal expansion coefficient

\(T,T_{\infty }\) :

Temperature, ambient temperature

\(\Lambda ^{**}\) :

Solutal expansion coefficient

\(C,C_{\infty }\) :

Concentration, ambient concentration

\(D_\textrm{B}\) :

Brownian diffusion

\(\tau =\frac{\left( \rho c\right) _{p}}{\left( \rho c\right) _{f}}\) :

Ratio of heat capacity

\(D_\textrm{T}\) :

Thermophoresis

\(\left( \rho c\right) _\textrm{p}\) :

Effective heat capacity of nanoparticles

\(\lambda _\textrm{t}\) :

Heat-flux relaxation time

\(\left( \rho c\right) _{f}\) :

Liquid heat capacity

\(\lambda _{c}\) :

Mass-flux relaxation time

\(K_{1}\) :

Reaction rate

\(M=\frac{\sigma B_{0}^{2}}{c\rho _{f}}\) :

Hartman number

\(\lambda =\frac{Gr_{x}}{\text {Re}_{x}^{2}}\) :

Thermal buoyancy variable

\(\text {Re}_{x}=\frac{xu_\textrm{w}}{\nu }\) :

Reynolds number

\(N=\frac{Gr_{x}^{*}}{Gr_{x}}\) :

Buoyancy ratio variable

\(Gr_{x}=\frac{g\Lambda ^{*}({T}_\textrm{w}-T_{\infty })x^{3}}{\nu ^{2}}\) :

Thermal-Grashof number

\(N_\textrm{b}=\frac{\tau D_\textrm{B}\left( C_\textrm{w}-C_{\infty }\right) }{\nu }\) :

Brownian motion variable

\(Gr_{x}^{*}=\frac{g\Lambda ^{**}\left( C_\textrm{w}-C_{\infty }\right) x^{3}}{\nu ^{2}}\) :

Solutal-Grashof number

\(\beta _{1}=\frac{\alpha _{1}^{*}c}{\mu },\) \(\beta _{2}=\frac{\alpha _{2}^{*}c}{\mu },\) \(\beta =\frac{\beta _{3}^{*}c^{2}}{\mu }\) :

fluid parameters

\(N_\textrm{t}=\frac{\tau D_\textrm{T}\left( T_\textrm{w}-T_{\infty }\right) }{T_{\infty }\nu }\) :

Thermophoresis variable

\(\gamma _{1}=\lambda _\textrm{t}c\) :

Thermal-relaxation variable

\(\gamma =\frac{K_{1}}{c}\) :

Chemical reaction variable

\(\Pr =\frac{\nu }{\alpha }\) :

Prandtl number

\(\textrm{Sc}=\frac{\nu }{D_\textrm{B}}\) :

Schmidt number

\(\gamma _{2}=\lambda _{c}c\) :

Solutal-relaxation variable

\(C_{f_{x}}\) :

Skin-friction coefficient

\(\tau _\textrm{w}\) :

Wall shear-stress

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Acknowledgements

This work was partially supported by the Natural Science Foundation of Jiangxi Province (No. 2920224BAB201018), the Natural Science Foundation of Xuzhou (No. KC22056) and Research Project of Science and Technology Plan of Jiangxi Provincial Department of Education (No. GJJ2203204 & GJJ2203201).

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Authors and Affiliations

  1. College of Education, Nanchang Normal College of Applied Technology, Nanchang, 330108, China

    Fuzhang Wang

  2. School of Mathematics, Xuzhou University of Technology, Xuzhou, 221018, China

    Fuzhang Wang

  3. NUTECH School of Applied Sciences and Humanities, National University of Technology, Islamabad, 44000, Pakistan

    M. Waqas

  4. Department of Mechanical Engineering, Lebanese American University, Beirut, Lebanon

    M. Waqas

  5. Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, 80207, 21589, Jeddah, Saudi Arabia

    W. A. Khan

  6. Mechanical Engineering Department, College of Engineering and Islamic Architecture, Umm Al-Qura University, 5555, 21955, Makkah, Saudi Arabia

    Basim M. Makhdoum

  7. Center of Research, Faculty of Engineering, Future University in Egypt, New Cairo, 11835, Egypt

    Sayed M. Eldin

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Wang, F., Waqas, M., Khan, W.A. et al. Cattaneo–Christov heat-mass transfer rheology in third-grade nanoliquid flow confined by stretchable surface subjected to mixed convection. Comp. Part. Mech. 10, 1645–1657 (2023). https://doi.org/10.1007/s40571-023-00579-w

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