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An effect of thermal radiation on inclined MHD flow in hybrid nanofluids over a stretching/shrinking sheet

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Abstract

The current work shows an incline MHD and Casson fluid flow with a mixed convective boundary layer with hybrid nanofluid Cu–Al2O3/water flow over a stretching/shrinking sheet. The present study is analyzed using an Al2O3–Cu/H2O hybrid nanofluid with a fixed Prandtl number of 6.8. The governing equation of highly nonlinear partial differential equations is converted into ordinary differential equations using exact similarity transformations. Moreover, the radiation effects are also permitted with help of Rosseland’s approximation. The subsequent system of equations is then investigated analytically with appropriate boundary conditions. The outcomes of this topic can be addressed using a graphical representation with many parameters like radiation, heat source/sink, stretching/shrinking mass transpiration so on. The research shows that the solution depicts a unique explanation for stretching/shrinking sheets and that the explanation demonstrates the dual flora focused on some stretching/shrinking sheet parameters. The nanoparticles are disseminated in water, which serves as the base fluid. Graphs are also used to study the effects of the magnetic parameter, mass transpiration, and heat source/sink parameter on the velocity profile. It has a wide range of uses in the polymer sector, power generators, flow meters, and pumps, among others. The results indicate that the solution illustrates an inimitable solution for the stretching sheet and that the explanation manifests the dual flora aimed at some parameters for the stretching/shrinking sheet. The hybrid nanofluid has significant features improving the heat transfer process and is extensively developed for manufacturing industrial uses. It was found that the basic similarity equations admit two phases for both stretching/shrinking surfaces.

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Abbreviations

Al2O3 :

Aluminum oxide

B. Cs:

Boundary conditions

Cu:

Copper

HNF:

Hybrid nanofluid

MHD:

Magnetohydrodynamic

a :

Stretching/shrinking sheet (s−1)

\(B_{0}\) :

Strength of magnetic field (W m−2)

\(b_{0}\) :

Constant

\(b > \,0\,\) :

Heated plate

\(b\, < \,0\,\,\) :

Cooled plate

C P :

Constant pressure (W kg−1 K−1)

d :

Stretching/shrinking parameter

f :

Similarity variable

\({\text{Gr}}_{{\text{x}}}\) :

Grashof number

\(\vec{g}\) :

Gravitational acceleration

\(k^{*}\) :

Mean absorption coefficient (m−2)

\(N_{{\text{I}}}\) :

Heat source/sink constraint (\({\text{W}}\,{\text{m}}^{ - 3} \,{\text{K}}^{ - 1}\))

\(N_{{\text{I}}} \, > \,0\) :

Heat source parameter

\(N_{{\text{I}}} \, < \,0\) :

Heat sink parameter

Pr:

Prandtl number

Q :

Chandrasekhar’s formulation

\(Q_{0} \,\,\) :

Heat source/sink of coefficient (\({\text{W}}\,{\text{m}}^{ - 3} \,{\text{K}}^{ - 1}\))

\(Q_{0} > 0\) :

Heat source

\(Q_{0} < 0\) :

Heat sink

\(q_{{\text{r}}}\) :

Radiative heat flux (J s−1 m−2)

\(q_{{\text{w}}}\) :

Heat flux at the wall (J s−1 m−2)

\(q_{0} \,\,\) :

Constant

\({\text{Rd}}\) :

Radiation number \(\left( { = \frac{{4\,\sigma *\,T_{\infty }^{3} }}{{3\,\kappa_{f} \,k*}}} \right)\)

\({\text{Re}}_{{\text{x}}}\) :

Reynolds number \(\left( { = \;\frac{{xU_{{\text{w}}} (x)}}{{\nu_{{\text{f}}} }}} \right)\)

T :

Temperature (K)

\(T_{\infty }\) :

Ambient temperature (K)

\(T_{{\text{w}}}\) :

Surface temperature (K)

\(U_{{\text{w}}}\) :

Stretching velocity of the sheet

u, v :

Velocity component (m s−1)

\(V_{{\text{w}}}\) :

Mass flux \(\left( { = \,\, - \,\left( {\frac{{3\left| a \right|\,\nu_{{\text{f}}} }}{2}} \right)^{1/2} V_{{\text{c}}} } \right)\) (m s−1)

V C :

Suction/injection velocity (m s−1)

\(V_{{\text{c}}} > 0\) :

Suction velocity (m s−1)

\(V_{{\text{c}}} < 0\) :

Injection velocity (m s−1)

x, y :

Coordinate along the sheet (m)

α:

Thermal diffusivity (m2 s−1)

\(\beta\) :

Solution domain

\(\Lambda\) :

Mixed parameter \(\left( { = \;\frac{2}{3}\,\frac{{G_{{{\text{rx}}}} }}{{{\text{Re}}_{{\text{x}}}^{2} }}} \right)\)

\(\Lambda \, > \,0\,\) :

Heated plate for assisting flow

\(\Lambda \, < \,0\,\) :

Cooled plate for opposing flow

\(\varepsilon\) :

Casson fluid

\(\kappa\) :

Thermal conductivity (mol m−3)

\(\nu\) :

Kinematic viscosity (\({\text{m}}^{2} \,{\text{s}}^{ - 1}\))

\(\eta\) :

Similarity variable

\(\mu\) :

Dynamic viscosity (\({\text{kg}}\,{\text{m}}^{ - 1} \,{\text{s}}^{ - 1}\))

\(\rho\) :

Density (\({\text{kg}}\,{\text{m}}^{ - 3}\))

\(\sigma^{*}\) :

Stefan–Boltzmann constant (\({\text{Wm}}^{ - 2} {\text{K}}^{ - 4}\))

\(\tau\) :

Inclined angle of magnetic field respect to x-axis (rad)

\(\nu\) m :

Magnetic permeability

\(\phi\) :

Volume fraction of nanoparticle

\(\psi\) :

Stream function (m2 s−1)

S:

Solid particle involved in HNF

f:

Base fluid

nf:

Nanofluid

w:

Wall condition

\(\infty\) :

For from the sheet

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Acknowledgements

The author K. N. Sneha is thankful to the National Fellowship and Scholarship for Higher Education for ST (NFST), New Delhi, India for financial support in the form of Junior Research Fellowship: Awardee No. 202021-NFST-KAR-01224, 13/08/2021.

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Authors and Affiliations

  1. Department of Mathematics, Davangere University, Shivagangothri, Davangere, 577007, India

    K. N. Sneha & U. S. Mahabaleshwar

  2. Department of Mechanical Engineering, Birla Institute of Technology and Science Pilani, Pilani Campus, Vidhyavihar, Pilani, Rajasthan, 333031, India

    Suvanjan Bhattacharyya

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  1. K. N. Sneha
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Sneha, K.N., Mahabaleshwar, U.S. & Bhattacharyya, S. An effect of thermal radiation on inclined MHD flow in hybrid nanofluids over a stretching/shrinking sheet. J Therm Anal Calorim 148, 2961–2975 (2023). https://doi.org/10.1007/s10973-022-11552-9

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