Study On The Flow Of Viscoelastic Fluids In Ducts With Non-circular Cross Section

The flows of fluid in ducts have been concerned by many researchers because of its widespread applications in industry and daily living.The flow of fluid becomes oscillating due to the action of periodic pressure gradient,which is called oscillatory flow.The curvature of the curved duct generates centrifugal forces,which cause the secondary flow of the fluid.This paper focuses on the flow of Maxwell fluid in the ducts with different shapes,including rectangular cross-sectional straight duct,isosceles right triangle cross-sectional straight duct and rectangular cross-sectional curved duct.In the first part,the oscillatory flow of Maxwell fluid in a long tube with rectangular cross section is considered.The analytical expressions for velocity profile and phase difference are obtained.Furthermore,the numerical solutions for the velocity field are obtained by using a finite difference scheme method.The stability of this finite difference scheme method is also discussed.The effects of relaxation time and Deborah number on the velocity profile and phase difference are discussed numerically and graphically.It turns out that the the oscillatory flow is induced by the elastic effect of Maxwell fluid,the intensity of which is positively related to relaxation time.The oscillatory flow will not be developed due to the limitation of the size and aspect ratio of cross section.And when the size of cross section is sufficiently large,the variation of the axial velocity field results from the interaction of relaxation time and Reynolds number,both of which are used to measure the relative magnitude of elastic,viscous and inertial effect.For the second study,the oscillatory flow of Maxwell fluid in a long tube with isosceles right triangular cross section is considered.The analytical expressions for the velocity and phase difference are obtained explicitly for the flow driven by the periodic pressure gradient.The numerical solutions are calculated by using a high-order compact finite difference method.The effects of relaxation time and Deborah number on the velocity and phase difference are discussed numerically and graphically.It is found that the distribution of velocity is always symmetrical with respect to the median of the hypotenuse of the triangular cross section.The increased relaxation time or the Deborah number strengthens the elastic effect of the Maxwell fluid,which leads to the enhancement of oscillatory flow.However,the dominant elasticity results in the dissipation of energy,which postpones the flow of the Maxwell fluid.In the third part,the incompressible flow of a Maxwell fluid through a curved duct with rectangular cross section is numerically investigated over a wide range of Dean number and curvature of the duct.Unsteady solutions,such as periodic,multi-periodic and chaotic solutions,are obtained by using the spectral method.The combined effects of large Dean number,Deborah number and curvature on fluid flow behaviors are discussed in detail.It is found that increasing Deborah number accelerates the occurrence of the four-cell structure of secondary flow,no matter what the Dean number is.Periodic solutions are found to appear for the case with a smaller Dean number due to the presence of elasticity.The periodic solution turns to a chaotic solution if the Dean number is further increased.The chaotic solution is weak for smaller Deborah number,while it becomes strong for larger Deborah number.