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Analysis of the nanoscale heat transport and Lorentz force based on the time-dependent Cross nanofluid

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Abstract

Control of transport of energy by means of external force effects is one of the most important problems in modern applied mathematics. Magnetic force has an influence on heat transport phenomena and has various applications in industrial, engineering, and medical sciences. The root theme of this work is to study MHD flow with stagnation point and flow of Cross nanofluid in contracting/extracting cylinder. The impacts of variable thermal conductivity, Lorentz’s force on unsteady Cross nanofluid and the cylindrical coordinate are investigated using the behavior of expanding/contracting cylinder. Discovering the impacts of physical parameters on movement, energy exchange and mass transport visibility of Cross nanofluid flow with respect to region (shear thinning/thickening) and on the basis of geometry (contracting/extracting) is most interesting and beauty of this attempt. Smooth debate on fluid behavior in light of numerical outcome classifying shear thinning/thickening and contracting/extracting of geometry is disclosed comprehensively. By keeping the idea of shooting methodology, the nonlinear higher order differential equations are converted into a first-order system of ordinary differential equations. Furthermore, bvp4c Matlab built-in command is used for comparison of the numerical solutions to solve these linear ordinary differential equations. The numerical solutions are plotted in figures as well as tabulated in some tables.

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Data availability statement

Data available on request from the authors.

Change history

Abbreviations

\(b_{0}\) :

Constant of dimension length \(t\)

\(K(T)\) :

Variable thermal conductivity

\(\beta\) :

Constant of contraction–expansion

\(U_{w} (x,t)\) :

Initial velocity

\(U_{e} (x,t)\) :

Free stream velocity

\(T_{w} ,C_{w}\) :

Temperature and concentration of cylinder

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

Temperature and concentration of cylinder

\(a,\,c\) :

Constant of dimension \(({\text{time}})^{ - 1}\)

\(\tau\) :

Cauchy stress tensor

\(k\) :

Thermal conductivity

\(x,\,\,r\) :

Space variable

\(s\) :

Suction parameter

\(k_{\infty }\) :

Conductivity away for

Pr :

Prandtl number

\(\xi\) :

Chemical rate reaction parameter

\(\lambda\) :

Velocity ratio parameter

\(\mu_{0}\) :

Zero shear rate viscosity

\(A_{1}\) :

First Rivilin–Erickson tensor

\(\mu_{\infty }\) :

Infinite shear rate viscosity

\(p\) :

Pressure

\(I\) :

Identity tensor

\(\Gamma\) :

Relaxation time constant

\(n\) :

Power law index

Sc:

Schmidt number

\(\mu\) :

Viscosity

\(q_{w}\) :

Wall shear stress

\(\tau_{rx \, }\) :

Heat flux

\(M\) :

Magnetic parameter

\(u,\,\,v\) :

Velocity component

\(c_{f}\) :

Skin friction

\(B_{0}\) :

Magnetic field strength constant

\(\mathop \gamma \limits^{.}\) :

Shear strain

\(B\) :

Magnetic field strength

\(c_{p}\) :

Specific heat

D:

Solute diffusivity

\(R_{r}\) :

Chemical reaction parameter

\(\sigma\) :

Reaction rate parameter

\(\rho\) :

Density

C:

Concentration profile

T:

Temperature profile

V:

Velocity profile

\(\theta_{w}\) :

Temperature ratio parameter

A:

Unsteadiness parameter

We:

Weissenberg

\(\alpha_{m}\) :

Thermal diffusivity

Re:

Reynolds number

Nu:

Nusselt number

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Hazara University, Manshera, Pakistan

    Assad Ayub, Zulqurnain sabir & Hafiz A. Wahab

  2. Department of Industrial Engineering, Faculty of Engineering, King Abdulaziz University, Jeddah, Saudi Arabia

    Mohammed Balubaid

  3. GRC Department, Faculty of Applied Studies, King Abdulaziz University, Jeddah, Saudi Arabia

    S. R. Mahmoud

  4. Faculty of Engineering and Technology, Future University, Cairo, Egypt

    Mohamed R. Ali

  5. Department of Basic Science, Faculty of Engineering at Benha, Benha University, Benha, 13512, Egypt

    Mohamed R. Ali

  6. Department of Physics and Engineering Mathematics, Faculty of Engineering, Zagazig University, Zagazig, Egypt

    R. Sadat

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  1. Assad Ayub
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Ayub, A., sabir, Z., Wahab, H.A. et al. Analysis of the nanoscale heat transport and Lorentz force based on the time-dependent Cross nanofluid. Engineering with Computers 39, 2089–2108 (2023). https://doi.org/10.1007/s00366-021-01579-1

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