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The significance of an externally applied magnetic field and an imposed negative temperature gradient on the onset of natural convection in a thin horizontal layer of alumina-water nanofluid for various sizes of spherical alumina nanoparticles (e.g., 30 , 35 , 40 , 45) and volumetric fractions (e.g., 0.01, 0.02, 0.03, 0.04) is explored and analyzed numerically in this paper. The generalized Buongiorno's mathematical model with the simplified Maxwell's equations and the Oberbeck-Boussinesq approximation were adopted to simulate the two-phase transport phenomena, in which the Brownian motion and thermophoresis aspects are taken into account. Moreover, the rheological behavior of alumina-water nanofluid and related flow are assumed to be Newtonian, incompressible and laminar. Based on the linear stability theory, the perturbed partial differential equations (PDEs) of magnetohydrodynamic convective nanofluid flow are firstly simplified formally using the normal mode analysis technique and secondly converted to a generalized eigenvalue problem considering more realistic boundary conditions, in which the thermal Rayleigh number is the associated eigenvalue. Additionally, the resulting eigenvalue problem was solved numerically using powerful collocation methods, like Chebyshev-Gauss-Lobatto Spectral Method (CGLSM) and Generalized Differential Quadrature Method (GDQM). Furthermore, the thermo-magneto-hydrodynamic stability of the nanofluidic system and the critical size of convection cells are highlighted graphically in terms of the critical thermal Rayleigh and wave numbers, for various values of the magnetic Chandrasekhar number, the volumetric fraction and the diameter of alumina nanoparticles.

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In the current study, the effects of radial magnetic field, slip and jump conditions on the steady two-dimensional free convective boundary layer flow over an external surface of an isothermal sphere for an electro-conductive polymer are numerically studied. It is assumed that the studied fluid has a non-Newtonian rheological behavior and follows the Carreau fluid model. In this investigation, the formulation of the Carreau fluid model has been used first time for describing the present boundary layer problem, and then the resulting partial differential equations are transformed to ordinary differential equations by using non-similarity transformations. The obtained ordinary differential equations are solved numerically by a well-known method named as Keller-Box method. The finding results show that a weak elevation in temperature is accompanied with the increase in the Carreau fluid parameter, whereas a significant acceleration in the flow is computed near the sphere surface. It is shown also that an increase in the thermal slip parameter allows to strongly decrease both the skin friction coefficient and the local Nusselt number. The skin friction coefficient is also depressed with increasing magnetic body force parameter. Moreover, it is observed that an increase in the momentum slip parameter allows to decrease the skin friction coefficient, whereas the local Nusselt number is reduced with the increase in the Carreau fluid parameter. It is found also that the skin friction coefficient is increased with greater stream-wise coordinate, whereas the local Nusselt number is reduced with the increase in this parameter.

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In this paper, the MHD peristaltic flow inside wavy walls of an asymmetric channel is investigated, where the walls of the channel are moving with peristaltic wave velocity along the channel length. During this investigation, the electrical conductivity both in Lorentz force and Joule heating is taken to be temperature dependent. Also, the long wavelength and low Reynolds number assumptions are utilized to reduce the governing partial differential equations into a set of coupled nonlinear ordinary differential equations. The new set of obtained equations is then numerically solved using the generalized differential quadrature method (GDQM). This is the first attempt to solve the nonlinear equations arising in the peristaltic flows using this method in combination with the Newton-Raphson technique. Moreover, in order to check the accuracy of the proposed numerical method, our results are compared with the results of built-in Mathematica command NDSolve. Taking Joule heating and viscous dissipation into account, the effects of various parameters appearing in the problem are used to discuss the fluid flow characteristics and heat transfer in the electrically conducting fluids graphically. In presence of variable electrical conductivity, velocity and temperature profiles are highly decreasing in nature when the intensity of the electrical conductivity parameter is strengthened.

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