Linear stability of nanofluid boundary-layer flow over a flat plate

Abstract

The linear stability of nanofluid boundary-layer flow over a flat plate is investigated using a two-phase model that incorporates Brownian motion and thermophoresis, building upon the earlier work of Buongiorno (2006). Solutions to the steady boundary-layer equations reveal a thin nanoparticle concentration layer near the plate surface, with a characteristic thickness of $O({\Rey}^{-1/2}{Sc}^{-1/3})$, for a Reynolds number $\Rey$ and Schmidt number $Sc$. When Brownian motion and thermophoresis are neglected, this nanoparticle concentration layer disappears, resulting in a uniform concentration across the boundary layer. Neutral stability curves and critical conditions for the onset of the Tollmien--Schlichting wave are computed for a range of nanoparticle materials and volume concentrations. Results indicate that while the effects of Brownian motion and thermophoresis are negligible, the impact of nanoparticle density is significant. Denser nanoparticles, such as silver (Ag) and copper (Cu), destabilise the Tollmien--Schlichting wave, whereas lighter nanoparticles, like aluminium (Al) and silicon (Si), establish a small stabilising effect. Additionally, stability characteristics are influenced by the viscosity model. Finally, a high-Reynolds number asymptotic analysis is undertaken for the lower branch of the neutral stability curve.

Figures (16)

• Figure 1: Diagram of a nanofluid flow, composed of a base fluid ($bf$) and nanoparticles ($np$) over a flat plate. Here, $\delta^*$ represents the boundary-layer thickness.
• Figure 2: ($a$) Non-dimensional dynamic viscosity $\mu$ as a function of $\phi_{\infty}$, for the Brinkman1952, Batchelor1977, PakCho1998, and Maiga2004 models. ($b$) Non-dimensional density $\rho$, specific heat capacity $c$, and thermal conductivity $k$ as a function of $\phi_{\infty}$, for copper (Cu) nanoparticles in water. Refer to table \ref{['Table1']} for fluid and nanoparticle properties.
• Figure 3: Steady base flow profiles for variable $\phi_{\infty}$ and $T_w = 2$, for copper (Cu) nanoparticles in water. (a) Streamwise velocity $U_B=f'(\xi)$, (b) $U_B'=f"(\xi)$, (c) temperature $T_B=\theta(\xi)$, (d) $T_B'=\theta'(\xi)$, (e) nanoparticle volume concentration $\phi_B=\varphi(\xi)$, and (f) $\phi'_B=\varphi'(\xi)$. Dotted lines depict the equivalent solutions in the instance $Le\rightarrow\infty$ and $Sc\rightarrow\infty$.
• Figure 4: (a) Displacement thickness $\delta_1$, (b) momentum thickness $\delta_2$, and (c) shape factor $H$ as functions of the free-stream nanoparticle volume concentration $\phi_{\infty}$, for different nanoparticle materials.
• Figure 5: (a) Thermal displacement thickness $\delta_1$ and (b) concentration displacement thickness $\delta_{\phi}$ as functions of the free-stream nanoparticle volume concentration $\phi_{\infty}$, for different nanoparticle materials.