High Prandtl Number Fluids

For the case of high Prandtl number fluids, the kinematic viscosity of the fluid is much greater than its thermal diffusivity, so viscous force has a significant effect on the momentum equation and the effect of inertial force is negligible. The viscous force in fact balances the buoyancy force, and so from eq. $\begin{matrix} {{\left( \frac{L}{{{\delta }_{t}}} \right)}^{4}}\text{Ra}_{L}^{-1}{{\Pr }^{-1}}, & {{\left( \frac{L}{{{\delta }_{t}}} \right)}^{4}}\text{Ra}_{L}^{-1}, & 1 \\ \text{Inertia} & \text{Viscous} & \text{Buoyancy} \\ \end{matrix}$ from Scale Analysis, one obtains the following:

${{\delta }_{t}}\sim L\text{Ra}_{L}^{-1/4} \qquad \qquad(1)$

Substituting eq. (1) into eqs. $u\sim \alpha \frac{L}{\delta _{t}^{2}}$ and $v\sim \frac{\alpha }{{{\delta }_{t}}}$ from Scale Analysis , the scales of the velocity components become:

$u\sim \frac{\alpha }{L}\text{Ra}_{L}^{1/2} \qquad \qquad(2)$

$v\sim \frac{\alpha }{L}\text{Ra}_{L}^{1/4} \qquad \qquad(3)$

The scale of the heat transfer coefficient can be obtained by analyzing eq. ${{h}_{x}}=-\frac{k}{{{T}_{w}}-{{T}_{\infty }}}{{\left( \frac{\partial T}{\partial y} \right)}_{y=0}}$ from Governing Equations for Natural Convection, as follows:

${{h}_{x}}=\frac{-k{{\left( \frac{\partial T}{\partial y} \right)}_{y=0}}}{{{T}_{w}}-{{T}_{\infty }}}\sim \frac{k(\Delta T/{{\delta }_{t}})}{\Delta T}=\frac{k}{{{\delta }_{t}}}$

The scale of Nusselt number is therefore:

$\text{Nu}=\frac{{{h}_{x}}L}{k}\sim \frac{L}{{{\delta }_{t}}}\sim \text{Ra}_{L}^{1/4} \qquad \qquad(4)$

which will be confirmed by exact solution in the next section.

For the case of high Prandtl number fluids the momentum boundary layer thickness, δ, is much greater than the thermal boundary layer thickness, δt (see Fig. 1 from Governing Equations for Natural Convection). While the above scale analysis indicated that flow within the thermal boundary layer is dominated by balance between the buoyancy force and the viscous force, buoyancy force does not exist beyond the thermal boundary layer. Therefore, the velocity that is developed outside the thermal boundary layer but within the momentum boundary layer results from the viscous drag of the thermal boundary layer. Therefore, the flow between the thermal and momentum boundary layers is dominated by the balance between the viscous force and the inertial force. Since the thermal boundary layer is very thin for the case of $\Pr \gg 1$, there exists a balance between inertia and viscosity in the momentum equation over the entire momentum boundary layer.

Refer to Fig. 1 from Governing Equations for Natural Convection once again and consider the momentum equation of the momentum boundary layer (y˜δ) for the entire flat plate (x˜L). The thickness of the momentum boundary layer is much smaller than the length of the vertical plate, i.e. $\delta \ll L$. The force balance in the momentum boundary layer between inertial and viscous forces gives us:

$\frac{{{u}^{2}}}{L}\sim \nu \frac{u}{{{\delta }^{2}}}$

where the vertical velocity component, u, is induced by the thin thermal boundary layer. Substituting eq. (2) into the above equation yields the scale of the momentum boundary layer:

$\delta \sim L\text{Ra}_{L}^{-1/4}{{\Pr }^{1/2}} \qquad \qquad(5)$

Comparing eqs. (5) and (1) gives us the following relationship between the thicknesses of the momentum and thermal boundary layers:

$\frac{\delta }{{{\delta }_{t}}}\sim {{\Pr }^{1/2}}>1 \qquad \qquad(6)$

It is evident that the greater the Prandtl number, the greater the ratio of δ / δt. This means that the region of unheated fluid which is being driven vertically by the heated layer due to viscous action is thicker for fluids with a higher Prandtl number.