Despite the continuing interest in the transport of biological fluid within contracting or expanding vessels, our knowledge is yet to be fully developed, even in the two-dimensional case. For example, explicit solutions and close approximations to these models remain unknown, and the physical problem has been restricted to the “slow” expansion or contraction of the walls. Thus, the purpose of this short communication is to partially address such challenges and gaps by generating explicit solutions and improving approximations to the flow problem without the “slowness” restriction being imposed. We show that when the Reynolds number is zero (i.e., the inviscid case), the corresponding homogeneous differential equation under consideration may be completely solved. We then illustrate how this exact solution may be leveraged to form more precise approximations to the flow via perturbation techniques when the Reynolds number is small. Our perturbation approach is only in one parameter (the Reynolds number) instead of the usual two parameters (the Reynolds number and wall dilation rate), and thus we make no restriction regarding the “slowness” of wall expansion or contraction for our general perturbation scheme. Our act of “shining new light through old windows” improves and extends the results of Majdalani, Zhou and Dawson and, moreover, our method has significant potential to be applied by researchers to form more precise one-parameter perturbation approximations to flow problems in contrast to the limitations of the traditional two-parameter perturbation approaches that have dominated the literature.