In this paper we analytically investigate the electroviscous effect and electrokinetic energy conversion in the time periodic pressure-driven flow of an incompressible viscous Newtonian liquid through a parallel-plate nanochannel with surface charge-dependent slip. Analytical and semi-analytical solutions for electric potential, velocity and streaming electric field are obtained and are utilized to compute electrokinetic energy conversion efficiency. The results show that velocity amplitude and energy conversion efficiency are reduced when the effect of surface charge on slip length is considered. The surface charge effect increases with zeta potential and ionic concentration. In addition, the energy conversion efficiency is large when the ratio of channel half-height to the electric double layer thickness is small. The boundary slip results in a large increase in energy conversion. Higher values of the frequency of pressure pulsation lead to higher values of the energy conversion efficiency. We also obtain the energy conversion efficiency in constant pressure-driven flow and find that the energy conversion efficiency in periodical pressure-driven flow becomes larger than that in constant pressure-driven flow when the frequency is large enough.

In this paper, the separation of variables method is applied to investigate the effects of solution pH, background salt concentration and AC electric field frequency on time periodic electroosmotic flow in a pH-regulated parallel-plate nanochannel. The surface charge is generated by the protonation and deprotonation of the functional group SiOH. The background salt is KCl. The pH value of the solution is adjusted by HCl and KOH. Analytical and semi-analytical solutions for electric potential and velocity distributions are obtained. The results show that the electric potential caused by the electric double layer depends greatly on the solution pH and background salt concentration. The amplitudes of the velocity and flow rate of the time periodic electroosmotic flow decrease with the background salt concentration and increase with the deviation of the solution pH from the isoelectric point. In a nanochannel having a height less than 100nm, the electroosmotic velocity amplitude is not affected by the AC electric field frequency because the oscillating Reynolds number is much less than unity.

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Time-periodic pressure-driven slip flow and electrokinetic energy conversion efficiency in a nanotube are studied analytically. The slip length depends on the surface charge density. Electric potential, velocity and streaming electric field are obtained analytically under the Debye–Hückel approximation. The electrokinetic energy conversion efficiency is computed using these results. The effects of surface charge-dependent slip and electroviscous effect on velocity and electrokinetic energy conversion efficiency are discussed. The main results show that the velocity amplitude and the electrokinetic energy conversion efficiency of the surface charge-dependent slip flow are reduced compared with those of the surface charge-independent slip flow.

We study Sturm-Liouville (SL) problems on an infinite number of intervals, adjacent endpoints are linked by means of boundary conditions, and characterize the conditions which determine self-adjoint operators in a Hilbert space which is the direct sum of the spaces for each interval. These conditions can be regular or singular, separated or coupled. Furthermore, the inner products of the summand spaces may be multiples of the usual inner products with different spaces having different multiples. We also extend the GKN Theorem to cover the infinite number of intervals theory with modified inner products and discuss the connection between our characterization and the classical one with the usual inner products. Our results include the finite number of intervals case.