Bioconvective flow of fourth-grade nanofluid via a stretchable porous surface with microbial activity and activation energy

The current work scrutinizes electro-magnetized mixed convective flow of radiative fourth-grade nanofluid over a stretchable surface with bioconvection mechanism, thermophoretic body force, Brownian motion and activation energy for both nanoparticles (NPs) and motile microorganisms. Also, features of viscous dissipation, fluctuating fluid viscosity, Joule heating, suction/injection, heat source/sink and convective heat-mass-microbial conditions are included in the flow model. By employing suitable dimensionless variables, the set of model equations has been transformed into non-dimensional ordinary differential equations, which are subsequently tackled using quasilinearization-based spectral collocation approach applied in overlapping grids. For different flow parameters, numerical results of flow profiles and quantities of engineering intrigue are discussed. The results reveal that the inclusion of Biot numbers for heat, mass and microbes transfer helps to enhance flow profiles along with quantities of engineering interest. The material parameters enhance the magnitude of velocity, which is also higher for the fourth-grade nanomaterial model. Better heat transfer attributes can be achieved through strong thermophoretic force, thermal radiation, heat generation, viscous dissipation and NP Brownian motion. Activation energy increases NP concentration, while a decreasing trend is notable for higher chemical reaction and NP Brownian diffusion parameters. Bioconvection parameters reduce the motile microbes concentration, while enhancing the rate of motile microbes transfer. Microbial Brownian motion and microbial reaction play a crucial role in the dynamics of microorganisms, and that is confirmed by their noticeable effect on the flow properties

Numerical solution of time-dependent Emden-Fowler equations using bivariate spectral collocation method on overlapping grids

In this work, we present a new modification to the bivariate spectral collocation method in solving Emden-Fowler equations. The novelty of the modified approach is the use of overlapping grids when applying the Chebyshev spectral collocation method. In the case of nonlinear partial differential equations, the quasilinearisation method is used to linearize the equation. The multi-domain technique is applied in both space and time intervals, which are both decomposed into overlapping subintervals. The spectral collocation method is then employed in the discretization of the iterative scheme to give a matrix system to be solved simultaneously across the overlapping subintervals. Several test examples are considered to demonstrate the general performance of the numerical technique in terms of efficiency and accuracy. The numerical solutions are matched against exact solutions to confirm the accuracy and convergence of the method. The error bound theorems and proofs have been considered to emphasize on the benefits of the method. The use of an overlapping grid gives a matrix system with less dense matrices that can be inverted in a computationally efficient manner. Thus, implementing the spectral collocation method on overlapping grids improves the computational time and accuracy. Furthermore, few grid points in each subinterval are required to achieve stable and accurate results. The approximate solutions are established to be in excellent agreement with the exact analytical solutions