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A numerical study of micropolar flow inside a lid-driven triangular enclosure

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Abstract

A study is carried out to analyze the mixed convection flow and heat transfer inside a lid-driven triangular conduit under the effects of micro-gyration boundary conditions. The micropolar constitutive equation characterizes the fluid inside the cavity. The lower boundary is at a uniform temperature and sliding in its plane with constant velocity u0, while the inclined walls are cold. Dual cases are considered here, namely the intense concentration (d) and the weak concentration of microelements (\(m = 0.5\)). The governing nonlinear equations are simulated employing the Galerkin finite element method, where the pressure term is handled via the Penalty approach. Using the numerical data, graphical results are produced to illustrate the effects of physical parameters. Specifically, this refers to the effects of the Grashof number (Gr), Prandtl number (Pr), Reynolds number (Re) and vortex viscosity parameter (K) on the streamlines, mid-section velocity profiles, temperature contours, and local and average Nusselt numbers on the cold and heated boundaries of the conduit. Particular emphasis is given on the identification of the set of parameters for which simultaneous symmetry in streamlines and isotherms prevails. The grid independence test is also performed by comparing the average Nusselt numbers (on the hot and cold boundaries of the conduit) for various mesh sizes, and the optimal solution is found. Moreover, the results are also benchmarked with the previously published data.

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Abbreviations

\(Pr\) :

Prandtl number

\(Gr\) :

Grashof number

\(R\) :

Residual of weak form

\(N\) :

Dimensional micro-rotation vector

\(Nu\) :

Local Nusselt number

\(Nu_{C}\) :

Nusselt number at inclined cold walls of cavity

\(Nu_{H}\) :

Nusselt number at hot wall of cavity

\(m\) :

Micro-gyration parameter

\(p\) :

Dimensional pressure

T :

Dimensional temperature

\(T_{H}\) :

Temperature of heated hot surface

\(T_{C}\) :

Temperature of cold walls

\(u\) :

Dimensional longitudinal velocity

\(v\) :

Dimensional transverse velocity

\(\bar{u}\) :

Dimensionless longitudinal velocity component

\(\bar{v}\) :

Dimensionless transverse velocity component

K :

Ratio of vortex and dynamic viscosities

\(\kappa\) :

Vortex viscosity

\(k\) :

Thermal conductivity of fluid

\(\bar{p}\) :

Dimensionless pressure

\(\bar{N}\) :

Dimensionless micro-rotation vector

\(j\) :

Micro-rotation per unit mass

\(Re\) :

Reynolds number

g :

Magnitude of gravitational acceleration

\(J\) :

Jacobian matrix of residual system

\(\alpha\) :

Viscosity gradient coefficient

β :

Viscosity gradient coefficient

γ :

Viscosity gradient coefficient

\(\beta_{0}\) :

Volume expansion coefficient

\(\nu\) :

Kinematic viscosity

\(\rho\) :

Density of fluid

\(\mu\) :

Dynamic viscosity

\(\bar{\theta }\) :

Dimensionless temperature

\(\psi\) :

Stream function

\(j\) :

Residual number

\(\delta\) :

Penalty parameter

\(\phi\) :

Shape function

\(b\) :

Bottom wall of the cavity

\(C\) :

Cold wall of the cavity

\(H\) :

Hot wall of the cavity

\(i\) :

Node number

\(s\) :

Sidewall of the cavity

\(n\) :

Total number of nodes

\(e\) :

Element number

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Authors and Affiliations

  1. Department of Mathematics and Statistics, International Islamic University, Islamabad, 44000, Pakistan

    N. Ali, Mubbashar Nazeer & T. Javed

  2. Department of Mathematics and Computer Sciences, Stetson University, Deland, FL, 32723, USA

    F. Abbas

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  1. N. Ali
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  2. Mubbashar Nazeer
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  3. T. Javed
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  4. F. Abbas
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Correspondence to Mubbashar Nazeer.

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Ali, N., Nazeer, M., Javed, T. et al. A numerical study of micropolar flow inside a lid-driven triangular enclosure. Meccanica 53, 3279–3299 (2018). https://doi.org/10.1007/s11012-018-0884-5

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  • DOI: https://doi.org/10.1007/s11012-018-0884-5

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