Laplace transform solutions for arbitrarily unsteady laminar flow and wall displacement in long, straight, flexible tubes

The problem of arbitrarily unsteady laminar flow in long, straight tubes with linearly elastic walls, that develops from a known initial state, has been solved using Laplace transformation. It is solved under the assumptions that unsteady effects propagate axisymmetrically as long-wavelength disturbances, at a wave speed that is constant or changes slowly with the Laplace-transform-domain variable s. Solutions for quantities such as wall displacement, pressure/pressure gradient, and wall shear stress are expressed as convolution integrals of the flow rate history and weighting functions, which are found by numerical inverse transformation. Since such solutions cannot be found in the physical domain without solving a system of coupled differential equations, this Laplace transform approach provides efficient solutions for pulsatile periodic and aperiodic flows that may come to rest before restarting, such as those in some blood vessels.