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Rayleigh–Bénard convection in a horizontal layer of nanofluid in the presence of uniform vertical magnetic field is investigated by using Galerkin weighted residuals method. The model used for the nanofluid describes the effects of Brownian motion and thermophoresis. Linear stability theory based upon normal mode analysis is employed to find expressions for Rayleigh number and critical Rayleigh number. The boundaries are considered to be free–free, rigid–rigid and rigid–free. The influence of magnetic field on the stability is investigated and it is found that magnetic field stabilizes the fluid layer. It is also observed that the system is more stable in the case of rigid–rigid boundaries and least stable in case of free–free boundaries. The expression for Rayleigh number for oscillatory convection has also been derived for free–free boundaries.

In this study, we consider problems of thermal ignition and flame front propagation. Ordinary and partial differential equations with proper initial and boundary conditions are solved for generic cases. Numerical schemes are tested and results are discussed. Comparisons with the literature for benchmark cases are favorable.

Bioconvection has shown significant promise for environmentally friendly, sustainable “green” fuel cell technologies. The improved design of such systems requires continuous refinements in biomathematical modeling in conjunction with laboratory and field testing. Motivated by exploring deeper the near-wall transport phenomena involved in bio-inspired fuel cells, in the present paper, we examine analytically and numerically the combined free-forced convective steady boundary layer flow from a solid vertical flat plate embedded in a Darcian porous medium containing gyrotactic microorganisms. Gyrotaxis is one of the many taxes exhibited in biological microscale transport, and other examples include magneto-taxis, photo-taxis, chemotaxis and geo-taxis (reflecting the response of microorganisms to magnetic field, light, chemical concentration or gravity, respectively). The bioconvection fuel cell also contains diffusing oxygen species which mimics the cathodic behavior in a proton exchange membrane (PEM) system. The vertical wall is maintained at iso-solutal (constant oxygen volume fraction and motile microorganism density) and iso-thermal conditions. Wall values of these quantities are sustained at higher values than the ambient temperature and concentration of oxygen and biological microorganism species. Similarity transformations are applied to render the governing partial differential equations for mass, momentum, energy, oxygen species and microorganism species density into a system of ordinary differential equations. The emerging eight order nonlinear coupled, ordinary differential boundary value problem features several important dimensionless control parameters, namely Lewis number (Le), buoyancy ratio parameter i.e. ratio of oxygen species buoyancy force to thermal buoyancy force (Nr), bioconvection Rayleigh number (Rb), bioconvection Lewis number (Lb), bioconvection Péclet number (Pe) and the mixed convection parameter ($𝜀)$ spanning the entire range of free and forced convection. The transformed nonlinear system of equations with boundary conditions is solved numerically by a finite difference method with central differencing, tridiagonal matrix manipulation and an iterative procedure. Computations are validated with the symbolic Maple 14.0 software. The influence of buoyancy and bioconvection parameters on the dimensionless temperature, velocity, oxygen concentration and motile microorganism density distribution, Nusselt, Sherwood and gradient of motile microorganism density are studied. The work clearly shows the benefit of utilizing biological organisms in fuel cell design and presents a logical biomathematical modeling framework for simulating such systems. In particular, the deployment of gyrotactic microorganisms is shown to stimulate improved transport characteristics in heat and momentum at the fuel cell wall.