Scaling laws for external fluid flow induced by controlled periodic heating of a solid boundary

We demonstrate that considerable variation of mean Prandtl number ($\mathrm{P}{\mathrm{r}}_0$) from unity brings in an additional length scale (called the viscous penetration depth, ${\delta }_{\mathrm{v}}$) into the dynamics of instantaneous as well as time-averaged (mean) flow induced by thermoviscous expansion along a periodically heated solid wall. We investigate the limiting cases of high and low Prandtl numbers ($\mathrm{P}{\mathrm{r}}_0\gg 1$ and $\mathrm{P}{\mathrm{r}}_0 \ll$ 1) through detailed order-of-magnitude analysis. Our study reveals that the viscous penetration depth scales universally with $\mathrm{P}{\mathrm{r}}_0$ so long as such depth remains small compared to the wavelength of the applied thermal wave. While a high $\mathrm{P}{\mathrm{r}}_0$ is found to obstruct the mean flow, the converse is not necessarily true. Subsequent analysis clearly shows that a low-$\mathrm{P}{\mathrm{r}}_0$ flow can induce negative thermoviscous force within the thermal boundary layer and thus retard the mean motion, leading to a nontrivial reduction of net mass flow along the plate. Numerical prediction of friction factor variation with $\mathrm{P}{\mathrm{r}}_0$ agrees well with the scaling estimates for both high-$\mathrm{P}{\mathrm{r}}_0$ and low-$\mathrm{P}{\mathrm{r}}_0$ fluids. The findings may very well act as fundamental design basis for engineering devices that may potentially be developed for thermal molecular trapping and particle sorting and accumulation based on unsteady heating.

Figure 1

(a) Instantaneous flow field induced by thermoviscous expansion along a traveling temperature wave applied on a flat plate. Two consecutive cycles with periodically altering hot (red) and cold (blue) regions are portrayed. Positive and negative $x$ velocities (parallel to the plate) exhibit difference in magnitude (arrow length) between the hot and cold regions. Resulting mean or time averaged flow direction is also shown. (b) Temperature wave, traveling along the plate with constant speed $c$ from right to left.

Figure 2

Oscillation of normalized $x$ velocity during the course of one cycle (time period $t_u=\lambda /c$). Every figure depicts ten velocity profiles capturing ten different instants of the cycle at equal time interval of $0.1t_u$. (a) $\mathrm{P}{\mathrm{r}}_0=54$ case; (b) enlarged view of the velocity profiles in (a) near the wall; (c) $\mathrm{P}{\mathrm{r}}_0=0.054$ case; (d) magnified view of the velocity profiles in (c) near the wall.

Figure 3

Oscillation of dimensionless $y$ velocity during the course of one cycle (time period $t_u=\lambda /c$). Every figure depicts ten velocity profiles capturing ten different instants of the cycle at equal time interval of $0.1t_u$. (a) $\mathrm{P}{\mathrm{r}}_0=54$ and (b) $\mathrm{P}{\mathrm{r}}_0=0.054$. Note the symmetry of the profiles about the midpoint ($v=0$), which actually demonstrates that the mean transverse velocity $V_{\mathrm{avg}}=0$ in the entire domain, irrespective of the Prandtl number.

Figure 4

Oscillation of dimensionless temperature $\theta =\left(T-T_0\right)/\mathrm{\Delta }T$ during the course of one cycle (time period $t_u=\lambda /c$). Every figure depicts ten temperature profiles capturing ten different instants of the cycle at equal time intervals of $0.1t_u$. (a) $\mathrm{P}{\mathrm{r}}_0=54$ and (b) $\mathrm{P}{\mathrm{r}}_0=0.054$. Note that the extents of thermal boundary layer are the same ($\simeq 2.66{\delta }_t$) for both the Prandtl numbers ($\mathrm{P}{\mathrm{r}}_0=54$ and 0.054), which are obtained by altering the thermal conductivity of water.

Figure 5

Variation of viscous boundary layer thickness relative to thermal boundary layer thickness for a wide range of mean Prandtl numbers. White circles represent CFD simulation data for water with changed values of mean viscosity and/or thermal conductivity. Since $\mathrm{P}{\mathrm{r}}_0$ is increased beyond 100 by increasing the viscosity (keeping $\lambda$ the same), the dependence of viscous boundary layer thickness (${\delta }_{\mathrm{v}}$) on $\mathrm{P}{\mathrm{r}}_0$ decreases substantially.

Figure 6

Maximum mean velocity (normalized) vs Prandtl number in the high-$\mathrm{P}{\mathrm{r}}_0$ regime. Mean velocity scale $U_s$ is taken from Eq. (21). White circles represent CFD solutions for water with modified values of mean viscosity and/or thermal conductivity.

Figure 7

(a) Time evolution of mean velocity profile near the wall for a low Prandtl number ($\mathrm{P}{\mathrm{r}}_0=0.054$) case. Mean velocity scale $U_s$ is taken from Eq. (22). (b) Net thermoviscous force distribution over the flat plate at $t=100\mathrm{ms}$ for the same case ($\mathrm{P}{\mathrm{r}}_0=0.054$). Here, ${\delta }^{*}\simeq 2.7{\delta }_t$ and ${\delta }_{\mathrm{v}}\simeq 0.24{\delta }_t$. To show the negative force zone, the force axis (horizontal) is clipped.

Figure 8

Dimensionless wall shear (time-averaged) variation with Prandtl number for a constant ${\beta }_T{\eta }_T{\left(\mathrm{\Delta }T\right)}^2$. White circles represent CFD simulation data for water with changed values of mean viscosity and/or thermal conductivity. Lines simply represent slopes ($\frac{1}{2}$ and $1\frac{1}{2}$).