The above video depicts fluid velocity and displacement resulting from fully developed oscillatory flow in a cylindrical vessel.  It is intended to conjure an impression of a mass of fluid that is being accelerated within a tube.  I use the term acceleration here in the technical sense which is literally a change in velocity -- specifically the time rate of change of velocity.   We are all familiar with the term "accelerator" applied to the foot pedal on our vehicles to make them go faster ( up to a point).  However the brake pedal is also an accelerator and so is the steering wheel.  Acceleration is a vector quantity with both magnitude and direction.  Acceleration can be in the direction of motion ( thereby increasing velocity ), opposite the direction of motion ( decreasing velocity ), or perpendicular to the direction of motion ( changing the direction of motion ).  The acceleration vector can have components in the direction of motion and perpendicular to it, changing both speed and direction of motion.

\( \Large \textbf{F} = m \, \textbf{a}\)

Inertance is the property of the circulation that impedes a change in flow rate.   Harking back to the Bernoulli equation, inertance relates to the term involving an integration of \(dv/dt\) along a streamline.  

\( \LARGE p_1+\frac{\rho}{2} v_1^2 + \rho g h_1 = p_2+\frac{\rho}{2} v_2^2 + \rho g h_2 + L_{\mu} + \rho \int_1^2  \frac{dv}{dt} ds \)

Consider a parcel of fluid within a circular tube.  In the figure below, the parcel of fluid in blue has a volume \(A \Delta z\) where \(A\) is the cross-sectional area of the tube (\(\pi r_0^2\)).  


The mass of the blue parcel is it's volume multiplied by density (\(\rho A \Delta z\)).  We'll take other forces into account later, but consider for the moment just the pressure force acting on this parcel.  There is the upstream pressure \(p|_{z}\) at location \(z\), and the downstream pressure \(p|_{z+\Delta z}\) at location \(z+\Delta z\).  Now that we're talking about a dynamic system, you have to remember that the pressure in the tube varies with time. Imagine now that the fluid parcel moves as a solid cylindrical body, just as it's shown, at velocity \(v\). Then the \(\bar{F} = m\bar{a}\) equation looks like this:

\(\Large A (p|_{z}-p|_{z+\Delta z}) = \rho A \Delta z \frac{dv}{dt} \)

The term \(dv/dt\) is the rate of change of velocity, the acceleration of the parcel. According to convention however, we want to replace the velocity in this expression with the flow rate, \(q\).  The average velocity of the fluid over the cross section is \(q/A\) so \(dv/dt = dq/dt/A\). 

 \(\Large A (p|_{z}-p|_{z+\Delta z}) =  \rho A \Delta z \frac{1}{A} \frac{dq}{dt} = \rho \Delta z \frac{dq}{dt}  \)

Finally, we divide through by \(A\) to put this equation into the same physical units as in the Bernoulli equation - pressure.  


\(\Large  (p|_{z}-p|_{z+\Delta z}) =  \frac{\rho \Delta z}{A} \frac{dq}{dt}  \)

The quantity that multiplies the time rate of change of flow, \(\rho \Delta z/A\) is called the inertance.  It has to do specifically (SPECIFICALLY) with the aspect of the circulation that impedes "flow acceleration".  Clearly it has to do with the mass of the fluid in the tube and the fact that flow rate will not change unless a force acts to change it.  

The inertance is the constant by which \(dq/dt\) must be multiplied to give the force per unit area (pressure) associated with the acceleration of the fluid.

I think it's difficult to understand what inertance is without reading the equation and seeing how it comes about.  You'll read in the literature that the term inertance has been applied to cardiac valves that don't open very well.  I think this is a misnomer.


The electrical analog for inertance is an inductor. This is a gadget that usually consists of a coil of wire so that a magnetic field is generated when the rate of current flow through the inductor is changing.  The schematic symbol for the inductor is


and the characteristic equation is:

\(\Large  \Delta v = L \frac{di}{dt}\)

\(\Delta v\) is the voltage across the inductor, \(L\) is the inductance, and \(di/dt\) is the rate of change of current through the inductor. An inductor stores energy in the electromagnetic field generated.  Inductance (\(L\)) has dimension of voltage-time/current and an inductance of 1 henry subjected to a changing current of 1 ampere/second induces a 1 volt electromotive force.

