On fully developed mixed convection with viscous dissipation in a vertical channel and its stability

Abstract

The fully developed regime of mixed convection in a vertical plane channel with symmetric and uniform temperature prescribed on the bounding walls is studied. The effect of viscous dissipation is taken into account and the Oberbeck-Boussinesq approximation is adopted by choosing the average fluid temperature as the reference temperature. The viscous dissipation effect induces the existence of dual branches of stationary solutions. A nonlinear stability analysis versus fully-developed modes of perturbation is carried out showing that the second branch of dual stationary solutions is unstable.

Bistable fully developed mixed convection flow with viscous dissipation in a vertical channel

“…Equation ( 2.1 ) is often stated [ 6 , 10 , 11 ] with a mean temperature in place of the fixed ambient temperature T a . Different choices produce different quantative results, but the qualitative behaviour remains the same [ 5 , 6 ]. The nonlinear optically thin thermal radiation approximation σ ( T 4 − T 4 a ) in equation ( 2.2 ) has been used before when studying the heat transfer in a medium between parallel walls [ 12 – 14 ].…”

“…If ε (or r ) is decreased, the second turning point E moves to the right towards infinity. When ε = 0, the no radiation case, there is no second turning point and no third branch and we are left with the mixed convection flows studied before [ 5 , 6 ]. As ε → 0, the flow marked by D on figure 1 a , b approaches the flow called completely passive natural convection flow with w (0) = 6.111 and θ (0) = 13.155 [ 15 ]—which are not too far from the values given in table 2 for D. However, the transition ε → 0 is rather complicated far from the origin as the radiation term is a singular perturbation at high temperatures.…”

“…Figure 4 shows that velocities monotonically increase all the way from A to F. In the no radiation case, back flows were found [ 5 , 6 ]; however, in the region of the values of parameters considered here, there are no back flows. When moving from B towards F in figure 1 , velocity profiles start having inflection points at D, just as in the no radiation case, even though this is not obvious in figure 4 .…”

“…If ε (or r ) is decreased, the unstable bump on figure 5 moves up. When ε = 0, the leading eigenvalue simply increases after the turning point [ 5 , 6 ]. Introduction of radiation ε > 0 has a stabilizing effect, which is rather obvious in view of equation ( 2.6 ).…”

“…Barletta et al [ 4 ] found dual solutions in vertical channels and the existence of a critical stress (minimum pressure gradient). Instability of one branch of flows in vertical channels was established by Miklavčič [ 5 ] and by Barletta & Miklavčič [ 6 ]. These unstable dual solutions are due to the nonlinear viscous term in the energy balance equation and are discussed in detail by Barletta [ 7 ].…”