A general form of starting solutions of linear viscoelastic fluids for Couette flow

Couette flow, namely the flow of a fluid caused by moving boundary, is one of the fundamental problems in non-Newtonian fluid mechanics. Based on Laplace transform and the orthogonal basis function expansion, we provide a general method to solve the velocity field of the generalized linear viscoelastic fluid in Couette flow. This method can be applicable to the generalized cross sections and the generalized linear viscoelastic fluids. Through the method, the solution of Couette flow for the rectangular pipe and the generalized Burgers fluid can be obtained, which were rarely discussed before. The analytical solution consists of a quasi-steady solution and an approaching part, which is expressed by the sum of approaching terms. The analytical solution is composed of oscillatory solutions of multiple spatial modes. For example, the multiple resonance frequencies are found in Couette flow between parallel plates. The n-th resonance frequency is close to the oscillation frequency of n-order spatial mode. Low order spatial modes generally have stronger resonance.