Study On The Stability Of Liquid Film Flowing Down A Tube

The dynamics of falling films is a ubiquitous and important phenomenon that occurs extensively in nature and some industrial applications,so it has attracted much attention.Motivated by the airway closure dynamics in pulmonary capillaries,this investigation addresses the dynamics of annular viscoelastic films flowing down a flexible tube.Since the mucus(film)can be treated as a viscoelastic fluid and the free surface usually contaminated by an insoluble pulmonary surfactant,the effect of viscoelasticity and the effect of surfactant are focused.The instability characteristics of a viscoelastic liquid film coating the inner surface of a tube are studied first.It turns out that the effect of surface tensions destabilizes long wavelength perturbations but stabilizes short wavelength perturbations.The presence of an insoluble surfactant will weaken the effect of surface tensions.The fluid viscoelasticity stabilizes long wavelength perturbations but destabilizes short wavelength perturbations.Therefore,a window of parameters can be found in the presence of base flow,in which the flow is linear stable.Physically,this can be attributed to the competition and interaction between the instability mechanisms caused by fluid viscoelasticity and surface tension.A complementary long wavelength approximation is also carried out to find the necessary condition for the linear stable film flow.Based on Shkadov's integral boundary layer method,a set of nonlinear evolution equations is derived that is valid for flows with moderate Reynolds numbers.The linear stability property of the system is examined by using normal-mode analysis.The results indicate that the fluid viscoelasticity plays an unstable role in the stability of the annular film flow.The tube flexibility,which includes wall damping and wall tension,plays a dual role.A bifurcation analysis is performed,and the families of steady traveling waves are cataloged.It is found that the stiffness of the tube tends to stimulate the interfacial capillary ripples.The fluid viscoelasticity acts to strengthen the dispersion of the interfacial waves but weakens the interfacial capillary ripples.The spatio-temporal evolutions of the system are also solved numerically.When the tube radius is small enough,airway closure can be observed due to the Plateau-Rayleigh instability.The fluid viscoelasticity acts to promote airway closure while the tube radius is relatively small.Considering the flexibility of the tube wall,a modified lubrication method is proposed.And a set of nonlinear evolution equations is derived that is valid for flows with low Reynolds numbers.The linear stability analysis is carried out,and it turns out that the surfactant plays a dual role in the stability of film flow.A stable island is identified in some Marangoni number,which is attributed to the combination of the effects of surfactant mode and surface mode.Moreover,a numerical simulation is carried out under periodic boundary conditions.The regular pattern for the bifurcation of the periodic solution is presented.The effect of surfactant has a significant influence on the pattern,particularly,the bifurcation is promoted with moderate Marangoni number.However,the influence of the wall flexibility to the pattern is not obvious.