You see now the analogy with the above derivation for inertance:

\(\Large  \Delta p = L \frac{dq}{dt}\)

We saw in a previous article that we can transform this equation to the Fourier domain.

\(\Large  \Delta P(j\omega) =  L j\omega Q(j\omega) \)

The time domain (time-varying) functions representing \(p\) and \(q\) are now represented as \(P\) and \(Q\), each being a sinusoid of specific frequency \(\omega\).   Now we can write the representation of the impedance (\(P/Q\)) of an inertance:

\(\Large \frac{\Delta P}{Q} = Z=  j\omega L\)

\(Z\) is the symbol for impedance. The ratio of the 2 sinusoids increases linearly with increasing frequency (\(\omega\)) and proportionally to \(L\).  For a given inertance, increasing the oscillation frequency will require a proportionally larger oscillatory driving pressure to generate a given oscillatory flow rate.  Alternatively a pressure oscillation of fixed magnitude will result in a decreasing flow oscillation as frequency increases.  The \(j\) in the equation is a thing that shifts the sinusoid by 90°.  Put in a cosine pressure oscillation and you'll get a sine wave flow.  A pure inductance has no impedance at constant flow rate since \(\omega = 0\).  It just let's current pass unimpeded.  Similarly there is no acceleration associated with pulsatility when flow rate is constant.

The figures below illustrate the relationship between pressure difference across an inertance (labeled pressure gradient here) and flow through it.  The first figure depicts the phase relationship for a sinusoid of a particular frequency.  It's apparent that a sinusoidal pressure difference (input) generates a sinusoidal flow (output) of the same frequency.  This is an important characteristic of a linear system!   For a sinusoidal input, we don't have any new frequencies generated in the output.   This allows us to treat (or calculate) each sinusoidal frequency independently of the others.  Any input can be broken down into frequency components (sinusoids, by Fourier transform), the output due to each determined, and the outputs added back together to determine the total output (inverse Fourier transform).   Each sinusoid can be represented by a complex number having both modulus (magnitude) and phase.  For a pure inertance and a specific frequency, the ratio of the pressure modulus to the flow modulus is a constant, \(\omega L\).  The phase is \(\pi/2\) radians (90 degrees) meaning that the force (pressure difference) "leads" the flow by \(\pi/2\) radians.  We know from Newton's laws that force is "in phase" with acceleration.  But velocity (flow here) "lags" acceleration.    At the beginning of the strip (far left), the pressure force is at a peak, but flow is actually 0.  The pressure force begins to decrease, but flow rate continues to increase as the pressure force is still acting to increase velocity.   Several strategically placed vertical lines allow you to compare and conceptualize relationships between pressure force and flow rate.


The following figure shows what happens when an oscillatory ( sinusoidal ) pressure force of constant magnitude but increasing frequency is applied.  An inertance impedes changes in flow rate due to the applied force.  Each frequency shift suggested in the figure corresponds to a doubling of the oscillation frequency and results in a halving of the flow sinusoid magnitude according to the \(\Delta P = j\omega L Q\) relationship.  


Similarly, a constant magnitude flow sinusoid requires a greater pressure force ( sinusoid modulus ) to drive the flow.  Each shift in frequency below again corresponds to a doubling of frequency and a doubling of the sinusoidal pressure forced to drive the flow.  Not readily apparent is the fact that the sinusoids remain shifted by 90° throughout as was shown above at the higher sweep rate and lower frequency.



Effect of Velocity Profile on Inertance

 Consider the 2 velocity profiles that follow.  


Above we calculated the inertance associated with the upper figure, a purely flat (uniform) velocity profile (side view of a conduit with circular cross-section).  Well this isn't a bad approximation for some circumstances, but let's consider whether the shape of the velocity profile affects the inertance. Both of the flows depicted represent the exact same amount of flow, just a different shape to the profile.

