## Abstract

Heat-exchanger tubes subjected to cross-flow experience a fluid elastic instability (FEI) at a critical flow velocity. Baffle plates are often used to provide additional stiffness to long tubes. A cladding is provided between the baffle plates and the tubes to allow for thermal expansion, and the cladding acts like a soft spring. In this paper, we study the contact between the cladding and the tube when the stiffness of the cladding is asymmetric. The tube is modeled as a cantilever Euler–Bernoulli beam, and the fluid forces are represented using an added mass, added damping and a time-delayed displacement term. This results in a partial delay differential equation model for the tube vibration. Using the Galerkin approximation and considering only a single mode, a delay differential equation (DDE) model is developed. However, due to the presence of the delay, the single mode model is also infinite-dimensional. This model contains sign function nonlinearity due to the asymmetry in the contact stiffness of the cladding. The DDE model is essentially nonlinear—that is, no linear approximation exists. However, its solutions are scalable: if q(t) is a solution, then $α q(t)$ is also a solution for any $α>0$. By exploiting this scalability property of solutions, we use Lyapunov-like exponents to probe the stability of the nonlinear DDE. Further, by numerical continuation, we numerically demonstrate the existence of periodic solutions on the stability boundary. Our stability charts will be beneficial to the designers of heat-exchangers.

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