Sodium-alginate (SA) based nanofluids represent a new generation of fluids with improved performances in terms of heat transfer. This work examines the influence of the nanoparticle shape on a non–Newtonian viscoplastic Cu–nanofluid pertaining to this category. In particular, a stretching/shrinking sheet subjected to a transverse magnetic field is considered. The proposed Cu–nanofluid consists of four different nanoparticles having different shapes, namely bricks, cylinders, platelets, and blades dispersed in a mixture of sodium alginate with Prandtl number Pr = 6.45. Suitable similarity transformations are employed to reduce non–linear PDEs into a system of ODEs and these equations and related boundary conditions are solved numerically by means of a Runge–Kutta–Fehlberg (RKF) method. Moreover, analytical solutions are obtained through the application of a MAPLE built–in differential equation solver (Dsolve). The behavior of prominent parameters against velocity and temperature is analyzed. It is found that the temperature increases for all shapes of nanoparticles with the viscoplastic parameter and the Eckert number.

The basic idea of dispersing particles into liquids can be traced back to the published theoretical work of Maxwell [_{2}O_{3}–water nanofluid inside a vertical micro–annulus with two different types of heat fluxes imposed at the walls. They noted that both temperature and concentration dependent buoyancy forces affect flow fields and nanoparticle migration. Sheikholeslami et al. [_{2}O_{3}–water/ethylene glycol [_{3}O_{4}–water [_{2}–water [

Literature review revealed that most articles on nanofluids mainly focus on the impacts of nanoparticle dimensions, types, dispersibility of nanoparticles, and the base fluids. Some recent work suggests that the shapes of nanoparticles have a significant impact on the thermo–physical properties of nanofluids. Elias et al. [_{2}–EG nanofluid with platelets shape nanoparticles gave the highest heat transfer enhancement. The shape effects of nano–size particles in Cu–H_{2}O nanofluid on entropy generation was analyzed by Ellahi et al. [

Monfared et al. [_{3}O_{4}–nanofluids. Their results showed that platelet shape nanoparticles should be used to obtain the highest Nusselt number. Hosseinzadeh et al. [_{2}–GO hybrid nanofluid flow through an upright cylinder with different shapes of nanoparticles was investigated by Chu et al. [_{2}–GO hybrid nanofluid had the maximum temperature. Khashi’ie et al. [_{2}O_{3} water based nanofluid flow in respect of the two important of parameters of EMHD and thermal radiation. Their results indicated that the maximum heat transfer enhancement occured in the case of blade shape nanoparticles. Anwar et al. [_{2}–Co hybrid nanofluid. They used spherical, cylindrical, blade, platelet, and brick shape nanoparticles and found that maximum heat transfer rate occured in the case of blade shape nanoparticles. Shahsavar et al. [_{2}O_{3}–H_{2}O nanofluid filled in a square cavity with multiple circular-, square-, and triangular-shaped obstacles. They found a higher heat transfer rate in the cavity with triangular obstacles. Cao et al. [

In the above work, Newtonian fluids were taken as a base fluid. However, few researchers have considered non–Newtonian based nanofluids in their research. It is well recognized that non–Newtonian fluids are encountered in numerous transport processes such as central heating systems, molten polymers, and nuclear reactors [_{2}O_{3}–nanofluid with accretion/ablation was illustrated by Hussanan et al. [

Cu | 8933 | 385 | 400 | |

SA | 989 | 4175 | 0.6376 |

The steady 2D flow of SA based Cu–nanofluids with five shapes of nanoparticles is considered over a stretching sheet having a sheet velocity assumed as

The rheological behavior of Cu–nanofluid followed by [

where

The viscosity of SA based Cu–nanofluid with different shapes of nanoparticle, except spherical, is calculated from the following correlation:

Viscosity coefficients | Platelets | Blades | Cylinders | Bricks |
---|---|---|---|---|

37.1 | 14.6 | 13.5 | 1.9 | |

612.6 | 123.3 | 904.4 | 471.4 |

In the case of spherical nanoparticle shape, viscosity of SA based Cu–nanofluid is

Pak et al. [

The electrical conductivity of SA based Cu–nanofluid given by Sheikholeslami [

Many relationships proposed for the thermal conductivity of a two–phase mixture, in which Maxwell [

Sphericity | Platelets | Blades | Cylinders | Bricks |
---|---|---|---|---|

0.52 | 0.36 | 0.62 | 0.81 | |

5.7 | 8.6 | 4.9 | 3.7 |

To simplify the

Using

where

The closed–form solution of

where

The following closed–form solution for velocity is obtained

By substituting

Taking

Closed–form solution of

where

Using boundary conditions

Note that above solution of temperature

The non–linear differential

The effect of different physical parameters, namely, the viscoplastic parameter

The present article examined nanoparticles shape effects on sodium alginate-based Cu–nanofluid over a stretching/shrinking sheet. The vital findings are

Velocity field decreases for all shapes of Cu nanoparticles for magnetic parameter and viscoplastic parameter.

Velocity field increases for all shapes of Cu nanoparticles for stretching sheet parameter.

Temperature field increases for different shape of Cu nanoparticles for viscoplastic parameter and Eckert number.

Temperature field decreases for all shapes of Cu nanoparticles for Prandtl number.

The temperature of non–Newtonian sodium alginate-based Cu–nanofluids is higher than kerosene and engine-oil-based Cu–nanofluids.

Constant

Magnetic field intensity

Eckert number

Dimensionless velocity

Magnetic parameter

Shape factor

Prandtl number

Temperature of nanofluid

Velocity component in x–direction

Velocity component in y–direction

Base fluid heat capacity

Heat capacity of nanofluid

Base fliud thermal conductivity

Nanofluid thermal conductivity

Nanoparticle thermal conductivity

Stretching parameter

Viscoplastic parameter

Similarity variable

Dimensionless parameter

Nanofluid volume fraction

Base fluids density

Nanofluid density

Nanoparticles density

Base fluids dynamic viscosity

Nanofluid dynamic viscosity

Base fluid electric conductivity

Nanofluid electric conductivity

Nanoparticles electric conductivity

Nanofluid

Base fluid

Nanoparticle

Wall

Infinity

The fourth author would like to thank University of Education, Lahore, Pakistan for the financial support.

The authors received no specific funding for this study.

The authors declare that they have no conflicts of interest to report regarding the present study.

_{2}O

_{3}–water nanofluid inside a vertical microannulus in the presence of heat generation/absorption

_{2}O nanofluid flow in a porous channel with magnetic field using mesoscopic method

_{2}O

_{3}–water/ethylene glycol with effective prandtl number impacts

_{3}O

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