To motivate the thought process, we have to ask ourselves whether both flows carry the same amount of momentum.  Mr. Newton (Sir) determined that momentum is a quantity that is conserved.  A force has to act on a body to change its momentum.  Momentum is a vector quantity, \(m\bar{v}\) where \(m\) is the mass, \(\bar{v}\) is the velocity ( also a vector quantity ).  So the question goes like this: Which of these 2 velocity profiles results in the most momentum flow through the tube.

The figure below suggests flow across the surface, represented by the larger oval, where velocity varies over the surface.  Our job is to figure out how much momentum is flowing across this surface.  The concentration of momentum at each point is the fluid density multiplied by the velocity, \(\rho \bar{v}\).  In the somewhat simplified situation below, all the velocities are in the same direction – let's call it direction \(z\).  In the more general situation we have 3 separate equations, one for each of the orthogonal directions ( \(\rho v_x\), \(\rho v_y\),  and \(\rho v_z\)).  Here we have only \(M_z = \rho v_z\).  

 \(M_z\) crosses the smaller oval surface at the velocity \(v_z\) so the rate at which momentum crosses the larger oval is the sum of all the smaller ovals:

\(\Large \int_A M_z v_z dA = \int_A \rho v_z^2 dA\)

The integration \(dA\) means we are adding up all the values over the area.  \(v_z\) is called the flux velocity – the component of the velocity that carries momentum ( or any other quantity ) across the surface; it's the component of the velocity that is directed perpendicular to the surface.  Again in this case we have a somewhat simplified situation in that the area is already perpendicular to the direction of the momentum and the flux velocity.  Rest assured that an equally rigorous mathematical expression can be written for the more general case.

Now all we have to do is perform the above integration for the 2 cases above.  However I'm going to be a little more general and perform this process for a wide range of velocity profiles all at once.  We did this once before using a range of velocity profiles represented by the following:

\(\Large v =\frac{q}{\pi r_0^2}\frac{n+2}{n} \left[1-\frac{r^n}{r_0^n}\right] \)

 This formula gives a range of velocity profiles depending on the value of \(n\). \(n=2\) gives the parabolic (Poiseuille) profile and \(n \rightarrow \infty\) is a perfectly flat profile with everything in between depending on \(n\).  \(q/(\pi r_0^2)\) is the average velocity and \((n+2)/n\) adjusts the magnitude of the velocities so that the flow rate is exactly \(q\).  

To perform the above integration we have:

\(\Large \int_A \rho v_z^2 dA = \int_0^{2\pi}\int_0^{r_0} \rho v_z^2 \, r\, dr \,d\theta = 2\pi \int_0^{r_0}\left[ \frac{q}{\pi r_0^2}\frac{n+2}{n} \left(1-\frac{r^n}{r_0^n}\right) \right]^2 r\,dr = \frac{n+2}{n+1}\frac{\rho q^2}{\pi r_0^2}\)

With \(n=2\) (Poiseuille, parabola), the fraction \((n+2)/(n+1)\) is \(4/3\) and it tends to \(1\) as \(n \rightarrow \infty\) with the flat profile.  The parabola carries \(4/3\) times as much momentum (same flow rate and tube size) as the flat profile.

 Next we'll figure out what happens if inertances are added.  We want the equivalent inertance if 2 are placed in series ..

or in parallel.  


The pressure across \(L_1\) for the series arrangement (top fig) is \(\Delta p_1 = L_1 dq/dt\).  For \(L_2\) it's \(\Delta p_2 = L_2 dq/dt\).  Hence the total pressure change:

\(\Large \Delta p = \Delta p_1 + \Delta p_2 = L_1 \frac{dq}{dt} + L_2 \frac{dq}{dt} = (L_1+L_2) \frac{dq}{dt}\)

\(\Large L_e = L_1+L_2\)

Placed in series, the equivalent inertance is just the sum and of the 2 inertances; the same result we had for resistances in series.  Without belaboring the point, we would find that the equivalent inertance is for the parallel arrangement is:

\(\Large L_e = \frac{1}{\frac{1}{L_1}+\frac{1}{L_2}}\)

Just as was found for the resistors.

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