The nonstationary flow of a conducting fluid in a plane pipe in the presence of a transverse magnetic field

We consider the nonstationary flow of an incompressible viscous conducting fluid in the plane pipe of infinite length in the presence of a transverse magnetic field. Using the Laplace transformation we obtain the expressions for the fluid flow velocity and the electric and magnetic field intensities when the conductivity values of the fluid and pipe walls are arbitrary. Solutions are expressed in terms of complex integrals which are calculated for the particular case of ideally conducting walls.