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Modeling and analysis of von Kármán swirling flow for Oldroyd-B nanofluid featuring chemical processes

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Abstract

In the current era of emerging technologies, the desired demand is the outstanding efficiency to be achieved by the virtue of standard design and the selection of constituents. Out of these, a key role is the transmission of heat across the systems, and the enhancement of this transmission is the fundamental requirement to be addressed. An addition of nanoparticles with the selected liquids supported the progress toward the optimum level. For the same goal, here we design and discuss a 2D model of Oldroyd-B nanoliquid regarding the rheological aspects taking the thermophoretic and Brownian moment with the consideration of MHD. In this model, various involved physical quantities remained under discussion including chemical processes, convective heat transportation mechanism and heat source–sink aspects. The mathematical differential model non-dimensionalized using the suitable transformations obeys the fundamental laws. For the purpose of solution for the raised nonlinear ordinary system, we adopted homotopy analysis method-based algorithm, which is proved to be the best available technique for analytic solution. The outcomes are displayed graphically for various dimensionless physical quantities. Our analytical analysis indicates that \(\left( {f^{\prime}\left( \eta \right),g\left( \eta \right)} \right)\) liquid velocities deteriorate via higher estimation of \(\beta_{1}\) (Deborah number), while \(h(\eta )\) intensifies Reynolds number, radiation parameter and magnetic parameter. Moreover, nanoliquid temperature rises for larger values of Brownian moment parameter.

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Abbreviations

\({\mathbf{V}}\) :

Velocity vector

\(\left( {r,\theta ,z} \right)\) :

Polar cylindrical coordinates

\(\rho\) :

Density of fluid

\(p\) :

Liquid pressure

\(\mu\) :

Dynamic viscosity

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

Retardation–relaxation times

\(B_{0}\) :

Magnetic field

\(J\) :

Current density

\(\tau\) :

Ratio of heat capacity

\(\left( {C,T} \right)\) :

Concentration/temperature of liquid

\(N_{b}\) :

Brownian moment

\(S\) :

Extra stress tensor

\(D_{\text{T}}\) :

Thermophoresis effect

\(\frac{D}{Dt}\) :

Upper-convected derivative

\(A_{1}\) :

First Rivlin–Ericksen tensor

\(k_{1}\) :

Chemical reaction rate constant

\(\left( {h_{\psi } ,h_{f} } \right)\) :

Mass heat transfer coefficients

\(\left( {C_{f} ,T_{f} } \right)\) :

Concentration/temperature of heated fluid under the sheet

\(\left( {v_{r} ,v_{\theta } ,v_{z} } \right)\) :

Velocity components

\(\varOmega\) :

Swirl rate

\(\eta\) :

Dimensionless variable

\(\left( {f,g,h} \right)\) :

Dimensionless velocities

\(\Pr\) :

Prandtl number

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

Deborah numbers

\(M\) :

Magnetic field

\(\lambda\) :

Heat generation–absorption parameter

\(\left( \gamma \right)\) :

Biot number

\(N_{b}\) :

Brownian motion parameter

\(N_{t}\) :

Thermophoresis parameter

\({\text{Le}}\) :

Lewis number

Sc:

Schmidt number

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

Local Nusselt number

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

Reynolds number

\(\omega\) :

Rotation strength parameter

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Acknowledgements

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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Authors and Affiliations

  1. School of Mathematics and Statistics, Beijing Institute of Technology, Beijing, 100081, China

    S. Z. Abbas, W. A. Khan & H. Sun

  2. Department of Mathematics and Statistics, Hazara University, Mansehra, 21300, Pakistan

    S. Z. Abbas & M. Ali

  3. Department of Mathematics, Mohi-ud-Din Islamic University, Nerian Sharif, Azad Jammu and Kashmir, 12010, Pakistan

    W. A. Khan

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

    M. Waqas

  5. Department of Mathematics, Quaid-I-Azam University, Islamabad, 44000, Pakistan

    M. Irfan

  6. Department of Mathematics, Abbottabad, UST, Abbottabad, Pakistan

    S. Ahmad

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  1. S. Z. Abbas
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Correspondence to S. Z. Abbas or W. A. Khan.

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Technical Editor: Pedro Manuel Calas Lopes Pacheco, D.Sc.

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Abbas, S.Z., Khan, W.A., Sun, H. et al. Modeling and analysis of von Kármán swirling flow for Oldroyd-B nanofluid featuring chemical processes. J Braz. Soc. Mech. Sci. Eng. 41, 556 (2019). https://doi.org/10.1007/s40430-019-2050-y

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