Laboratory experiments have shown that thermal gradients in silicate melts can lead to isotopic fractionation; this is known as the Richter effect. However, it is perplexing that the Richter effect has not been documented in natural samples as thermal gradients commonly exist within natural igneous systems. To resolve this discrepancy, theoretical analysis and calculations were undertaken. We found that the Richter effect, commonly seen in experiments with wholly molten silicates, cannot be applied to natural systems because natural igneous samples are more likely to be formed out of partially molten magma and the presence of minerals adds complexity to the behaviour of the isotope. In this study, we consider two related diffusion-rate kinetic isotope effects that originate from chemical diffusion, which are absent from experiments with wholly molten samples. We performed detailed calculations for magnesium isotopes, and the results indicated that the Richter effect for magnesium isotopes is buffered by kinetic isotope effects and the total value of magnesium isotope fractionation can be zero or even undetectable. Our study provides a new understanding of isotopic behaviour during the processes of cooling and solidification in natural magmatic systems.

In this investigation, we intend to present the influence of the prominent viscous dissipation and Soret effects on mixed convection heat and mass transfer over the vertical frustum of a cone in a nanofluid. The model used for the nanofluid incorporates the effects of Brownian motion and thermophoresis. In addition, the uniform wall nanoparticle condition at the surface is replaced with the zero nanoparticle mass flux condition to execute physically applicable results. The governing equations of a nanofluid flow in the dimensional form are reduced to a system of partial differential equations in the non-dimensional form by using suitable non-similarity variables and then solved by using a recently introduced spectral method named as Bivariate Pseudo-Spectral Local Linearisation Method (BPSLLM). The convergence and error analysis tests are conducted to examine the accuracy of the spectral method. To validate the method, the present numerical results are compared with the existing results in some special cases and the outcomes are observed to be in very good agreement. The effects of Brownian motion, thermophoresis, Eckert number, Soret number, nanoparticle and regular buoyancy parameters on the dimensionless surface drag, heat, nanoparticle mass and regular mass transfer rates over the vertical frustum of a cone are discussed and illustrated graphically.

We have explored the combined long-wave Marangoni and Rayleigh instability ofthe quiescent state of a binary- liquid layer heated from below or from above in the presenceof the Soret effect. We found that in the case of small Biot numbers there are two long-wave regions of interest k ~ Bi 1/2 and k ~ Bi 1/4. The dependence of both monotonic andoscillatory thresholds of instability in these regions on both the Soret and dynamic Bondnumbers has been investigated. The complete linear stability analysis reveals the diversityof instability types in the long-wave region, and a need in the development of the nonlineartheory of the discovered phenomena becomes obvious.

Our aim here is to study the thermal diffusion phenomenon in a forced convective flow. A system of nonlinear parabolic equations governs the evolution of the mass fractions in multicomponent mixtures. Some existence and uniqueness results are given under suitable conditions onstate functions. Then, we present a numerical scheme based on a "mixed finite element"method adapted to a finite volume scheme, of which we give numerical analysis. In a last part, we apply an homogenization technique to the studied equations in order to obtain an efficient modelling of Soret effect and adsorption in a porous medium at a macroscopic scale.