Kelvin’s theorem implies that irrotational flow will remain irrotational if the following four restrictions are satisfied:

• There are no net viscous forces along C. If C moves into regions where there are net viscous forces such as within a boundary layer that forms on a solid surface, then the circulation changes. The presence of viscous effects causes diffusion of vorticity into or out of a fluid circuit and consequently changes the circulation.
• The body forces are conservative. Conservative body forces such as gravity act through the center of mass of a fluid particle and therefore do not generate torques that cause fluid particle rotation.
• The fluid density must depend on pressure only (barotropic flow). A flow will be barotropic if the fluid is homogenous and one of the two independent thermodynamic variables is constant. Isentropic, isothermal, and constant density conditions lead to barotropic flow. Flows that are not barotropic are called baroclinic. Here fluid density depends on the pressure and the temperature, composition, salinity, and/or concentration of dissolved constituents.
  

  Consider fluid elements in barotropic and baroclinic flows.
  
  For the barotropic element, lines of constant \(p\) are parallel to lines of constant \(\rho\), which implies that the resultant pressure forces pass through the center of mass of the element.
  
  For the baroclinic element, the lines of constant \(p\) and \(\rho\) are not parallel. The net pressure force does not pass through the center of mass, and the resulting torque changes the vorticity and circulation.

• The frame of reference must be an inertial frame. As described in fluid motion in a noninertial frame of reference, the conservation of momentum equation includes extra terms when the frame of reference rotates and accelerates, and these extra terms were not considered in the short proof given above.