A note on activation energy and magnetic dipole aspects for Cross nanofluid subjected to cylindrical surface


Our main emphasis in this manuscript was to scrutinize the aspects of activation energy for magnetized Cross nanofluid subjected to cylindrical surface. Formulation for energy expression is developed through heat sink-source phenomenon. More specifically, Velocity of Cross liquid is deliberated by considering infinite shear rate viscosity and Lorentz’s force. The considered Cross nanoliquid expression (Buongiorno relation) comprises thermophoretic and Brownian movement mechanisms. Moreover, Chemical processes are deliberated subjected to appliance of activation energy. Bvp4c algorithm is implemented to tackle the nonlinear structure. Outcomes for Sherwood number, Nusselt number, skin fraction, temperature, concentration and velocity are presented in this manuscript. Our results revealed that temperature of Cross nanoliquid intensifies for larger thermophoretic parameter. Moreover, nanoliquid concentration dwindles for greater estimation of activation energy parameter.

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\(u,v\) :

Velocity components

\(x,r\) :

Cylindrical coordinates

\(\rho\) :

Fluid density

\(m\) :

Fitted rate constant

\(D_{B}\) :

Brownian diffusion coefficient

a, c :


\(\varGamma\) :

Material parameter

\(T\) :

Temperature of fluid

\(T_{w}\) :

Surface temperature

\(T_{\infty }\) :

Ambient fluid temperature

\(C\) :

Concentration of nanofluid

\(n\) :

Power law index

\(\mu\) :

Generalized Newtonian viscosity

\(c_{p}\) :

Specific heat

\(C_{w}\) :

Surface concentration

\(u_{w}\) :

Stretching velocity

\(\nu\) :

Kinematic viscosity

\(\mu_{0}\) :

Shear viscosity

\(D_{T}\) :

Thermophoretic force

\(\tau\) :

Ratio parameter

\(C_{\infty }\) :

Ambient nanoparticle concentration

\(k^{*}\) :

Mean absorption coefficient

\(\beta_{0}\) :

Magnetic field strength

\(\alpha_{m}\) :

Thermal conductivity

\(\sigma^{*}\) :

Stefan Boltzmann

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

Capacity of heat for base liquid

\(\tau_{w}\) :

Surface shear stress

\(\eta\) :

Non-dimensional variable

\(\psi\) :

Stream function

\({\text{We}}\) :

Local Weissenberg number

\(M\) :

Parameter for magnetic field

\(\gamma\) :

Curvature parameter

\({ \Pr }\) :

PRANDTL number

\(\sigma\) :

Reaction rate parameter

\(A\) :

Time-dependent parameter

\(E\) :

Parameter for activation energy

\(N_{t}\) :

Thermophoresis parameter

\(\delta\) :

Temperature difference parameter

\(N_{b}\) :

Parameter of Brownian moment

\(\theta_{f}\) :

Temperature ratio parameter

\(N_{R}\) :

Radiation parameter

\(\gamma_{\text{1}}\) :

Biot number

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

Local Reynolds number

\(f\) :

Non-dimensional velocity

θ :


ϕ :


\(\tau_{rx}\) :

Wall shear stress

\(q_{w}\) :

Wall heat flux

\({\text{Nu}}\) :

Local Nusselt number

\(C_{f}\) :

Drag force

\({\text{Sc}}\) :

Schmidt number

\(\left( {h,h^{*} } \right)\) :

Temperature-dependent/space dependent heat sink/source coefficients

\(Q^{'''}\) :

Non-uniform heat sink/source

\(\left( {\delta_{1} ,\delta_{2} } \right)\) :

Dimensionless space–time dependent heat sink/source


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This project was funded by the postdoctoral international exchange program for incoming postdoctoral students, at Beijing Institute of Technology, Beijing, China.

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Khan, W.A., Ali, M., Shahzad, M. et al. A note on activation energy and magnetic dipole aspects for Cross nanofluid subjected to cylindrical surface. Appl Nanosci 10, 3235–3244 (2020). https://doi.org/10.1007/s13204-019-01220-0

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  • Non-uniform heat absorption-generation
  • Nanofluid
  • Non-Newtonian fluid
  • Activation energy