Interdisciplinary Applied Mathematics

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with strong electrostatic forces. If a voltage difference is applied along a microchannel, flow is initiated very close to the wall within a distance of less than 100 nm. This situation can be modeled with a slip velocity proportional to the electric field, and a continuum approach suffices to describe this flow.

1.5 Modeling of Nanoflows

Fluid flows in nanometer scale channels and pores, referred to as nanoflows, play an important role in determining the functional characteristics of many biological and engineering devices and systems. In this section, we first introduce a few applications in which nanoflows play an important role and then discuss issues in modeling nanoflows. In Chapters 10-13 we present fundamental aspects of nanoflows.

Ionic channels are naturally occurring nanotubes found in the membranes of all biological cells (Hille, 2001). They are defined as a class of proteins, and each channel consists of a chain of amino acids folded in such a way that the protein forms a nanoscopic water-filled pore controlling the transport of ions in and out of the cell, and in and out of compartments inside cells like mitochondria and nuclei, thereby maintaining the correct internal ion composition that is crucial for cell survival and function. Each channel carries a strong and steeply varying distribution of permanent charge, which depends on the particular combination of channel and prevalent physiological conditions. The narrowest diameter of an ion channel can vary from a few angstroms to tens of angstroms. For example, shown in Figure 1.25 is a gramicidin ion channel, which is one of the smallest ion channels, with a critical diameter of 4 A and a length of 25 A. Many ion channels have the ability to selectively transmit or block a particular ion species, and most exhibit switching properties similar to electronic devices. Malfunctioning channels cause or associate with many diseases, and a large number of drugs act directly or indirectly on ion channels. The possibility to incorporate ion channel structures in electronic circuits as sensing, memory, or even as computational elements opens up exciting new opportunities and great challenges. Given the physiological importance of ion channels, it is important to understand the flow of water and electrolytes in naturally existing nanoscopic pores in the presence of a strong permanent charge; see Chapters 12 and Section 13.2.

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