The Effect of Magnetic Field on Compressible Boundary Layer by Homotopy Analysis Method

We analyse the effect of applied magnetic field on the flow of compressible fluid with an adverse pressure gradient. The governing partial differential equations are solved analytically by Homotopy analysis method (HAM) and numerically by finite difference method. A detailed analysis is carried out for different values of the magnetic parameter, where suction/ injection is imposed at the wall. It is also observed that flow separation is seen in boundary layer region for large injection. HAM is a series solution which consists of a convergence parameter h which is estimated numerically by plotting <em>h</em> curve. Singularities of the solution are identified by Pade approximation.

Classifying complete $$\mathbb {C}$$-subalgebras of $$\mathbb {C}[[t]]$$

We address the problem of classifying complete $$\mathbb {C}$$ C -subalgebras of $$\mathbb {C}[[t]]$$ C [ [ t ] ] . A discrete invariant for this classification problem is the semigroup of orders of the elements in a given $$\mathbb {C}$$ C -subalgebra. Hence we can define the space $$\mathcal {R}_{\Gamma }$$ R Γ of all $$\mathbb {C}$$ C -subalgebras of $$\mathbb {C}[[t]]$$ C [ [ t ] ] with semigroup $$\Gamma $$ Γ . After relating this space to the Zariski moduli space of curve singularities and to a moduli space of global singular curves, we prove that $$\mathcal {R}_{\Gamma }$$ R Γ is an affine variety by describing its defining equations in an ambient affine space in terms of an explicit algorithm. Moreover, we identify certain types of semigroups $$\Gamma $$ Γ for which $$\mathcal {R}_{\Gamma }$$ R Γ is always an affine space, and for general $$\Gamma $$ Γ we describe the stratification of $$\mathcal {R}_{\Gamma }$$ R Γ by embedding dimension. We also describe the natural map from $$\mathcal {R}_{\Gamma }$$ R Γ to the Zariski moduli space in some special cases. Explicit examples are provided throughout.