Based on the statistical self-similar fractal characteristics of the microstructure of porous media, the total flow rate and permeability of Newtonian fluids in the rough fracture network and rough matrix pores are derived, respectively. According to the connection structure between fractures and pores, the permeability analysis model of fluids in a matrix-embedded fracture network is established. The comparison between the predicted values of the model and the experimental data shows that the predicted values of the permeability of the rough fracture network and the rough matrix pores decrease with the increase in the relative roughness of the fractures and matrix pores, and are lower than the experimental data. Meanwhile, the predicted total flow rate of a rough fractured dual porous media is lower than that of a smooth fractal model and experimental data. In addition, it is also found that the larger the average inclination angle and the relative roughness of the fracture network, the smaller the permeability of the fractured dual porous media, and the relative roughness of the fracture network has a far greater influence on fluid permeability in the fractured dual porous media than the relative roughness of the matrix pores.

The ratio of penumbral to umbra area of sunspots plays a crucial role in the solar physics fields, especially for understanding the origin and evolution of the solar activity cycle. By analyzing the recently digitized sunspot drawings observed from Yunnan Observatories (1957–2021), we investigate the long-term variation of the penumbral to umbra area ratio of sunspots. An automatic extraction method, based on the maximum between-class variance and the morphological discrimination, is used to accurately extract penumbra and umbra and to calculate the ratio over six solar cycles (cycle 19–24). The expected value of the ratio of penumbra to umbra area is found to be 6.63 ± 0.98, and it does not exhibit any systematic variation with sunspot latitudes and phases. The average ratio fluctuates from 5 to 7.5 per year and the overall trend has decreased after 1999 compared to the previous one. The ratio of sunspot penumbra to umbra area satisfies the log-normal distribution, implying that its variation is related to the evolution of the photospheric magnetic field. Our results are consistent with previous works.

To date, the rotational characteristics of the solar transition region remain unclear. In this work, by applying the flux modulation method to the images derived from the Solar Dynamics Observatory/Atmospheric Imaging Assembly between 2011 and 2022 at 304 Å wavelength, we have studied the rotation of the solar transition region, and the results obtained are as follows. The solar transition region rotates differentially, while, from the perspective of the entire time interval, the rotation coefficients A and B are 14.39 (±0.08) and −1.61 (±0.15), respectively, and we find no prominent asymmetry in the average rotation rate of the northern and southern hemispheres. The solar transition region rotates fastest during the solar cycle maximum, and the average rotation rate follows the overall trend of solar activity. Both the equatorial rotation rate (represented by coefficient A) and the latitudinal gradient (represented by coefficient B) of the solar transition region are smaller than that of the solar chromosphere and the corona, indicating the solar transition region rotates more slowly and more rigidly than the other two layers, and we speculate that the solar chromosphere and corona seem to restrain the rotation of the solar transition region at the same time.