Start-up Flow in a Pipe of a Double Distributed-Order Maxwell Fluid.

The distributed-order model is equivalent to weighting the fractional-order model on a given interval. In this paper, a double time-distributed order Maxwell model is developed and formulated to study the characteristics of a start-up pipe flow. The numerical solution of the distributed-order governing equation is obtained by the finite difference method. In order to test the convergence of numerical solutions, the truncation error of the proposed numerical method be calculated. The characteristics of the flow and the effects of fractional parameters and weight functions on the velocity distribution are discussed in detail. By comparing the velocity distribution diagrams, it can be inferred that the distributed-order model controlled by the weight function can describe the memory and elastic properties of fluid more flexibly.