The present work examines how plane waves are reflected within an isotropic, homogeneous, incompressible porous thermoelastic plane incorporating a spatial-temporal nonlocal elastic formulation together with hyperbolic two-temperatures (HTT) effects under impedance boundary conditions. The primary objective is to examine how nonlocal interactions and generalized heat conduction influence wave propagation and energy partition in porous thermoelastic media. The mathematical formulation is developed within the generalized Lord–Shulman thermoelastic framework combined with Klein–Gordon-type nonlocal elasticity. By employing Helmholtz decomposition and normal mode analysis, the governing equations are reduced, from which closed-form relations for the reflection coefficients and associated energy ratios are obtained. Numerical simulations demonstrate that the HTT parameter and both spatial and temporal nonlocal parameters significantly modify the distribution of reflected wave energy while satisfying the law of energy conservation. In particular, temporal nonlocality exerts a stronger influence on energy partition than spatial nonlocality, and the HTT model exhibits marked deviations from the classical one-temperature and two-temperature formulations. These findings provide a comprehensive multiscale framework for accurately characterizing wave reflection phenomena in porous thermoelastic media and offer valuable insights for applications in geomechanics, seismology, ocean engineering, petroleum engineering, biomedical imaging, and nanoscale thermoelastic analysis.
This paper investigates the properties and operator-theoretic representations of the heterogeneous r-Stirling numbers of the first and second kind. Building upon the recent introduction of heterogeneous Stirling sequences, we extend these concepts to the r-Stirling case and characterize them using the framework of boson creation (a +) and annihilation (a) operators. We derive explicit normal ordering results for the operators ⟨a+a⟩n,λ(a+)r ${\langle {a}^{+}a\rangle }_{n,\lambda }{({a}^{+})}^{r}$ and ⟨a+a+r⟩n,λ ${\langle {a}^{+}a+r\rangle }_{n,\lambda }$ , showing that their coefficients are given by the heterogeneous r-Stirling numbers of the second kind. Furthermore, we establish several new recurrence relations for both kinds of heterogeneous r-Stirling numbers using the underlying Heisenberg–Weyl algebra. These results provide a unified combinatorial and algebraic representation of these generalized sequences.
Container shape and operating parameters are among the important factors that determine the efficiency of phase change material integrated energy systems along with the material. In this study, phase transition performance of encapsulated phase change material in L, U and T shaped enclosures are numerically investigated under spatially varying magnetic field. The numerical analysis is conducted for a range of Hartmann number (Ha) values (from 0 to 80), magnetic field inclination (γ) (from 0 to 90), and non-uniform amplitude (Amp) (from 0 to 0.9). To obtain the maximum liquid fraction amount for a specified time period, an optimization study is conducted to determine the optimal magnetic field parameters for each of the cavities. It is observed that magnetic field controls the vortex number and its distribution for each of the cavities. Among different cavities, the lowest impact on transition time (t-C) is found for T-shaped cavity. By varying strength of magnetic field, t-C varies by roughly 21.4 % and 23.4 % for L and U-shaped cavities while they become 13 % and 11 % with varying inclination of magnetic field. The amplitude of non-linear part of magnetic field contributes to phase change dynamics. The highest amount of liquid fraction for L, U and T-shaped cavities is achieved at optimal values of (Ha, γ, Amp) = (33.33, 46.5, 0.825), (56.07, 6.35, 0.783), and (79.11, 80.03, 0.836), respectively by using optimization. When compared to the reference scenario of using pure fluid without magnetic field, the optimization provides the largest improvement in the liquid fraction amount for T-shaped cavities, followed by U and L-shaped cavities. The increment amounts for these enclosures are found as 18.3 %, 12.3 %, and 10.4 %. Using nanofluid is found to be beneficial for accelerating the process for all cavities. The outcomes show that enclosure type is very important for phase change process dynamics when varying operating parameters.
In the current investigation, comparison between a flow which combines a wall jet and an offset jet, named dual jets (DJ), and two free parallel jets (PJ) is performed considering the same jet’s spacing to jet width ratio d/w = 1. The number of Reynolds (Re) calculated based on the jet width w is considered equal to 10,000. Two-dimensional (2D) unsteady RANS equations have been adopted to simulate both the flow configurations. As clearly depicted from the mean flow finding in the current investigation, far away from the nozzle plate (x = 0), the two jets in PJ configuration merge into a single free jet, while in DJ configuration, they combine to form a single classical wall jet. For the two configurations, the time-dependent computation results show a periodic vortex shedding (PVS) phenomenon near the ejection plate. This phenomenon is like what would be observed behind the flow over 2D bluff body. The dimensionless Strouhal number (St) representing the frequency of vortex shedding in the case of PJ (St = 0.24) is noted to be higher than that for the DJ (St = 0.22).
This study investigates the effect of the Ni/Ti ratio on the thermo-structural and functional properties of NiTiCoB shape memory alloys with nominal compositions of Ni42−x Ti53+x (Co4.5B0.5) (x = 0–2.0 at.%), while maintaining constant Co and B contents. The alloys were produced by vacuum arc melting followed by homogenization at 900 °C for 24 h. Differential scanning calorimetry (DSC) was used to characterize phase transformation behavior and estimate apparent activation energies using the Kissinger and Ozawa methods. X-ray diffraction (XRD), optical microscopy, and Vickers microhardness measurements were employed for structural and microstructural characterization. The DSC results showed an overall tendency toward higher transformation temperatures with increasing nominal Ti content, although individual transformation temperatures exhibited non-monotonic variations. The apparent activation energies decreased overall from 395.7–380.87 kJ mol−1 to 269.2–260.57 kJ mol−1 according to the Kissinger and Ozawa methods, respectively. XRD revealed B2–NiTi, B19′–NiTi, NiTi2, and α–Ti-related phases, while the XRD-derived crystallite size generally decreased with increasing nominal Ti content. Microhardness exhibited an overall increase, reaching 325 ± 2 HV0.3. A preliminary single-specimen shape-recovery test indicated approximately 68 % recovery. These findings demonstrate that the nominal Ni/Ti ratio influences the thermal, structural, mechanical, and functional behavior of NiTiCoB alloys under fixed Co and B contents.
