Magnetic diffusion of time distributed-order Maxwell fluid in boundary layer under the action of induced magnetic field

Abstract

Hydrogel has been widely used in energy storage devices and flexible electronic equipment. To promote its applications, a laminar boundary layer model is proposed to analyze the velocity and temperature distributions in the mold. Based on the time distributed-order Maxwell and Cattaneo constitutive relations, the research establishes governing equations of unsteady two-dimensional incompressible viscoelastic electrically conducting hydrogel taking into account the induced magnetic field. When the magnetic Reynolds number is large enough, it is necessary to consider the magnetic diffusion in the boundary layer. The effects of thermal radiation and velocity slip are also considered at the same time. The Gauss quadrature rule is used to approximate the distributed-order integral, and then the numerical solution of the model is obtained by using the finite difference method and the L1-algorithm. The analytical solution is constructed to verify the effectiveness of the numerical solution. The results show that the fluid velocity decreases with the increment of magnetic parameter, while the induced magnetic field increases. The boundary layer thickness of velocity, induced magnetic field, and thermal becomes thinner with the enlargement of velocity and temperature relaxation time parameters. Moreover, the distributed-order and the fractional constitutive models are compared through different weight coefficients, and it is found that the fractional constitutive model obtains larger velocity and temperature distributions.