[Lecture] Stability of the aneurysm in a membrane tube with localized wall thinning filled with a fluid with a non-constant velocity profile
May. 13, 2022
Speaker: Andrej Il'ichev, professor, Steklov Mathematical Institute of RAS
Time: 17:00-18:00 p.m., May 13, 2022, Beijing time (12:00-13:00 Moscow time)
Venue: Zoom Meeting ID: 843 9536 9673 Password: 051554
We perform the stability analysis of bulging localized structures on the wall of a fluid-filled axisymmetric membrane elastic tube. The wall of the tube is assumed to be subjected to localized thinning. The problem has no translational invariance anymore, hence the stability of a bulging wave centered in the point of the localization of imperfection is essential, and not orbital stability up to a shift as in the case of translationally invariant governing equations. Localized bulging motionless wave solutions of the governing equations are called aneurysm solutions. We assume that the fluid is subjected to the power law for viscous friction of a non-Newtonian fluid, though the viscosity of the fluid does not play a significant role and can be neglected. The velocity profile remains not constant along the cross section of the tube (even in the absence of the viscosity) because no-slip boundary conditions are performed on the tube walls. Stability is established by demonstrating the non-existence of the unstable eigenvalues of the linearized problem with a positive real part. This is achieved by constructing the Evans function depending only on the spectral parameter, analytic in the right half of the complex plane Ω+ and which zeroes in Ω+ coincide with the unstable eigenvalues of the problem. The non-existence of the zeroes of the Evans function is performed using the argument principle from the analysis of complex variables. Finally, we discuss the possibility of applying the results of the present analysis to the aneurysm formation in damaged human vessels under the action of internal pressure.
Professor Andrej Il'ichev is currently a Leading Scientific Researcher at Steklov Mathematical Institute of RAS and a Professor at Bauman Moscow State Technical University. He graduated from the Faculty of Mechanics and Mathematics of Lomonosov Moscow State University in 1981 and got his Ph.D. degree there in 1986. In 1996, he became Doctor of Sciences. Professor Il'ichev’s research field includes nonlinear waves, dissipative and dispersive systems, Hamiltonian systems, dynamical stability of bound states, solitary waves, and qualitative theory of differential equations. Also, he has published more than 100 papers and 3 monographs.
Source: Sino Russian Mathematics Center