Pulsatile flow
The flow profiles was first derived by John R. Womersley (1907–1958) in his work with blood flow in arteries.
[1] The cardiovascular system of chordate animals is a very good example where pulsatile flow is found, but pulsatile flow is also observed in engines and hydraulic systems, as a result of rotating mechanisms pumping the fluid.
The pulsatile flow profile is given in a straight pipe by where: The pulsatile flow profile changes its shape depending on the Womersley number For
, viscous forces dominate the flow, and the pulse is considered quasi-static with a parabolic profile.
, the inertial forces are dominant in the central core, whereas viscous forces dominate near the boundary layer.
Thus, the velocity profile gets flattened, and phase between the pressure and velocity waves gets shifted towards the core.
[citation needed] The Bessel function at its lower limit becomes[2] which converges to the Hagen-Poiseuille flow profile for steady flow for or to a quasi-static pulse with parabolic profile when In this case, the function is real, because the pressure and velocity waves are in phase.
The Bessel function at its upper limit becomes[2] which converges to This is highly reminiscent of the Stokes layer on an oscillating flat plate, or the skin-depth penetration of an alternating magnetic field into an electrical conductor.
becomes large, the velocity profile becomes almost constant and independent of the viscosity.
Thus, the flow simply oscillates as a plug profile in time according to the pressure gradient, However, close to the walls, in a layer of thickness
Furthermore, the phase of the time oscillation varies quickly with position across the layer.
The exponential decay of the higher frequencies is faster.
For deriving the analytical solution of this non-stationary flow velocity profile, the following assumptions are taken:[3][4] 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 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 the Bessel function of first kind and order zero,
is the Bessel function of second kind and order zero,
is the dimensionless Womersley number.
The axisymmetric boundary condition (
Next, the wall non-slip boundary condition (
Hence, the amplitudes of the velocity profile of the harmonic
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, Flow rate is obtained by integrating the velocity field on the cross-section.
[5] It is important to notice that this formulation ignores the inertial effects.
For straight pipes, wall shear stress is The derivative of a Bessel function is Hence, If the pressure gradient
The measured velocity has only the real part of the full expression in the form of Noting that
, the full physical expression becomes at the centre line.
The measured velocity is compared with the full expression by applying some properties of complex number.
