Arabian Journal for Science and Engineering

, Volume 44, Issue 1, pp 531–540 | Cite as

Entropy Generation in Different Types of Fractionalized Nanofluids

  • Muhammad Saqib
  • Farhad AliEmail author
  • Ilyas Khan
  • Nadeem Ahmad Sheikh
  • Arshad Khan
Research Article - Physics


The study of classical nanofluid is limited to partial differential equations with integer-order neglecting memory effect. Fractionalized nanofluids, modeled by partial differential equations with Caputo time-fractional derivative, have the capability to address the memory effect. This article deals with the flow and entropy generation of electrically conducting different types of fractionalized nanofluids passing over an infinite vertical plate embedded in porous medium. The governing equations are transformed into dimensionless form, and then, a time-fractional model is generated using the Caputo approach. Two different nanoparticles (molybdenum disulfide and graphene oxide) are dispersed in three different base fluids (water, kerosene oil and methanol). The problem is solved for the exact solutions using the Laplace transformation technique. The impacts of fractional parameter \(\alpha \) and volume fraction of nanoparticles \(\varphi \) on velocity profile, entropy generation, Bejan number and the rate of heat transfer are exhibited in tabular form. Finally, the graphs are plotted for different types of nanoparticles and base fluids and discussed physically. Moreover, from present solutions, the well-known published results are recovered to validate the obtained results.


Fractionalized nanofluids Entropy generation Bejan number Caputo time-fractional derivatives Exact solutions 

List of symbols


Dimensional velocity


Dimensionless velocity


Fluid temperature

\(T_\mathrm{W} \)

Wall temperature

\(T_\infty \)

Ambient temperature

\(\theta \)

Dimensionless temperature


Acceleration due to gravity

\(c_\mathrm{p} \)

Specific heat at a constant pressure

\(k_\mathrm{f} \)

Thermal conductivity of the fluid


Laplace transforms parameter

\(\nu _\mathrm{f} \)

Kinematic viscosity of the fluid

\(\mu _\mathrm{f} \)

Dynamic viscosity

\(\rho _\mathrm{f} \)

Fluid density

\(\rho _\mathrm{s} \)

The density of the solid

\(U_0 \)

The amplitude of the velocity

\(\beta _\mathrm{T} \)

The volumetric coefficient of thermal expansion

\(\rho _\mathrm{nf} \)

Nanofluids density

\(\mu _\mathrm{nf} \)

Dynamic viscosity of nanofluids

\((\beta _\mathrm{T} )_\mathrm{nf} \)

Thermal expansion coefficient of nanofluids,

\((\rho c_p )_\mathrm{nf} \)

Specific heat capacity of nanofluids

\(k_\mathrm{nf} \)

The thermal conductivity of nanofluids.


Thermal Grasshof number


Prandtl number


Nusselt number

\(\varphi \)

Nanoparticles volume fraction

\(\alpha \)

Fractional order/fractional parameter


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Copyright information

© King Fahd University of Petroleum & Minerals 2018

Authors and Affiliations

  • Muhammad Saqib
    • 1
    • 2
    • 3
  • Farhad Ali
    • 1
    • 2
    • 3
    Email author
  • Ilyas Khan
    • 4
  • Nadeem Ahmad Sheikh
    • 1
    • 2
    • 3
  • Arshad Khan
    • 5
  1. 1.Computational Analysis Research GroupTon Duc Thang UniversityHo Chi Minh CityVietnam
  2. 2.Faculty of Mathematics and StatisticsTon Duc Thang UniversityHo Chi Minh CityVietnam
  3. 3.Department of MathematicsCity University of Science and Information TechnologyPeshawarPakistan
  4. 4.Basic Engineering Sciences DepartmentCollege of Engineering Majmaah UniversityMajmaahSaudi Arabia
  5. 5.Institute of Business and Computer SciencesThe University of AgriculturePeshawarPakistan

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