Transient flow and heat transfer mechanism for Williamson-nanomaterials caused by a stretching cylinder with variable thermal conductivity

  • Hashim Email author
  • Aamir Hamid
  • Masood Khan
Technical Paper


The utilization of nanometre-sized solid particles in working fluids has been seriously recommended due to their enhanced thermal characteristics. This suspension of solid particles in base fluids can significantly enhance the physical properties, such as, viscosity and thermal conductivity. They are widely used in several engineering processes, like, heat exchangers, cooling of electronic equipment, etc. In this exploration, we attempt to deliver a numerical study to simulate the nanofluids flow past a circular cylinder with convective heat transfer in the framework of Buongiorno’s model. A non-Newtonian Williamson rheological model is used to describe the behavior of nanofluid with variable properties (i.e., temperature dependent thermal conductivity). The leading flow equations for nanofluid transport are mathematical modelled with the assistance of Boussinesq approximation. Numerical simulation for the system of leading non-linear differential equations has been performed by employing versatile, extensively validated, Runge–Kutta Fehlberg scheme with Cash–Karp coefficients. Impacts of active physical parameters on fluid velocity, temperature and nanoparticle concentration is studied and displayed graphically. It is worth to mention that the temperature of non-Newtonian nanofluids is significantly enhanced by higher variable thermal conductivity parameter. One major outcome of this study is that the nanoparticle concentration is raised considerably by an increasing values of thermophoresis parameter.

List of symbols

a, β

Positive constants




Density of the fluid


Strength of magnetic field


Viscosity at infinite shear rate


Viscosity at zero shear rate


Viscosities ratio

x, r

Cylindrical polar coordinates

u, v

Velocity components


Stretching cylinder velocity


Relaxation time


Kinematic viscosity


Electrical conductivity


Temperature of hot fluid


Heat transfer coefficient


Temperature of the fluid


Concentration of nanoparticles


Free stream temperature


Free stream nanoparticles concentration


Brownian diffusion coefficient


Thermophoretic diffusion coefficient


Specific thermal capacity


Variable thermal conductivity


Ratio of effective heat capacities


Stream function


Dimensionless velocity


Dimensionless temperature


Dimensionless concentration


Dimensionless variable


Curvature parameter


Weissenberg number


Unsteadiness parameter


Prandtl number


Schmidt number


Thermophoretic parameter


Brownian motion parameter


Local Reynolds number


Heat capacity of the fluid


Heat capacity of nanoparticles


Surface shear stress


Skin friction coefficient


Nusselt number


Sherwood number


Surface heat flux


Surface mass flux


Thermal Biot number


Concentration Biot number



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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematics & StatisticsRiphah International UniversityIslamabadPakistan
  2. 2.Department of MathematicsQuaid-i-Azam UniversityIslamabadPakistan

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