Asymptotic models of flow in a pipe with compliant walls

For a rotationally symmetric vortex-free flow of a viscous incompressible fluid in a compliant tube based on the shallow water theory (Lagrangian approach), a nonlinear amplitude equation describing the behavior of finite perturbations in the vicinity of waves propagating along the characteristics is constructed. It is assumed that the flow occurs in an infinite cylindrical region having a free surface on which kinematic and dynamic conditions are satisfied, taking into account surface tension. The characteristic size of the cylindrical region in the axial direction is considered to be much larger than the characteristic size in the radial direction. It is found that in the case of the considered vortex-free flow (the Navier-Stokes equations), the flow equations do not contain terms that take into account viscosity (coincide with the equations of an ideal incompressible fluid - Euler equations). The influence of fluid viscosity is taken into account only due to the dynamic boundary condition at the boundary. The amplitude equation has the form of the Korteweg-de Vries--Burgers equation, the solution of which is well studied by analytical, asymptotic and numerical methods. The coefficients of the equation are calculated and, depending on their values, a qualitative analysis of the behavior of perturbations was carried out. The constructed amplitude equation and the quasi-linear hyperbolic equations arising in the process of construction as the main term of the asymptotics, as well as equations for finite perturbations, can be used to describe the flow of a fluid jet and/or blood flow in the aorta. In principle, both quasi-linear equations, and the amplitude equation, and equations for finite perturbations, obtained, as a rule, using the averaging method, are known and widely used, in particular, for modeling blood flow. However, when constructing well-known models using the averaging method, a large number of heuristic assumptions are used, often poorly substantiated. The method of constructing models proposed in the presented paper is mathematically more correct and does not contain any assumptions except for the requirement formulated in the problem statement about the vortex-free nature of the flow and the order of smallness of the parameters (viscosity, surface tension). In addition, a comparison of the obtained equations with the equations of the averaging method is given. From a mathematical point of view, the constructed flow models are equations for determining the main and subsequent terms of the asymptotics.