Electro-osmotic pumping through a bumpy microtube: Boundary perturbation and detection of roughness

To machine precision, a micro-duct cannot be fabricated without producing surface roughness. It is of essential importance to examine the effects and predict the level of roughness on electro-osmotic (EO) pumping for ducts of fundamental shapes. In this study, we consider a bumpy microtube with its wall shape modeled by the product of two sinusoidal functions. Boundary perturbation is carried out with respect to the amplitude roughness epsilon (relative to the Debye length) up to the second-order by considering the Debye-Huckel approximation and viscous Stokes equation for the electrolyte transport. Besides the amplitude roughness epsilon, the key parameters include the azimuthal wave number n and the axial wave number alpha of the bumpiness, as well as the non-dimensional electrokinetic width K. It is shown that the EO pumping rate Q is modified by a second-order term -epsilon(2)pi chi, namely, Q = Q(0) - epsilon(2)pi chi, where Q(0) denotes the pumping rate through the smooth tube. The net effect chi = chi(1) + chi(2) comprises two components: chi(1) = chi(1)(K) < 0 increases with increasing K, representing a pure gain, while chi(2) has no definite sign and is a complex function of K, n, and alpha. In particular, chi is negative at small alpha whilst being positive at large alpha, and the dividing line of signs also depends on K. For small alpha (<1), chi increases with increasing n at all K, while for large alpha (>1), chi decreases with increasing n at large K (>20). For a given number of oscillations A(c) = n alpha (>1), there exists an intermediate n at which the EO pumping rate is maximized at small K (<20). Moreover, we identify a long-wavelength limit singularity in the EO pumping rate as alpha -> 0 for all n, i.e., in the longitudinal sense. In addition, the velocity component along the tube axis is modified by a second-order term of the roughness, though the same velocity component near the wavy wall exhibits periodic behaviors in phase with the wall roughness. Physical reasoning is given to all the derived mathematical results, and their implication in practical applications as a model for predicting tube roughness is explained. As the tube shape represents a conduit of practical use, a particular emphasis is placed upon potential applications of the derived result.