Transient or Pulsating Flows in Tubes

Transient flow in tubes under instantaneously applied pressure gradient. The problem on a transient laminar viscous flow in an infinite circular tube under a constant pressure gradient instantaneously applied at t = 0 is considered in [449]. The equation of motion in the cylindrical coordinates has the form 

Here Jo and J are the Bessel functions of the first kind. The volume rate of flow is given by the formula 

A similar problem for an annular channel with a homogeneous initial velocity field was solved in [296], For t 0 and b U a, we consider the equation of motion in cylindrical coordinates 

Here Jn(A) and Fn(A) are the Bessel functions of the first and second kind, respectively, [261], and the A, are the roots of the transcendental equation 

If 6 = 0 and U = 0, then the solution (1.9.21) turns into the expression (1.9.15) for a circular tube, note also that the solution tends to the steady-state solution (1.5.14) (with a =b and a2 = a) for annular tubes as t -4 oo. 

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