Iterative network model to predict the behaviour of a Sierpinski fractal network

The topography of rough surfaces strongly influences the conduction of heat and electricity between two surfaces in contact. Roughness measurements on a variety of surfaces have shown that their structure follows a fractal geometry whereby similar images of the surface appear under repeated magnification. Such a structure is characterized by the fractal dimension D, which lies between 2 and 3 for a surface and between 1 and 2 for a surface profile. This paper uses the fractal characterization of surface roughness to develop a new network model for analyzing heat conduction between two contacting rough surfaces. The analysis yields the simple result that the contact conductance h and the real area of contact At are related as h ~ AtD/2 where D is the fractal dimension of the surface profile. Contact mechanics of fractal surfaces has shown that At varies with the load F as At ~ Fη where η ranges from 1 to 1.33 depending on the value of D. This proves that the conductance and load are related as h ~ FηD/2 and resolves the anomaly in previous investigations, which theoretically and experimentally obtained different values for the load exponent. The analytical results agreed well with previous experiments although there is a tendency for overprediction.

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Spontaneous Imbibition in a Fractal Network Model with Different Wettabilities

Water ◽

10.3390/w13172370 ◽

2021 ◽

Vol 13 (17) ◽

pp. 2370

Author(s):

Shaobin Cai ◽

Li Zhang ◽

Lixin Kang ◽

Yongfei Yang ◽

Wenlong Jing ◽

...

Keyword(s):

Mathematical Model ◽

Network Model ◽

Radius Ratio ◽

Spontaneous Imbibition ◽

Length Ratio ◽

Shape Model ◽

Capillary Bundle ◽

The Relationship ◽

Fractal Network ◽

Optimal Radius

In this work, we derived a mathematical model for spontaneous imbibition in a Y-shaped branching network model. The classic Lucas–Washburn equation was used for modeling the imbibition process occurring in the Y-shape model. Then, a mathematical model for the Newtonian fluid’s imbibition was derived to reveal the relationship between dimensionless imbibition time and length ratio, radius ratio, and wetting strength. The dimensionless imbibition time in the model was adopted to compare with that of the capillary bundle model. Different length and radius ratios were considered in the adjacent two-stage channels, and different wettabilities were considered in the different branches. The optimal radius ratio, length ratio, and wetting strength were calculated under the condition of the shortest imbibition time. In addition, the shortest dimensionless imbibition time of the three-stage Y-shaped branching network model was calculated when the wettability changes randomly. The results indicate that the imbibition time changed mostly when the wettability of the second branch changed, and the second branch was the most sensitive to wettability in the model.

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