text: For deriving the analytical solution of this non-stationary flow velocity profile, the following assumptions are taken:
  * Fluid is homogeneous, incompressible and Newtonian;
  * Tube wall is rigid and circular;
  * Motion is laminar, axisymmetric and parallel to the tube's axis;
  * Boundary conditions are: axisymmetry at the centre, and no-slip condition on the wall;
  * Pressure gradient is a periodic function that drives the fluid;
  * Gravitation has no effect on the fluid.
  Thus, the Navier-Stokes equation and the continuity equation are simplified as
  and
  respectively. The pressure gradient driving the pulsatile flow is decomposed in Fourier series,
  where  is the imaginary number,  is the angular frequency of the first harmonic (i.e., ), and  are the amplitudes of each harmonic . Note that,  (standing for ) is the steady-state pressure gradient, whose sign is opposed to the steady-state velocity (i.e., a negative pressure gradient yields positive flow). Similarly, the velocity profile is also decomposed in Fourier series in phase with the pressure gradient, because the fluid is incompressible,
  where  are the amplitudes of each harmonic of the periodic function, and the steady component () is simply Poiseuille flow
  Thus, the Navier-Stokes equation for each harmonic reads as
  With the boundary conditions satisfied, the general solution of this ordinary differential equation for the oscillatory part () is
  where  is the Bessel function of first kind and order zero,  is the Bessel function of second kind and order zero,  and  are arbitrary constants, and  is the dimensionless Womersley number. The axisymmetric boundary condition () is applied to show that  for the derivative of above equation to be valid, as the derivatives  and  approach infinity. Next, the wall non-slip boundary condition () yields . Hence, the amplitudes of the velocity profile of the harmonic  becomes
  where  is used for simplification.
  The velocity profile itself is obtained by taking the real part of the complex function resulted from the summation of all harmonics of the pulse,