In fluid dynamics, a flow with periodic variations is known as pulsatile flow, or as Womersley flow. The flow profiles was first derived by John R. Womersley in his work with blood flow in arteries. 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 rotatingmechanisms pumping the fluid.
Equation
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. For, 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.
The Bessel function at its upper limit it becomes 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. On the surface, but the exponential term becomes negligible once 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, the velocity adjusts rapidly to zero. Furthermore, the phase of the time oscillation varies quickly with position across the layer. The exponential decay of the higher frequencies is faster.
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, and are the amplitudes of each harmonic. Note that, is the steady-state pressure gradient, whose sign is opposed to the steady-state velocity. 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 axisymetic 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,
Flow rate
is obtained by integrating the velocity field on the cross-section. Since, then
Velocity profile
To compare the shape of the velocity profile, it can be assumed that where is the shape function. It is important to notice that this formulation ignores the inertial effects. The velocity profile approximates a parabolic profile or a plug profile, for low or high Womersley numbers, respectively.
If the pressure gradient is not measured, it can still be obtained by measuring the velocity at the centre line. 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. For any product of complex numbers, the amplitude and phase have the relations and, respectively. Hence, and which finally yield
