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In this paper, a meshfree point collocation method, with a upwinding scheme, is presented to obtain the numerical solution of the coupled equations in velocity and magnetic field for the fully developed magnetohydrodynamic (MHD) flow through a straight pipe of rectangular section with insulated walls. The moving least-square (MLS) approximation is employed to construct shape functions in conjunction with the framework of point collocation method. Computations have been carried out for different applied magnetic field orientations and different Hartmann numbers from 5 to 1,000,000. As the adaptive upwinding local support domain is introduced in the meshless collocation method, numerical results show that the method can compute MHD problems not only at low and moderate values but also at high values of the Hartmann number with high accuracy and good convergence.

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In this paper, a meshless local Petrov-Galerkin (MLPG) method is given to obtain the numerical solution of the coupled equations in velocity and magnetic field for the steady magnetohydrodynamic (MHD) flow through a straight duct of rectangular section. Mehdi Dehghan has applied MLPG method to solve the MHD flow control equations at Hartmman numbers less than 40. The integration subdomain is properly adjusted to carry out the computations for Hartmman numbers from 5 to 400. Numerical results show that the MLPG method with adjusted integration subdomains can compute MHD problems not only at low values but also at moderate values of the Hartmann number with good accuracy and convergence.

Numerical simulation on the liquid metal mangetohydrodynamic (MHD) flow is a very attractive way to study the MHD flow problems. The two-level radial point interpolation method is applied to study the fully developed MHD flow in this work. With the method, numerical simulations of the MHD flows at Hartmann numbers up to 10 4 in a duct with arbitrary wall conductance ratios, namely, from thin conducting to insulated wall ducts, are conducted. The non-dimensional flowrate for different electrical conductivities of the duct wall are listed and compared with the exact solutions, showing a good accuracy. These results also indicate that the higher electrical conductivity of the duct wall, the lower non-dimensional flowrate, which is consistent with the well-known MHD flow behavior. Numerical simulations for insulated walls with conducting cracks or different orientation of external applied magnetic field are also carried out with the two-level radial point interpolation method.

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