A Study On The Stability Of Two-dimensional Flows Of Shear-thinning Fluids

The stability of two-dimensional flows of non-Newtonian fluids in plane or annular channelshas important value in both theoretical studies and practical applications. As one of the basic featuresof non-Newtonian fluids, the shear-thinning property has drawn lots of research focus during the lastdecades. In addition, studies on the flow stability under crossflow are significantly meaningful tounderstand the effects of blowing and suction through porous walls as an efficient flow controlmethod.In the present work, flows in plane channels under crosslfow as well as axial flows throughconcentric annuli are considered, which are driven by both the wall motion and the pressure gradient.Applied fluid models include widely used power-law fluids and Bingham fluids. Carreau fluids,which present characteristics closer to real fluids, are also taken into consideration. Exact solutionsfor plane Couette-Poiseuille flow of power-law fluids under uniform crossflow, as well as axialCouette-Poiseuille flow of Bingham fluids through concentric annuli, are derived. With the stabilityequations established for power-law fluids and Carreau fluids in plane channel flow situation, studieson the long-term development and short-term transient growth of flow disturbances are performedusing modal and non-modal approaches. Effects of the shear-thinning property and crossflow onmodal and non-modal stability are analyzed.As the exact solution for plane Couette-Poiseuille flow of power-law fluids under uniformcrossflow shows, the basic flow can maintain Couette velocity profile as long as a relation betweencrossflow and pressure gradient is satisfied. Therefore, the basic flow is not influenced by crossflowand shear-thinning property, which affect the flow stability only by additional inertial and viscousterms in the stability equation. Modal stability analyses demonstrate that the shear-thinning propertydestabilizes the flow, while crossflow destabilizes the flow first and then stabilizes it. The criticalcrossflow Reynolds number that makes the stable flow unstable tends towards a constant as thestreamwise Reynolds number increases. A linear relation between the constant for a power-law fluidand that for a Newtonian fluid is discovered. Non-modal stability analyses demonstrate that theshear-thinning property enhances transient growth, while crossflow weakens it.When the restriction between crossflow and pressure gradient is removed, the basic flow is nolonger uninfluenced by crossflow and shear-thinning property. As studies on the stability of planeCouette-Poiseuille flow of Carreau fluids under uniform crossflow shows, crossflow stabilizes theflow first, and then destabilizes it, and stabilizes it eventually. The alteration of the basic flow induces an extra lower critical crossflow Reynolds number that makes the unstable flow stable,which contains both long-wave and short-wave instability mechanism. The critical crossflowReynolds number that makes the stable flow unstable contains long-wave instability mechanism only.The competition between stabilization of altering the basic flow and destabilization of an additionalviscous term in the stability equation is discovered, both of which originate from the shear-thinningproperty. Non-modal stability analyses demonstrate that the shear-thinning property still enhancestransient growth, while crossflow enhances it first and then weakens it. The enhancement effect ofcrossflow comes from the alteration of the basic flow.As the exact solution for axial Couette-Poiseuille flow of Bingham fluids through concentricannuli shows, there are two flow cases existing in annular geometry, which vanish in plane geometry.The physical mechanism of these two flow cases is illustrated.