Plane-Parallel Advective Flow in a Horizontal Layer of an Incompressible Fluid with an Internal Linear Heat Source

We present a new exact solution of the Navier-Stokes equations in the Oberbeck-Boussinesq approximation describing a plane-parallel advective flow in a plane horizontal layer of an incompressible fluid with solid boundaries. At the boundaries, a linear temperature distribution is defined in the presence of an internal heat source that is linear with respect to the horizontal coordinate. Examples of such solutions are given. The possibility of an analytical determination of the velocity and temperature of such flows is demonstrated. The velocity profile has not a cubic profile, which is usual for advective flows, but a more complex form depending on the source type.