Steady flow in a channel or tube with an accelerating surface velocity. An exact solution to the Navier—Stokes equations with reverse flow

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Abstract
An exact solution to the Navier–Stokes equations for the flow in a channel or tube with an accelerating surface velocity is presented. By means of a similarity transformation the equations of motion are reduced to a single ordinary differential equation for the similarity function which is solved numerically. For the two-dimensional flow in a channel, a single solution is found to exist when the Reynolds number R is less than 310. When R exceeds 310, two additional solutions appear and form a closed branch connecting two different asymptotic states at infinite R. The large R structure of the solutions consists of an inviscid fluid core plus an O(R−1) thin boundary layer adjacent to the moving wall. Matched-asymptotic-expansion techniques are used to construct asymptotic series that are consistent with each of the numerical solutions.For the axisymmetric non-swirling flow in a tube, however, the situation is quite different. For R [Lt ] 10[sdot ]25, two solutions exist which form a closed branch. Beyond 10[sdot ]25, no similarity solutions exist within the range 10[sdot ]25 < R < 147. Once R exceeds 147, multiple solutions reappear, which form two closed branches that connect four different asymptotic states at infinite R. The possibility of an axisymmetric flow with swirl is considered, and two sets of swirling solutions are found to exist for all R > 0. These solutions, however, do not evolve from the R = 0 state nor do they bifurcate from the non-swirling solutions at any finite value of R.

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