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ABSTRACT
We study the unsteady natural convection of nanofluid between the outer and inner surfaces of a concentric annulus. The inner surface is heated sinusoidally with time about a fixed mean temperature but the outer boundary is kept at a constant temperature. The effects of Brownian motion and thermophoresis are taken into account. Numerical solutions are presented using the continuous finite-element method. Results indicate that the heat and mass transfer rates are significantly influenced by inner wall temperature oscillation, which presents periodic triangle fluctuation for the considered parameters of amplitude A, inner circle radius R, and frequency F.
Nomenclature
| a, A | = | amplitude, dimensionless amplitude |
| Cp | = | specific heat at constant pressure |
| DB | = | Brownian diffusion coefficient |
| DT | = | thermophoretic diffusion coefficient |
| F | = | dimensionless oscillating frequency |
| g | = | gravitational acceleration |
| k | = | thermal conductivity |
| L | = | gap between inner and outer boundaries of the enclosure (=r0−ri) |
| Le | = | Lewis number (=α/DB) |
| Nb | = | Brownian motion parameter (=(ρc)pDB(ϕh−ϕc)/((ρc)f α)) |
| Nt | = | thermophoresis parameter (=(ρc)pDT(Th−Tc)/((ρc)f αTc)) |
| Nu | = | local surface Nusselt number |
| Nuave | = | inner surface-averaged Nusselt number |
| Nuτ | = | time-averaged Nusselt number |
| Nr | = | buoyancy ratio number |
| p, P | = | pressure, dimensionless pressure |
| Pr | = | Prandtl number (=µ/ρfα) |
| r, R | = | inner circle radius, dimensionless inner circle radius |
| Ra | = | Rayleigh number |
| Sh | = | local surface Sherwood number |
| Shave | = | inner surface-averaged Sherwood number |
| Shτ | = | time-averaged Sherwood number |
| t | = | time |
| T | = | temperature |
| u, v | = | velocity components along x- and y-axes |
| U, V | = | dimensionless velocity components in the X- and Y-directions |
| x, y & X, Y | = | space coordinates and dimensionless space coordinates |
| α | = | thermal diffusivity |
| β | = | thermal expansion coefficient |
| ζ | = | angular location |
| Θ | = | dimensionless temperature |
| µ | = | dynamic viscosity |
| ν | = | kinematic viscosity |
| ρ | = | density |
| τ | = | dimensionless time |
| ϕ | = | concentration |
| Φ | = | dimensionless concentration |
| ψ | = | stream function |
| ω | = | oscillating frequency |
| Subscripts | = | |
| ave | = | average |
| c | = | cold |
| h | = | hot |
| i | = | inner |
| loc | = | local |
| o | = | outer |
| w | = | condition on the sheet |
| τ | = | time-averaged |
| ∞ | = | condition far away from the plate |
Nomenclature
| a, A | = | amplitude, dimensionless amplitude |
| Cp | = | specific heat at constant pressure |
| DB | = | Brownian diffusion coefficient |
| DT | = | thermophoretic diffusion coefficient |
| F | = | dimensionless oscillating frequency |
| g | = | gravitational acceleration |
| k | = | thermal conductivity |
| L | = | gap between inner and outer boundaries of the enclosure (=r0−ri) |
| Le | = | Lewis number (=α/DB) |
| Nb | = | Brownian motion parameter (=(ρc)pDB(ϕh−ϕc)/((ρc)f α)) |
| Nt | = | thermophoresis parameter (=(ρc)pDT(Th−Tc)/((ρc)f αTc)) |
| Nu | = | local surface Nusselt number |
| Nuave | = | inner surface-averaged Nusselt number |
| Nuτ | = | time-averaged Nusselt number |
| Nr | = | buoyancy ratio number |
| p, P | = | pressure, dimensionless pressure |
| Pr | = | Prandtl number (=µ/ρfα) |
| r, R | = | inner circle radius, dimensionless inner circle radius |
| Ra | = | Rayleigh number |
| Sh | = | local surface Sherwood number |
| Shave | = | inner surface-averaged Sherwood number |
| Shτ | = | time-averaged Sherwood number |
| t | = | time |
| T | = | temperature |
| u, v | = | velocity components along x- and y-axes |
| U, V | = | dimensionless velocity components in the X- and Y-directions |
| x, y & X, Y | = | space coordinates and dimensionless space coordinates |
| α | = | thermal diffusivity |
| β | = | thermal expansion coefficient |
| ζ | = | angular location |
| Θ | = | dimensionless temperature |
| µ | = | dynamic viscosity |
| ν | = | kinematic viscosity |
| ρ | = | density |
| τ | = | dimensionless time |
| ϕ | = | concentration |
| Φ | = | dimensionless concentration |
| ψ | = | stream function |
| ω | = | oscillating frequency |
| Subscripts | = | |
| ave | = | average |
| c | = | cold |
| h | = | hot |
| i | = | inner |
| loc | = | local |
| o | = | outer |
| w | = | condition on the sheet |
| τ | = | time-averaged |
| ∞ | = | condition far away from the plate |