Arabian Journal for Science and Engineering

, Volume 44, Issue 2, pp 1525–1541 | Cite as

Thermal Slip in Oblique Radiative Nano-polymer Gel Transport with Temperature-Dependent Viscosity: Solar Collector Nanomaterial Coating Manufacturing Simulation

  • R. MehmoodEmail author
  • Rabil Tabassum
  • S. Kuharat
  • O. Anwar Bég
  • M. Babaie
Research Article - Mechanical Engineering


Nano-polymeric solar paints and sol–gels have emerged as a major new development in solar cell/collector coatings offering significant improvements in durability, anti-corrosion and thermal efficiency. They also exhibit substantial viscosity variation with temperature which can be exploited in solar collector designs. Modern manufacturing processes for such nano-rheological materials frequently employ stagnation flow dynamics under high temperature which invokes radiative heat transfer. Motivated by elaborating in further detail the nanoscale heat, mass and momentum characteristics, the present article presents a mathematical and computational study of the steady, two-dimensional,non-aligned thermo-fluid boundary layer transport of copper metal-doped water-based nano-polymeric sol–gels under radiative heat flux. To simulate real nano-polymer boundary interface dynamics, thermal slip is analysed at the wall. A temperature-dependent viscosity is also considered. The conservation equations for mass, normal and tangential momentum and energy are normalized via appropriate transformations to generate a multi-degree, ordinary differential, nonlinear, coupled boundary value problem. Numerical solutions are obtained via the stable, efficient Runge–Kutta–Fehlberg scheme with shooting quadrature in MATLAB symbolic software. Validation of solutions is achieved with a variational iterative method utilizing Lagrangian multipliers. The impact of key emerging dimensionless parameters, i.e. obliqueness parameter, radiation–conduction Rosseland number (Rd), thermal slip parameter\((\alpha )\), viscosity parameter (m), nanoparticles volume fraction\((\phi )\), on non-dimensional normal and tangential velocity components, temperature, wall shear stress, local heat flux and streamline distributions is visualized graphically. Shear stress and temperature are boosted with increasing radiative effect, whereas local heat flux is reduced. Increasing wall thermal slip parameter depletes temperatures.


Non-orthogonal stagnation-point flow Thermal slip Variable viscosity Thermal radiation flux model Solar nano-polymer coating manufacture Copper volume fraction 



\(\hat{{x}}\)-component of velocity \(\left( {\hbox {m}/\hbox {s}}\right) \)


\(\hat{{y}}\)-component of velocity \(\left( {\hbox {m}/\hbox {s}}\right) \)


Temperature \(\left( \hbox {K}\right) \)


Wall temperature \(\left( \hbox {K}\right) \)

\(\hat{{T}}_\infty \)

Ambient temperature \(\left( \hbox {K}\right) \)


Pressure \(\left( {\hbox {N}/\hbox {m}^{2}}\right) \)


Cartesian coordinates along the stretching surface and normal to it \(\left( \hbox {m}\right) \)


Radiative thermal linearized heat flux


Rosseland mean absorption coefficient

\(\hat{\delta }^{*}\)

Stefan–Boltzmann constant

a / c

Stretching ratio


Reynolds viscosity variation exponent


Thermal slip factor


Variable viscosity parameter


Specific heat \(\left( {\frac{\hbox {J}}{\hbox {kg}\,\hbox {K}}}\right) \)


Prandtl number


Radiation–conduction parameter


Thermal conductivity \(\left( {\frac{\hbox {W}}{\hbox {m}\,\hbox {K}}}\right) \)

Greek Symbols

\(\alpha \)

Thermal slip parameter

\(\phi \)

Solid volume fraction of nanoparticles

\(\mu _0\)

Reference viscosity \(\left( {\frac{\hbox {kg}}{\hbox {m}\,\hbox {s}}}\right) \)

\(\hat{\nu }\)

Kinematics viscosity \(\left( {\hbox {m}^{2}/\hbox {s}}\right) \)

\(\hat{\rho }\)

Density \(\left( {\hbox {kg}/\hbox {m}^{3}}\right) \)

\(\gamma \)

Flow obliqueness parameter

\(\hat{\mu }\)

Dynamic viscosity \(\left( {\frac{\hbox {kg}}{\hbox {m}\,\hbox {s}}}\right) \)


\(\hbox {f}\)

Base fluid

\(\hbox {s}\)


\(\hbox {nf}\)



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The authors appreciate the reviewer comments which have improved the present work.

Compliance with Ethical Standards

Conflicts of interest

The authors declare that they have no conflict of interest with any individual or organization regarding this articles content.


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Copyright information

© King Fahd University of Petroleum & Minerals 2018

Authors and Affiliations

  • R. Mehmood
    • 1
    Email author
  • Rabil Tabassum
    • 1
  • S. Kuharat
    • 2
  • O. Anwar Bég
    • 2
  • M. Babaie
    • 3
  1. 1.Department of Mathematics, Faculty of Natural SciencesHITEC UniversityTaxila CanttPakistan
  2. 2.Propulsion, Hybrid Energy and Medical Engineering Sciences, Department of Mechanical/Aeronautical Engineering, School of Computing, Science and EngineeringSalford UniversityManchesterUK
  3. 3.Energy Sciences, Petroleum and Gas Engineering Division, School of Computing, Science and EngineeringSalford UniversityManchesterUK

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