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

  • S. Z. AbbasEmail author
  • W. A. KhanEmail author
  • H. Sun
  • M. Ali
  • M. Waqas
  • M. Irfan
  • S. Ahmad
Technical Paper


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.


Nanofluid Oldroyd-B fluid von Kármán swirling flow Heat sink–source 

List of symbols


Velocity vector

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

Polar cylindrical coordinates


Density of fluid


Liquid pressure


Dynamic viscosity

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

Retardation–relaxation times


Magnetic field


Current density


Ratio of heat capacity

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

Concentration/temperature of liquid


Brownian moment


Extra stress tensor


Thermophoresis effect


Upper-convected derivative


First Rivlin–Ericksen tensor


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


Swirl rate


Dimensionless variable

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

Dimensionless velocities


Prandtl number

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

Deborah numbers


Magnetic field


Heat generation–absorption parameter

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

Biot number


Brownian motion parameter


Thermophoresis parameter


Lewis number


Schmidt number


Local Nusselt number


Reynolds number


Rotation strength parameter



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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Copyright information

© The Brazilian Society of Mechanical Sciences and Engineering 2019

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsBeijing Institute of TechnologyBeijingChina
  2. 2.Department of Mathematics and StatisticsHazara UniversityMansehraPakistan
  3. 3.Department of MathematicsMohi-ud-Din Islamic UniversityNerian SharifPakistan
  4. 4.NUTECH School of Applied Sciences and HumanitiesNational University of TechnologyIslamabadPakistan
  5. 5.Department of MathematicsQuaid-I-Azam UniversityIslamabadPakistan
  6. 6.Department of MathematicsAbbottabad, USTAbbottabadPakistan

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