Consider a planar microchannel of width $h,$ as shown (it is actually very long in the $x$ direction and open at both ends A Cartesian coordinate system with its origin positioned at the center of the microchannel is used in the study. The microchannel is filled with a weakly conductive solution. When an electric current is applied across the two conductive walls, the current density, $\vec{J},$ transmitted through the solution is parallel to the $y$ axis. The entire device is placed in a constant magnetic field, $\vec{B},$ which is pointed outward from the plane (the $z$ direction $),$ as shown. Interaction between the current density and the magnetic field induces a Lorentz force of density $\vec{J} \times \vec{B}$. Assume that the conductive solution is incompressible, and since the sample volume is very small in lab-on-a-chip applications, the gravitational body force is neglected. Under steady state, the flow driven by the Lorentz force is described by the continuity (Eq. 5.1a) and Navier-Stokes equations (Eqs. 5.27 ), except the $x, y$, and $z$ components of the latter have extra Lorentz force components on the right. Assuming that the flow is fully developed and the velocity field $\vec{V}$ is a function of $y$ only, find the three components of velocity.