Dynamic Analysis of Multi-Stepped Functionally Graded Carbon Nanotube Reinforced Composite Plate with General Boundary Condition

This study presents the multi-stepped functionally graded carbon nanotube reinforced composite (FG-CNTRC) plate model for the first time, and its free and forced vibration is analyzed by employing the domain decomposition method. The segmentation technique is employed to discretize the structure along the length direction. The artificial spring technique is applied to the structural boundary and piecewise interface for satisfying the boundary conditions and the combined conditions between subplates. Based on this, the boundary conditions of subdomains could be considered as a free boundary constraint, reducing the difficulty in constructing the allowable displacement function. Since all the structures of subdomains are identical, the allowable displacement functions of them can be uniformly constructed using the two-dimensional ultraspherical polynomial expansion. The potential energy function of the plate is derived from the first-order shear deformation theory (FSDT). The allowable displacement function is substituted into the potential energy function, and then the natural frequencies and mode shapes of the multi-stepped FG-CNTRC plate are decided by using the Rayleigh–Ritz method. The accuracy and reliability of the proposed method are confirmed by the results of the previous literature and finite element method (FEM). On this basis, the influences of the geometric and material parameters on free and forced vibration of the multi-stepped FG-CNTRC plate are also studied.

Problem for an ordinary differential equation with a general boundary condition

For an ordinary differential equation of fractional order with a general boundary condition, a general representation of the solution of the equation is found, a condition of unique solvability is found, and an explicit representation of the solution is constructed.

Gas Flow in Microchannels and Nanochannels With Variable Cross Section for All Knudsen and All Mach Number Values

The analytical solution for steady viscous pressure-driven compressible isothermal gas flow through micro- and nanochannels with variable cross section for all Knudsen and all Mach number values is presented in this paper. The continuum one-dimensional governing equations are solved using the friction factor that is established in a special way to provide solutions for mass flow rate, pressure, and velocity distribution through the microchannels and nanochannels in the entire rarefaction regime. The friction factor, defined by the general boundary condition and generalized diffusion coefficient proposed by Beskok and Karniadakis (1999, “A Model for Flows in Channels, Pipes, and Ducts at Micro and Nano Scales,” J. Microscale Thermophys. Eng., 3, pp. 43–77), spreads the solution application to all rarefaction regimes from continuum to free molecular flow. The correlation between the product of friction factor and Reynolds number (Poiseuille number) and Knudsen number is established explicitly in the paper. Moreover, the obtained solution includes the inertia effect, which allows the application of the solution to both subsonic and supersonic gas flows, which was not shown earlier. The presented solution confirms the existence of the Knudsen minimum in the diverging, converging, and microchannels and nanochannels with constant cross section. The proposed solution is verified by comparison with experimental, analytical, and numerical results available in literature.

Decrease of Nozzle Clogging through Fluid Flow Control

Decreasing the clogging deposition rate of alumina inclusions in continuous casting nozzles is possible through three simultaneous measures: Flow modification, use of raw materials with low impurities contents, and smoothed internal surfaces. The control of the internal flow consists on avoiding dead regions and developing symmetric patterns. A mathematical model performed tests of the feasibility of these measures. The adherence of inclusions to the nozzle wall, using this model, employs a boundary condition based on the thickness of the sublaminar boundary instead of the conventional “trap” boundary condition. The use of the general boundary condition yields deposition rates that are unaffected by the inclusion size. The proposed boundary condition discriminates against the clogging deposition rate through the particle sizes. Plant trials complemented with water modeling, using these nozzles, proved that the present approach could considerably decrease the clogging occurrence.