Jeffery–Hamel flow of hybrid nanofluids in convergent and divergent channels with heat transfer characteristics


Described in this article is a numerical study on the Jeffery–Hamel flow of hybrid fluid consisting of Copper and Graphene oxide nanoparticles. The nanofluid are considered as single-phase fluid and their heat transport performance is also investigated. The flow is carried out in a convergent/divergent channel where the channel walls can stretch or shrink. Additionally, the impact of magnetic field on flow and heat transfer analysis are examined. The flow governing equations are modeled under the Boussinesq approximations using cylindrical polar coordinates. These partial differential equations have been changed into system of ordinary differential equations by means of dimensionless formulation. Later, the numerical solution of governing problem is obtained with a developed code in MATLAB software which employs boundary-value problem solver (bvp4c). The effects of eminent flow parameters on skin friction coefficient, Nusselt number, velocity, and temperature distributions in case of both convergent and divergent channels are plotted and investigated. It is concluded from current analysis that the local skin friction coefficient significantly reduces with higher magnetic parameter. We further observed that the fluid velocity increases with increasing values of Reynolds number in case of convergent channel, while an inverse is noted for divergent channel. The present review indicates that the rate of heat transfer has been enhanced due to greater Prandtl number.

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\(u\) :

Velocity component in radial direction

\(\left( {r,\,\theta ,\,z} \right)\) :

Cylindrical polar coordinates

\(p\) :


\(\alpha\) :

Inclination angle

\(B\left( r \right)\) :

Variable magnetic field

\(\,B_{0}\) :

Magnetic field strength

\(T\) :

Fluid temperature

\(\rho_{{{\text{hnf}}}}\) :

Hybrid nanofluid’s density

\(\mu_{{{\text{hnf}}}}\) :

Hybrid nanofluid’s dynamic viscosity

\(k_{{{\text{hnf}}}}\) :

Hybrid nanofluid’s thermal conductivity

\(\left( {c_{p} } \right)_{{{\text{hnf}}}}\) :

Hybrid nanofluid’s specific heat

\(U_{\max }\) :

Velocity at centerline of channel

\(T_{{\text{w}}}\) :

Surface temperature

\(\mu_{{\text{f}}}\) :

Base fluid’s viscosity

\(\rho_{{\text{f}}}\) :

Base fluid’s density

\(\sigma_{f}\) :

Base fluid’s electric conductivity

\(\varphi_{{{\text{Cu}}}}\) :

Copper’s volume fraction

\(\varphi_{{{\text{Go}}}}\) :

Graphene oxide volume fraction

\(x\) :

Similarity variable

\(f\left( x \right)\) :

Dimensionless velocity

\(g\left( x \right)\) :

Dimensionless temperature

\(f_{\max }\) :


\(Re\) :

Reynolds number

\(\Pr\) :

Prandtl number

\(M\) :

Magnetic parameter

\(C_{{{\text{fx}}}}\) :

Skin friction coefficient

\(Nu_{{\text{x}}}\) :

Nusselt number

\(\;\tau_{{{\text{w}}}}\) :

Surface shear stress

\(\;q_{{{\text{w}}}}\) :

Surface heat flux

\({\text{hnf}}\) :

Hybrid nanofluid

\({\text{f}}\) :

Base fluid


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Hafeez, M., Hashim & Khan, M. Jeffery–Hamel flow of hybrid nanofluids in convergent and divergent channels with heat transfer characteristics. Appl Nanosci (2020).

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  • Jeffery–Hamel flow
  • Hybrid nanofluid
  • Convergent/divergent channel
  • Heat transfer
  • Magnetic field
  • Numerical solution