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The application of the nanoparticles for the heat transfer augmentation has extensively increased in the scientific and industrial applications. In this research, semi-analytic method is used to disclose the heat transmission and flow feature of the fluid with nanoparticles among the two parallel sheets. In our model, one plate is warmed with specific heat flux while fluid is streamed from another plate which extends over times. Nanoparticles of Al₂O₃ are applied in the main fluid to obtain nanofluid flow. To obtained viscosity coefficient and heat conductivity of the base fluid with nanoparticles, Koo–Kleinstreuer–Li (KKL) formula is applied as reliable approach. Comprehensive investigations on different factors are done to disclose the impact of important aspects such as volume fraction of the nanoparticles, main stream velocity and expansion ratio on the main thermal and hydrodynamic characteristics of the nanofluid. It was found that the rate of the Nusselt number upsurges when the velocity of main stream, volume portion of the nanoparticles and power law index is increased. However, the increasing of the expansion ratio declines the heat transfer rate in our model. Our findings disclose that heat transfer rate is directly proportional with velocity of nanofluid as index of power law equals to zero.

The current analysis deals with the physiological flow of hybrid nanofluid within an expanding and contracting vertical annulus embedding with a permeable medium. Additionally, in the free convective viscous flow, the impact of thermal radiation on the heat transport phenomenon is vital. The significance of this research lies in the utilization of Hamilton–Crosser thermal conductivity to investigate the behavior of nanoparticles with different shapes. In this study, both oxide nanoparticles like Al₂O₃ and metal nanoparticles like Ag are incorporated into the blood, which serves as the base fluid. This incorporation aims to enhance the performance of hybrid nanofluids in influencing flow phenomena. The proposed mathematical model designed for the above-mentioned constraints is solved analytically following the transformation into their corresponding non-dimensional form. The statistical behavior of the diversified parameters is presented graphically and the numerical results for the rate coefficients are deployed via table. Further, the important characteristics of the results are depicted as follows: the combined effect of the particle concentration used in hybrid nanofluid favors in reducing the fluid velocity greatly than that of nanofluid and pure fluid.

This investigation aims to examine the resolution of the generalized nonlinear time fractional Harry Dym $(𝔾\mathbb{N}𝕋𝔽ℍ𝔻)$ equation through a combined application of analytical and numerical methodologies. The primary objective is to scrutinize the equation’s behavior and present proficient methodologies for its resolution. The Khater II analytical technique and numerical frameworks, specifically the Cubic-B-spline, Quantic-B-spline, and Septic-B-spline schemes, are proposed for this purpose. The outcomes of this inquiry yield substantial revelations regarding the characteristics of $𝔾\mathbb{N}𝕋𝔽ℍ𝔻$ equation. Both the analytical approach and numerical schemes demonstrate their efficacy in yielding precise solutions. These findings carry noteworthy implications across various disciplines, encompassing physics, mathematics, and engineering. The investigated model appears in plasma physics research on soliton theory and nonlinear waves. Soliton waves, which keep their shape and velocity while propagating, are found in plasma physics and other domains. In plasma environments, the Harry Dym equation describes these solitons and their behavior. Solitons are essential for understanding plasma dynamics, including nonlinear waves and structures. They help comprehend plasma dynamics, wave interactions, and other nonlinear processes. The principal deductions drawn from this study underscore the effectiveness and viability of the proposed techniques in resolving the $𝔾\mathbb{N}𝕋𝔽ℍ𝔻$ equation. This research introduces innovative contributions in terms of insights and methodologies pertinent to analogous nonlinear fractional equations. Its scope encompasses nonlinear dynamics, fractional calculus, and numerical analysis.

We present an analytical approach that is based on nonlinear programming to perform VLSI standard cell placement. Our method first clusters a netlist to reduce the number of cells and then performs quadratic optimization on the reduced netlist. Finally, it uses Nesterov’s method for solving nonlinear equations for the problem. The framework of our tool, Kapees3, is scalable and generates high quality results. The experimental results for Peko Suite 1 and Peko Suite 2 benchmarks show promising improvements. Our placement tool outperforms NTUPlace3, Dragon, Feng Shui, Capo10.5, by 46%, 57%, 48% and 25%, respectively, on PEKO Suite 1. For PEKO Suite 2, our placement tool outperforms NTUPlace3, Dragon, Feng Shui, Capo10.5 and mPL6 by 30%, 47%, 57%, 69% and 2.7%, respectively. On MMS benchmarks, we obtain wirelength improvement over Capo10.5 by 56.62%, FLOP by 7.84%, FastPlace by 11.55%, ComPLx by 4.58%, POLAR by 23.67%, mPL6 by 9.96% and NTUPlace3-Unified by 2.96%.