# Role of the Induced Magnetic Field on Dispersed CNTs in Propylene Glycol Transportation Toward a Curved Surface

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## Abstract

This article addresses velocity slip and induced magnetic field effects on heat transfer and nanofluid flow toward a curved stretching sheet. For a comprehensive and realistic analysis most commonly, used fluids such as water and propylene glycol are taken as the base fluids and single- and multi-wall carbon nanotubes (CNTs) are treated as the nanoparticles. Carbon nanotubes are considered due to their unusual characteristics like extraordinary thermal conductivity, viscosity, heat capacity, density, etc., which are significant in nanotechnology, electronics and material sciences. Moreover, heat transfer is discussed in the presence of heat generation and radiative heat flux. The curvilinear coordinates system has opted for mathematical formulation. The governing system of equations is simplified by employing similarity analysis. Computational analysis is performed on a reduced system by means of shooting method in computational software MATLAB. The computed distributions of velocity, induced magnetic field and temperature are examined for pertinent emerging parameters graphically. Furthermore, bar charts are drawn for the skin friction coefficient and Nusselt number. Some of the major findings include: velocity increased with an increase in surface bendiness. Moreover, the increase in velocity was reported to be slightly more in case of MWCNTs. Fluid flow decelerated with an increase in slip velocity. Reciprocal magnetic Prandtl number contributed to decelerating fluid flow. Moreover, magnetic parameter and \(\gamma \) played a significant role in increasing fluid velocity. The presence of CNTs contributed to accelerating fluid flow. Furthermore, the temperature was a decreasing function of surface bendiness. Moreover, the maximum temperature was observed in the case of SWCNTs–PG, while the minimum temperature was reported for MWCNTs–\({\text {H}}_2{\text {O}}\). Temperature rose for rising values of slip parameter. Reciprocal magnetic Prandtl number, heat generation parameter, thermal radiation and CNTs volumetric fraction contributed in up surging fluid temperature. Furthermore, the Nusselt number was an increasing function of \(\phi \), \(\beta \) and *Rd*, whereas the contrary results were noted for *k*, \(\kappa \) and *N*. Moreover, the maximum value of Nusselt number was reported for SWCNTs–\({\text {H}}_2{\text {O}}\). Also, skin friction increased for rising values of \(\phi \), whereas it decreased with an increase in *k*, \(\kappa \) and \(\beta \). Furthermore, maximum and minimum skin friction was noted for SWCNTs in case of both base fluids, respectively. The new-fangled results of the present investigation may be valuable in edifying research and in ceramic, plastic and polymer industry.

## Keywords

SWCNTs and MWCNTs Induced magnetic field Thermal radiation Internal heat generation Curvilinear transport \({\text {H}}_2{\text {O}}\) and PG base fluid## List of Symbols

- \(f^{\prime }\)
Dimensionless fluid velocity

- \(g^{\prime }\)
Dimensionless induced magnetic field

*P*Pressure

*Pr*Prandtl number

*N*Heat generation parameter

*Rd*Radiation parameter

*k*Curvature parameter

*u*,*v*Longitudinal and transverse components of velocity

- \(H_1\), \(H_2\)
Longitudinal and transverse induced magnetic field

- \(H_0 \)
Upstream uniform magnetic field at infinity

- \(q_\mathrm{r} \)
Radiative heat flux

*l*Slip length

- \(k_\mathrm{nf} \)
Thermal conductivity of nanofluid

*L*Dimensional constant

- \(C_\mathrm{p} \)
Specific heat

- \(Cf_\mathrm{s} \)
Local skin friction coefficient

- \(Nu_\mathrm{s} \)
Local Nusselt number

- \(Re_\mathrm{s} \)
Local Reynolds number

- SWCNT
Single-wall carbon nanotube

- MWCNT
Multi-wall carbon nanotube

- PG\(\left( {{\text {C}}_3 {\text{ H }}_8 {\text{ O }}_2 } \right) \)
Propylene glycol

- \({\text{ H }}_2 {\text{ O }}\)
Hydrogen oxide (water)

## Greek Symbols

- \(\eta \)
Dimensionless space variable

- \(\theta \)
Dimensionless temperature

- \(\lambda \)
Reciprocal magnetic Prandtl number

- \(\beta \)
Magnetic parameter

- \(\mu \)
Dynamic viscosity

- \(\gamma \)
Dimensionless parameter

- \(\sigma ^{*}\)
Stefan–Boltzmann constant

- \(\phi \)
Particle volume fraction

- \(\alpha \)
Thermal diffusivity

- \(\nu \)
Kinematic viscosity

- \(\rho \)
Density

- \(\kappa \)
Slip parameter

## Subscripts

- w
Wall condition

- \(\infty \)
Condition at infinity

- nf, f, s
Nanofluid, fluid, solid nanoparticles

- CNT
Carbon nanotube

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