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By considering a new discrete isospectral eigenvalue problem, a hierarchy of integrable positive and negative lattice models is derived. It is shown that they correspond to positive and negative power expansions of Lax operators with respect to the spectral parameter, respectively. And the equation in the resulting hierarchy is integrable in Liouville sense. Further, a Darboux transformation is established for the typical equations by using gauge transformations of Lax pairs, from which the exact solutions are given.

With the aim to optimize design, a simulation in system level has been presented for the square-wave micromixer in this article. The square-wave micromixer is divided into straight channels and square-wave units. The reduced-order model based on proper orthogonal decomposition is applied in calculating concentration of the sample in the straight channels, and numerical simulation is applied in calculating concentration of the sample in the square-wave units. The data can mutually be transferred between straight channels and square-wave units by data fitting and interpolation. The maximal relative deviation is 1.52% between simulation in system-level and only simulation. The computational efficiency will be improved significantly with the numbers of straight channels increasing. The Polymethyl methacrylate (PMMA) micromixer is fabricated with mill and hot bonding method. The mixing experiment of fluorescein sodium solution with different concentrations is carried out to verify simulation. The relative deviations between simulation in and experimental results are below 8.26%.

Transient response of phonon transport in a two-dimensional silicon thin film due to temperature disturbance at the film edge is investigated. Temperature oscillations with different frequencies are incorporated at the high-temperature edge of the film while uniform temperature is assumed initially in the film. The size of heat source, due to temperature oscillation, is varied at the film edge to investigate the coupling effect of oscillation frequency and the heat source size on the phonon transport in the film. Equivalent equilibrium temperature is introduced to assess the phonon transport characteristics for different temperature disturbance conditions. A numerical method incorporating the discrete ordinate method is used to solve the Boltzmann transport equation with the appropriate boundary conditions. It is found that equivalent equilibrium temperature oscillates in the region close to the high-temperature edge of the film; however, oscillation characteristics in terms of frequency and amplitude do not follow temperature oscillation at the film edge. Heat source size and oscillation frequency influence significantly equivalent equilibrium temperature in the film, which is more pronounced in the late heating periods.

In this paper, the numerical solutions for groundwater flow in unsaturated layered soil using the Richards equation are presented. A linearisation process for the nonlinear Richards equation to deal with groundwater flow in unsaturated layered soil is derived. To solve one-dimensional flow in the unsaturated zone of layered soil profiles, flux conservation and the continuity of pressure potential at the interface between two consecutive layers are considered in the numerical model. In addition, a novel method, named the dynamical Jacobian-inverse free method, incorporated with a two-side equilibration algorithm for solving ill-conditioned systems with extreme contrasts in hydraulic conductivity is proposed. The validity of the model is established in numerous test problems by comparing the numerical results with the analytical solutions. The results show that the proposed method can improve convergence and numerical stability for solving groundwater flow in unsaturated layered soil with extreme contrasts in hydraulic conductivity.

In the present paper, Lie symmetries of nonlinear shallow water equations with variable shapes of the bottom that include horizontal, inclined plane and a parabolic bottom are obtained. Exact particular solutions of the governing system are then obtained using the invariance of the system under these symmetries using Lie’s method. The evolutionary behaviour of the C1$${C^1}$$ discontinuity wave, influenced by the amplitude of the discontinuity wave and the geometry of the bottom, is discussed in detail and some contrasting observations are made.