"Nanoscale physics"

Generalized constitutive equations and coupled phenomena at the nanoscale: applications to nanofluidics
Kizilova N.N.1
1 Department of Aplied Mathematics, V.N. Karazin Kharkov National University. Svobody sq., 4, 61022 Kharkiv, Ukraine.

Physical processes at micro and nano scales are determined by general balance laws for mass, momentum, moment of force, energy, charge and entropy, but the constitutive equations needed for them are generalized modifications accounted for the nanoscale and fast phenomena [1], namely:  
- Guyer-Krumhansl law for heat transfer ;
- Unsteady Fick’s law for diffusion ;
- Unsteady Darcy law with Brinkman term for fluid percolation in porous materials ;
- Unsteady Ohm’s law for electric charge ;
where  are corresponding fluxes,  is the relaxation time,  is the second order scale length [2]. Besides, the first/second order velocity slip and temperature jump boundary conditions must be used in micro/nanofluidics instead of classical no-slip and continuity conditions that is also a source of novel physical phenomena and unique chemical and physical properties of micro/nanofluids [3]. 
In this paper a detailed review of the coupled mass/heat/charge/chemicals transfer physical phenomena at micro/nanoscales, in the high-frequency external electromagnetic fields, and their importance for nanotechnology is given. 

1.	Petersen N.O. Foundations for Nanoscience and Nanotechnology. CRC Press. - 2017.
2.	Yang C.C., Mai Y.-W. Thermodynamics at the nanoscale // Mat Sci Engin. – 2014. - 79, N5. – P. 1-40.
3.	Cherevko V., Kizilova N. Complex flows of micro/nanofluids with velocity slip boundary conditions // Nanophysics, Nanomaterials, Interface Studies, and Applications, Springer Proceedings in Physics, V.183. - 2017. - P. 207–230.