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Mathematical Modeling of Natural Convection in a Third-Grade Viscoelastic Micropolar Fluid from an Isothermal Inverted Cone

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Iranian Journal of Science and Technology, Transactions of Mechanical Engineering Aims and scope Submit manuscript

Abstract

A mathematical model is developed to investigate the nonlinear, isothermal steady-state free convection boundary layer flow in an incompressible third-grade viscoelastic micropolar fluid from a vertical isothermal cone. The non-Newtonian fluid exhibits both viscoelastic and micro-structural characteristics and is representative of certain polymers. The transformed conservation equations for mass, linear momentum, angular momentum and heat are solved numerically subject to realistic boundary conditions using the second-order accurate implicit finite-difference Keller-box method. The numerical code is validated with previous studies and also with a Nakamura tridiagonal method. Detailed interpretation of the computations is included. The present simulations are of interest in chemical engineering systems and solvent and low density polymer materials processing.

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Abbreviations

A :

Half angle of the cone

B :

Micropolar inertia density (material) parameter

C f :

Skin-friction coefficient

f :

Dimensionless stream function

Gr x :

Local Grashof number

g :

Acceleration due to gravity

j :

Microinertial density

K :

Vortex viscosity parameter

k :

Thermal conductivity of the fluid

N :

Angular velocity

Nu :

Local Nusselt number

Pr :

Prandtl number

r :

Local radius of the truncated cone

T :

Fluid temperature

u, v :

Dimensionless velocity components along the x- and y-directions, respectively

V :

Velocity vector

V w :

Arbitrary reference velocity

x :

Stream wise coordinate

y :

Transverse coordinate

\(\alpha\) :

Thermal diffusivity

\(\beta\) :

Coefficient of thermal expansion

\(\varepsilon_{1}\) :

First viscoelastic material fluid parameter

\(\varepsilon_{2}\) :

Second viscoelastic material fluid parameter

\(\beta_{3}\) :

Third-grade material parameter

\(\nu\) :

Kinematic viscosity

\(\rho\) :

Fluid density

\(\mu\) :

Newtonian dynamic viscosity

\(\eta\) :

Dimensionless radial coordinate

\(\theta\) :

Dimensionless temperature

\(\phi\) :

Third-grade dimensionless viscoelastic fluid parameter

\(\gamma^{*}\) :

Spin gradient viscosity

\(\xi\) :

Dimensionless tangential coordinate

\(\psi\) :

Dimensionless stream function

w:

Surface conditions on cone (wall)

∞:

Free stream conditions

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Authors and Affiliations

  1. Department of Mathematics, Salalah College of Technology, Salalah, Oman

    S. Abdul Gaffar

  2. Gort (Process/Aerospace Fluid Dynamics), Gabriel’s Wing House, 11 Rooley Croft, Bradford, BD61FA, West Yorks, UK

    O. Anwar Bég

  3. Department of Mathematics, School of Advanced Science, VIT University, Vellore, India

    V. Ramachandra Prasad

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  1. S. Abdul Gaffar
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  2. O. Anwar Bég
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  3. V. Ramachandra Prasad
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Correspondence to S. Abdul Gaffar.

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Gaffar, S.A., Bég, O.A. & Prasad, V.R. Mathematical Modeling of Natural Convection in a Third-Grade Viscoelastic Micropolar Fluid from an Isothermal Inverted Cone. Iran J Sci Technol Trans Mech Eng 44, 383–402 (2020). https://doi.org/10.1007/s40997-018-0262-x

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