The integrable Boussinesq equation and it’s breather,lump and soliton solutions

The fourth-order nonlinear Boussinesq water wave equation, which explains the propagation of long waves in shallow water, is explored in this article. We used the Lie symmetry approach to analyze the Lie symmetries and vector fields. Then, by using similarity variables, we obtained the symmetry reductions and soliton wave solutions. In addition, the Kudryashov method and its modification are used to explore the bright and singular solitons while the Hirota bilinear method is effectively used to obtain a form of breather and lump wave solutions. The physical explanation of the extracted solutions was shown with the free choice of different parameters by depicting some 2-D, 3-D, and their corresponding contour plots.

     

CFD and Population Balance Modeling of Crude Oil Emulsions in Batch Gravity Separation—Comparison to Ultrasound Experiments

Water-in-oil emulsion destabilization and separation in a batch gravity separator was investigated experimentally and by numerical modeling. A multiphase computational fluid dynamics (CFD) was used with a population balance model (PBM) to model separation behavior of crude oil emulsions. The inhomogeneous discrete method is used to solve the population balance equations. Closure kernels are applied to model droplet–droplet coalescence. To describe the increase in emulsion viscosity with water concentration, an emulsion viscosity model was selected that predicted emulsion stability and the denser emulsion layer forming above the coalescing interface, otherwise known as the dense packed zone or layer (DPZ). The results from a commercial CFD code are compared to experimental data of the water fraction vertical distribution measured by low-power ultrasound in the batch separator. The predicted time-dependent profiles of water fraction in the separator were found to be in good agreement with the experimental measurements for the range of water content from 6 to 50%. The model predicts the effect of water fraction on the separation kinetics and the evolution of the DPZ. Further studies are underway to apply the models to emulsions from different types of crude oils.     

Exact and approximate solutions of time-fractional models arising from physics via Shehu transform

In this present investigation, we proposed a reliable and new algorithm for solving time-fractional differential models arising from physics and engineering. This algorithm employs the Shehu transform method, and then nonlinearity term is decomposed. We apply the algorithm to solve many models of practical importance and the outcomes show that the method is efficient, precise, and easy to use. Closed form solutions are obtained in many cases, and exact solutions are obtained in some special cases. Furthermore, solution profiles are presented to show the behavior of the obtained results in other to better understand the effect of the fractional order.     

Iterative methods for solving fourth- and sixth-order time-fractional Cahn-Hillard equation

This paper presents analytical-approximate solutions of the time-fractional Cahn-Hilliard (TFCH) equations of fourth and sixth order using the new iterative method (NIM) and q-homotopy analysis method (q-HAM). We obtained convergent series solutions using these two iterative methods. The simplicity and accuracy of these methods in solving strongly nonlinear fractional differential equations is displayed through the examples provided. In the case where exact solution exists, error estimates are also investigated.