## THERMAL SCIENCE

International Scientific Journal

### Thermal Science - Online First

online first only
### On two-scale dimension and its applications

**ABSTRACT**

Dimension or scale is everything. When a thing is observed by different scales, different results can be obtained. Two scales are enough for most of practical problems, and a new definition of a two-scale dimension instead of the fractal dimension is given to deal with discontinuous problems. Fractal theory considers a self-similarity pattern, which cannot be found in any a real problem, while the two-scale theory observes each problem with two scales, the large scale is for an approximate continuous problem, where the classic calculus can be fully applied, and on the smaller scale, the effect of the porous structure on the properties can be easily elucidated. This paper sheds a new light on applications of fractal theory to real problems.

**KEYWORDS**

PAPER SUBMITTED: 2019-04-08

PAPER REVISED: 2019-04-09

PAPER ACCEPTED: 2019-04-09

PUBLISHED ONLINE: 2019-04-14

- Mandelbrot, B.B. The fractal geometry of nature. Macmillan Publishers, New York, 1983.
- Liu, Y.Q., et al. Air permeability of nanofiber membrane with hierarchical structure, Thermal Science, 22(2018), 4, pp. 1637-1643
- Wang, F.Y., et al., Improvement of air permeability of Bubbfil nanofiber membrane, Thermal Science, 22(2018), 1A , pp.17-21
- Li, X.-X., et al., A fractal modification of the surface coverage model for an electrochemical arsenic sensor, Electrochimica Acta, 296(2019),pp. 491-493
- Wang, Q.L., et al. Fractal analysis of polar bear hairs, Thermal Science, 19(2015), pp. S143-S144
- Wang, Q. L., et al. Fractal calculus and its application to explanation of biomechanism of polar bear hairs, Fractals, 26(2018), article number 1850086
- Liu, F. J., et al., A delayed fractional model for cocoon heat-proof property, Thermal Science, 21 (2017),4, pp. 1867-1871
- Wang,Y. & Deng, Q. Fractal derivative model for tsunami travelling, Fractals, 27(2019), article number 1950017
- Wang, Y. & An, J.Y. Amplitude-frequency relationship to a fractional Duffing oscillator arising in microphysics and tsunami motion, Journal of Low Frequency Noise, Vibration & Active Control, doi.org/10.1177/1461348418795813
- Hu,Y., He,J.H. On fractal space and fractional calculus, Thermal Science, 20 (2016), pp. 773-777
- He, J.H., et al. A new fractional derivative and its application to explanation of polar bear hairs, Journal of King Saud University Science, 28(2016), 2, pp.190-192
- He, J.H. Fractal calculus and its geometrical explanation, Results in Physics, 10(2018), 272-276
- He, J.H., et al., Geometrical explanation of the fractional complex transform and derivative chain rule for fractional calculus, Physics Letters A, 376(2012) 257-259.
- He,J.H., Li, Z.-B. Converting fractional differential equations into partial differential equations, Thermal Science, 16 (2012), pp. 331-334.
- Li, X.X., He, J.H.. Along the evolution process, Kleiber's 3/4 law makes way for Rubner's surface law: A fractal approach. Fractals, 27(2019), 1, article number 1950015
- Tian, D., et al. Hall-Petch effect and inverse Hall-Petch effect: A fractal unification, Fractals,26 (2018),6, article number 1850083
- Zhou, C.J., et al. What factors affect lotus effect? Thermal Science, 22(2018), 4, pp.1737-1743
- Liu, P. & He J.H. Geometric potential: an explanation on of nanofibers wettability, Thermal Science, 22 (2018), 1A, pp. 33-38.
- Li, X.X., & He J.H. Nanoscale adhesion and attachment oscillation under the geometric potential Part 1: the formation mechanism of nanofiber membrane in the electrospinning, Results in Physics, 12(2019), pp.1405-1410