Effective Prandtl Number Model Influences on the \(\gamma {\hbox {Al}}_2 {\hbox {O}}_3\)\({\hbox {H}}_2 {\hbox {O}}\) and \(\gamma {\hbox {Al}}_2 {\hbox {O}}_3\)\({\hbox {C}}_2 {\hbox {H}}_6 {\hbox {O}}_2 \) Nanofluids Spray Along a Stretching Cylinder


The flow of common fluids (water, oils and ethylene glycol etc.) is diluted by adding different small particles of metals, and their oxides are more powerful to reduce the scientific issues related to quicker heat transfer. According to this indication, we have contemplated finite film of \(\gamma {\hbox {Al}}_2 {\hbox {O}}_3\)\({\hbox {H}}_2 {\hbox {O}}\) and \(\gamma {\hbox {Al}}_2 {\hbox {O}}_3\)\({\hbox {C}}_2 {\hbox {H}}_6 {\hbox {O}}_2 \) nanoliquid sprayed on an extending cylinder. In this scenario, uniform magnetic field \(B_0\) and constant reference temperature are employed on the stream of thin film nanofluid. The impact of effective Prandtl number, viscosity and thermal conductivity is derived from the experimental data (Sheikhzadeh et al. in J Appl Fluid Mech 10:209–219, 2017; Lee et al. in J Heat transf 121:280–289, 1992; Wang et al. in J Thermo Phys Heat Transf 13:474–480, 1999; Hamilton and Crosser in Ind Eng Chem Fundam 1:187–191, 1962; Maiga et al. in Super Lattices Microstruct 35:543–55, 2004; Hayat et al. in J Mol Liq. https://doi.org/10.1016/j.molliq.2018.06.029, 2018). The model problem is excellently converted into a set of proper self-comparable forms with the assistance of possible transformations. Analytical results of velocity and thermal profile are computed using homotopy analysis method for both \(\gamma {\hbox {Al}}_2 {\hbox {O}}_3\)\({\hbox {H}}_2 {\hbox {O}}\) and \(\gamma {\hbox {Al}}_2 {\hbox {O}}_3\)\({\hbox {C}}_2 {\hbox {H}}_6 {\hbox {O}}_2 \) nanoliquid. Furthermore, during coating analysis, rate of spray, pressure distribution, skin friction coefficient (surface drag force) \(C_{\mathrm{f}} \) and Nusselt number (the rate of heat transfer) Nu for both nanofluids are also intended. The impact of additional ingrained quantities like magnetic parameter M, volume fraction of nanoparticles \(\varphi \), Grashof number Gr, fluid thickness parameter \(\beta \), Prandtl number Pr and Reynolds number Re is portrayed numerically and graphically for both alumina particles. The key observation indicates that the temperature of \(\gamma {\hbox {Al}}_2 {\hbox {O}}_3\)\({\hbox {C}}_2 {\hbox {H}}_6 {\hbox {O}}_2\) nanoliquid leading on \(\gamma {\hbox {Al}}_2 {\hbox {O}}_3\)\({\hbox {H}}_2 {\hbox {O}}\) nanoliquid during the study. Due to greater viscosity and thermal conductivity, \({\hbox {C}}_2 {\hbox {H}}_6 {\hbox {O}}_2\)-based nanofluid is observed as upgraded common base fluid assimilated to \({\hbox {H}}_2\)O.

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\(u, v \, w\) :

Velocities components \(\left( {{\hbox {ms}}^{-1}} \right) \)

\(B_0 \) :

Magnetic field strength \(\left( {{\hbox {NmA}}^{-1}} \right) \)

fg :

Dimensional velocity profiles

T :

Fluid temperature (K)

\(T_w \) :

Cylinder surface temperature (K)

\(T_\delta \) :

Free surface temperature (K)

M :

Magnetic parameter

p :

Pressure distribution

C :

Stretching parameter

Pr :

Prandtl number

Re :

Local Reynolds number

Ec :

Eckert number

Gr :

Grashof number

Nu :

Nusselt number

\(C_{\mathrm{f}}\) :

Skin friction coefficient

\(W_w \) :

Stretching velocity

\(U_w \) :

Suction/injection speed

\(\left( {C_p } \right) _{\mathrm{f}}\) :

Specific heat of base fluid \(\left( {\hbox {J/kgK}} \right) \)

\(k_{\mathrm{nf}}\) :

Thermal conductivity (\({\hbox {Wm}}^{-1}K^{-1})\)

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

Dynamic viscosity of nanofluid (mPa)

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

Thermal expansion coefficient

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

Nanofluid density (Kgm\(^{-3}\))

\(\upsilon _{\mathrm{nf}} \) :

Kinematic Viscosity

\(\xi \) :

Similarity variable

\(\varphi \) :

Nanoparticle volume fraction

\(\Theta \) :

Dimensional heat profiles

\(\sigma _{\mathrm{nf}} \) :

Electrical conductivity

\(\beta \) :

Non-dimensional thickness

\(\tau \) :

Surface shear stress

h :

Auxiliary constant






Base fluid


Solid nanoparticles


Homotopy asymptotic method




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The authors are very thankful to the CUSIT and AWKUM for providing them with the opportunity of funding for this study.

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Correspondence to Taza Gul.

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The authors state that they have no competing interest.

Authors’ contributions

The model of the problem was designed by TG using the available data from the experimental approach of \(\gamma {\hbox {Al}}_2 {\hbox {O}}_3\)\({\hbox {H}}_2 {\hbox {O}}\) and \(\gamma {\hbox {Al}}_2 {\hbox {O}}_3\)\({\hbox {C}}_2 {\hbox {H}}_6 {\hbox {O}}_2 \). TG and SN solved the problem, and SI, ZS and MAK participated in the results and discussion. All the authors read and approved the final manuscript.

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Gul, T., Nasir, S., Islam, S. et al. Effective Prandtl Number Model Influences on the \(\gamma {\hbox {Al}}_2 {\hbox {O}}_3\)\({\hbox {H}}_2 {\hbox {O}}\) and \(\gamma {\hbox {Al}}_2 {\hbox {O}}_3\)\({\hbox {C}}_2 {\hbox {H}}_6 {\hbox {O}}_2 \) Nanofluids Spray Along a Stretching Cylinder. Arab J Sci Eng 44, 1601–1616 (2019). https://doi.org/10.1007/s13369-018-3626-z

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  • Coating phenomena
  • HAM
  • MHD
  • Nanofluid thin film
  • Stretching cylinder
  • \(\gamma {\hbox {Al}}_2 {\hbox {O}}_3\)\({\hbox {H}}_2 {\hbox {O}}, \gamma {\hbox {Al}}_2 {\hbox {O}}_3\)\({\hbox {C}}_2 {\hbox {H}}_6 {\hbox {O}}_2\)