This paper reports an analytical and numerical study of the natural convection in a horizontal porous layer filled with a binary fluid. A uniform heat flux is applied to the horizontal walls while the vertical walls are impermeable and adiabatic. The solutal buoyancy forces are assumed to be induced either by the imposition of constant fluxes of mass on the horizontal walls (double-diffusive convection, $a\,{=}\,0$) or by temperature gradients (Soret effects, $a\,{=}\,1$). The governing parameters for the problem are the thermal Rayleigh number, $R_T$, the Lewis number, $Le$, the solutal Rayleigh number, $R_S$, the aspect ratio of the cavity, $A$, the normalized porosity of the porous medium, $\varepsilon$, and the constant $a$. The onset of convection in the layer is studied using a linear stability analysis. The thresholds for finite-amplitude, oscillatory and monotonic convection instabilities are determined in terms of the governing parameters. For convection in an infinite layer, an analytical solution of the steady form of the governing equations is obtained by assuming parallel flow in the core of the cavity. The critical Rayleigh numbers for the onset of supercritical, $R_{\hbox{\scriptsize\it TC}}^{\hbox{\scriptsize\it sup}}$, or subcritical, $R_{\hbox{\scriptsize\it TC}}^{\hbox{\scriptsize\it sub}}$, convection are predicted by the present theory. A linear stability analysis of the parallel flow pattern is conducted in order to predict the thresholds for Hopf bifurcation. Numerical solutions of the full governing equations are obtained for a wide range of the governing parameters. A good agreement is observed between the analytical prediction and the numerical simulations.
It is well known that the presence of large heterogeneous textured regions in forged near alpha titanium alloys could lead to large variations of mechanical properties when fatigue and creep cycles are applied at room temperature. On the other hand, experimental studies and microtexture investigations are complex to set up, lengthy and costly, and one cannot expect to understand the alloy behavior by relying only on empirical approaches. Hence, numerical methods are excellent alternatives for analyzing the influence of microscopic and macroscopic heterogeneities on mechanical properties in shorter times and with minimum need for experimentation. In the present investigation, a cellular automata (CA) method was used to simulate the effect of texture heterogeneities, on both local and global mechanical properties. A 2D array of cells was used and the stresses and strains developed in various heterogeneous regions were evaluated using the Eshelby theory. Using the CA method, various types of microstructures were modeled and compared with each other to quantify the influence of processing parameters on mechanical properties. The results predict, and are used to explain, the experimentally phenomena observed in creep responses during cold fatigue/creep tests of near alpha titanium samples.
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