Foam drainage equation in fractal dimensions: breaking and instabilities.

This paper is concerned with the construction of a phenomenological model for drainage of a liquid in foam in fractal dimensions. Our model is based on the concepts of "product-like fractal measure" introduced to model dynamics in porous media and "complex fractional transform" which converts a fractal space on a small scale to a smooth space with a large scale. The solution of the fractal foam drainage equation has been approximated using the He's homotopy perturbation method. Qualitative analysis shows that the behavior of the solitonic wave in fractal dimensions differ from the behavior in integer dimensions. This deformation generates instabilities in the foam dynamics, dispersion and spontaneous breaking of the solitonic wave.

Numerical Analysis of an Unsteady, Electroviscous, Ternary Hybrid Nanofluid Flow with Chemical Reaction and Activation Energy across Parallel Plates.

Despite the recycling challenges in ionic fluids, they have a significant advantage over traditional solvents. Ionic liquids make it easier to separate the end product and recycle old catalysts, particularly when the reaction media is a two-phase system. In the current analysis, the properties of transient, electroviscous, ternary hybrid nanofluid flow through squeezing parallel infinite plates is reported. The ternary hybrid nanofluid is synthesized by dissolving the titanium dioxide (TiO2), aluminum oxide (Al2O3), and silicon dioxide (SiO2) nanoparticles in the carrier fluid glycol/water. The purpose of the current study is to maximize the energy and mass transfer rate for industrial and engineering applications. The phenomena of fluid flow is studied, with the additional effects of the magnetic field, heat absorption/generation, chemical reaction, and activation energy. The ternary hybrid nanofluid flow is modeled in the form of a system of partial differential equations, which are subsequently simplified to a set of ordinary differential equations through resemblance substitution. The obtained nonlinear set of dimensionless ordinary differential equations is further solved, via the parametric continuation method. For validity purposes, the outcomes are statistically compared to an existing study. The results are physically illustrated through figures and tables. It is noticed that the mass transfer rate accelerates with the rising values of Lewis number, activation energy, and chemical reaction. The velocity and energy transfer rate boost the addition of ternary NPs to the base fluid.

Emergence of lump-like solitonic waves in Heimburg-Jackson biomembranes and nerves fractal model.

The aim of this study is to extend the soliton propagation model in biomembranes and nerves constructed by Heimburg and Jackson for the case of fractal dimensions. Our analyses are based on the product-like fractal measure concept introduced by Li and Ostoja-Starzewski in their attempt to explore anisotropic fractal elastic media and electromagnetic fields. The mathematical model presented in the paper is formulated for only a part of a single nerve cell (an axon). The analytical and numerical envelop soliton of this equation are reported. The results obtained prove the emergence of lump-type solitonic waves in nerves and biomembranes. In particular, these waves decay algebraically to the background wave in space direction. This scenario is viewed as a particular class of rational localized waves which are solutions of the integrable Ishimori I equation and the (2 + 1) Kadomtsev-Petviashvili I equation. The effects of fractal dimensions are discussed and were found to be significant to some extents.

Fractal Pennes and Cattaneo-Vernotte bioheat equations from product-like fractal geometry and their implications on cells in the presence of tumour growth.

In this study, the Pennes and Cattaneo-Vernotte bioheat transfer equations in the presence of fractal spatial dimensions are derived based on the product-like fractal geometry. This approach was introduced recently, by Li and Ostoja-Starzewski, in order to explore dynamical properties of anisotropic media. The theory is characterized by a modified gradient operator which depends on two parameters: R which represents the radius of the tumour and R0 which represents the radius of the spherical living tissue. Both the steady and unsteady states for each fractal bioheat equation were obtained and their implications on living cells in the presence of growth of a large tumour were analysed. Assuming a specific heating/cooling by a constant heat flux equivalent to the metabolic heat generation in the tissue, it was observed that the solutions of the fractal bioheat equations are robustly affected by fractal dimensions, the radius of the tumour growth and the dimensions of the living cell tissue. The ranges of both the fractal dimensions and temperature were obtained, analysed and compared with recent studies. This study confirms the importance of fractals in medicine.

Significant Involvement of Double Diffusion Theories on Viscoelastic Fluid Comprising Variable Thermophysical Properties.

This report examines the heat and mass transfer in three-dimensional second grade non-Newtonian fluid in the presence of a variable magnetic field. Heat transfer is presented with the involvement of thermal relaxation time and variable thermal conductivity. The generalized theory for mass flux with variable mass diffusion coefficient is considered in the transport of species. The conservation laws are modeled in simplified form via boundary layer theory which results as a system of coupled non-linear partial differential equations. Group similarity analysis is engaged for the conversion of derived conservation laws in the form of highly non-linear ordinary differential equations. The solution is obtained vial optimal homotopy procedure (OHP). The convergence of the scheme is shown through error analysis. The obtained solution is displayed through graphs and tables for different influential parameters.