Blundell Thermal Physics · Problem 10.5

The thermal diffusion equation

Physics — Statistical Mechanics · prandtl-number-penetration-depths

Problem

For oscillations at frequency

\omega

, a viscous penetration depth

\delta_v

can be defined by

\delta_v = \left(\frac{2\eta}{\rho\omega}\right)^{1/2}, \tag{10.76}

analogously to the thermal penetration depth

\delta = \left(\frac{2\kappa}{\rho c_p \omega}\right)^{1/2} \tag{10.77}

defined in this chapter. Show that

\left(\frac{\delta_v}{\delta}\right)^2 = \sigma_p, \tag{10.78}

where

\sigma_p

is the Prandtl number (see eqn 10.59).