ahmed rashed

One of the most important natural phenomena that has been studied extensively in engineering, oceanography,meteorology, and other fields is called fluid turbulence (FT). FT stands for irregular flow of fluid. Scientists detected models to describe this phenomenon, among these models is the (3+1)-dimensional Vakhnenko-Parkes (VP) equation. In this research, the high-frequency waves’ dynamical behavior through the relaxation medium is explored by considering two semi-analytic methods, the $(G^'/G)$ and the tanh-coth (TC) expansion methods. Nineteen different solutions have been detected and some of these solutions have been illustrated graphically. Figures show a range of degenerate, periodic, and complex propagating soliton wave solutions.

This study utilizes two robust methodologies to examine the precise solutions of the Dirac integrable system. The Homogeneous Balance Method (HB) is initially employed to generate an accurate solution. The system of equations for the quasi-solution is solved, where all the equations are of the same nature. The quasi-solution of the traveling wave results in the solitary wave solution of the system. The singular manifold method (SMM) is utilized following the Lie reduction of the Dirac system in order to search for the traveling wave solutions of the system. Both approaches demonstrate the existence of traveling wave solutions inside the system. The precise solutions of the Dirac system are shown in three-dimensional graphs. We have created solutions to the examined problem, including bright solutions, periodic soliton solutions, and complicated solutions.

This work was motivated by studying the behavior of nanofluid adjacent to a moving vertical plate. A non-homogeneous distribution of nanoparticles inside the boundary layer was considered with variable Brownian and thermal diffusion coefficients throughout the layer. Employing group similarity transformation method transformed the governing mathematical model into a system of ordinary differential equations. The resultant system was numerically solved using shooting method. The numerical investigation was carried out for different parameters namely: Prandtl number, Pr, temperature difference ratio, Υ, and the ratio of nanoparticles volumetric fraction difference, Υφ, and the attained results were illustrated graphically to examine their effect on different fluid characteristics. The results showed that increasing Pr values decreased the nanofluid velocity, shear stress, temperature distribution and nanoparticles volumetric fraction, while it increased the heat flux and nanoparticles gradient inside the boundary layer. On the other hand, increasing Υ values increased the nanofluid velocity, shear stress and heat flux but it decreased the temperature distribution. Also, increasing Υφ values decreased the nanofluid velocity, shear stress and temperature distribution but it increased the heat flux. The characteristics of nanofluids were studied to enhance the thermal conductivity and the efficiency of heat transfer systems. A comparison between the obtained results and the previous published results indicated an excellent agreement.