Integrating the following steady-state energy balance equation for fluid flow over the total volume of the liquid in the microtube (Eq. 4 in their paper)
Display Formula

$\rho Cpvz,\u2113(r)(\u2202T\u2202z)=\mu \xaf(\u2202vz,\u2113\u2202r)2$

(1)

the following equation for the temperature rise is derived (Eq.

5 in their paper):

Display Formula$8\pi \mu \xafL(1+8(\beta /D))2v\xafz,\u21132=(\pi 4D2)\rho v\xafz,\u2113Cp\Delta T$

(2)

However, in Eq.

1, the heat conduction term is omitted. The exact expression for the energy equation is

Display Formula$\rho Cpvz,\u2113(r)(\u2202T\u2202z)=\lambda 1r\u2202\u2202r(r\u2202T\u2202r)+\mu \xaf(\u2202vz,\u2113\u2202r)2$

(3)

Integrating Eq.

3 with the boundary condition of

$(\u2202T/\u2202r)r=D/2=0$ and

$(\u2202T/\u2202r)r=0=0$, the heat conduction term vanishes and Eq.

2 is obtained. However, Eq.

2 is not the equation for the temperature rise for the adiabatic slip flow. This is the equation for the temperature rise for flow in a tube whose wall is “moving” with velocity,

$vz,slip$ [

2]. To derive the equation for the temperature rise for the adiabatic slip flow, Eq.

3 should be integrated with the following boundary condition [

2]:.

Display Formula$\lambda (\u2202T\u2202r)r=D2=-\mu vz,slip(\u2202vz,\u2113\u2202r)r=D2$

(4)

where

$vz,slip$ and

$\lambda $ are the slip velocity and the thermal conductivity of the fluid. In the case of the slip flow, the energy is transported out from the fluid to the wall even though the fluid temperature and the wall temperature are identical. The difference between the kinetic energy of molecules which arrive on the wall and that of molecules which leave from the wall results in the transported energy. Therefore, the same amount of energy, which is transported out, has to come back to the fluid when the wall is adiabatic. The right hand side of Eq.

4 expresses the energy which is transported out from the fluid to the wall, as the form of the boundary work due to shear. However, note that no work is done by the wall since wall is stationary. The correct equation for the temperature rise for the adiabatic slip flow is as follows:

Display Formula$8\pi \mu \xafL(1+8(\beta /D))v\xafz,\u21132=(\pi 4D2)\rho v\xafz,\u2113Cp\Delta T$

(5)