Micromixers based on chaotic advection - Micromixers Fundamentals Design and Fabrication - page 227

FIGURE 6.35
Model of the thermocapillary actuation concept: (a) heater arrangement; (b) capillary cross-section; and
(c) heating schemes.

Neglecting radiation and heat transfer to the liquid plug, the transient energy equation is formulated
for a glass capillary as:

2

vq

vt ¼ a v

q
vx 2

2 hR o
rc R o R i q

(6.30)

where q is the temperature difference relative to the ambient temperature, R o and R i are the outer and
inner radii of the capillary, a , r , and c are the thermal diffusivity, density, and specific heat capacity of
the capillary material, respectively. The heat transfer coefficient due to free convection at the outer
surface of the capillary is estimated as h ¼ 0.631 k a /(2 R o ) with k a , the thermal conductivity of air [44] .
The periodic boundary conditions of the two heaters as depicted in Fig. 6.35 (c) are:

x¼ 0 ¼ x¼D h ¼ 0

q 0 0

d d x d d x

0 t < T h = 2

:

k ;

x¼ 0 ¼ 0 x¼D h ¼ (6.31)

q 0 0
k

d d x d d x

T= 2 < t T h :

;