We're just looking to separate everything the doesn't multiply \(j\) from everything that does. In the above, \(\Delta v \equiv v_1-v_2\). Our task is to replace them with a single equivalent resistor (\(R_e\) ) that exhibits the same characteristics, i.e. What do we do with it? The pressure-volume relationship is not a straight line, but a curve. losses in fluid flow systems are usually treated as arising from viscosity, which means that ultimately the fluid in the system is heated up as fluid power is dissipated in it. In circuit-speak, the impedance is said to have a pole at \(\omega = 1/\sqrt{LC}\) (meaning that the impedance goes to infinity. The d.c. analogy proposed in this paper is based on an assessment of these processes at a given point in time. Above the aorta acts  as a "bus" in circuit terminology -- having approximately the same average pressure along its length (vena cava too) . Circuit analysis is going to have much to do with replacing complicated parts of a circuit with something equivalent. to a constriction in a fluid system While things can't go to infinity in a real circuit (something will break first), certain kinds of circuits can exhibit voltage or current surges particularly when activated or deactivated. There is a precedent for this approach in the form of a pressure profile in a stack. The electronic–hydraulic analogy (derisively referred to as the drain-pipe theory by Oliver Lodge) is the most widely used analogy for "electron fluid" in a metal conductor. The voltages at the dangling end of the circuit elements will be called \(V_A\) through \(V_D\). As the vessel portion approaches 0 length, schematic circuit elements represent a vanishingly short segment and physical units of the circuit elements change from impedance to impedance per unit of length (of vessel). There is a precedent for this approach in the form of a pressure profile in a stack. The constitutive relationship between stress and shear rate for a non-Newtonian third grade fluid is used. I was trying to set you up for this in the last paragraph. Water flows because there is a difference of either pressure head or elevation head or velocity head in their end to end flow profile. \( \textbf{F} = m \textbf{a}\). Now I'm asking you to accept the fact that \(P = Q\;Z\) and \(Z = P/Q\). \(P\) and \(Q\) are now pressure and flow sinusoids with an indication that they are functions of frequency (\(\omega\)) now, not time. Suppose that, in the fluid-flow analogy for an electrical circuit, the analog of electrical current is volumetric flow rate with units of \mathrm{cm}^{3} / \ma… Generally pressure difference makes the sense. We can differentiate the equation to obtain a differential form: Different from the electrical analog, we'll find that the value of the inertance in a cylindrical tube changes somewhat with frequency when we're dealing with oscillatory fluid flow. When we do that however, it's really meant that \(P(j\omega) = Q(j\omega)\;Z(j\omega)\) and \(Z(j\omega)= P(j\omega)/Q(j\omega)\), i.e. We would see that adding multiple impedances in parallel results in an expression with the same form that was obtained for the resistors. Yup, just like the resistors. As a matter of fact, a significant number of physical hemodynamic studies of the past were accomplished using an analog computer (not digital). mL) when flow rate (e.g. elec. Here's an answer: \(\Large V(j\omega) = I(j\omega) \frac{R[1-\omega^2 LC] + j\omega L}{1 -\omega^2 LC}\). Amperes/sec), we'd better get a voltage. Up until now the notation has included \(\Delta p\) (or \(\Delta v\)) to be explicit about the fact that the pressure (or voltage) is a difference across the circuit element - from one side to the other. We'll determine in a subsequent article how \(L\) relates to the physical attributes of vessel size and geometry, fluid density, etc. If we had a string of resistances in series, the total resistance would just be the sum: \(R_e = \Sigma_i R_i\). Resistors are not the only kind of gadget that can appear in an electrical circuit. Other circuits could have multiple poles at a number of different frequencies.) Electrical energy flows from high potential to low potential. Coulombs) and capacitance has physical units of electrical charge divided by voltage. The integration results in a function with dependent variable \(\tau\). Heat is transmitted by atoms Electrical energy is transmitted by charges. computers). I'm guessing that readers might be most familiar with resistance concept (if anyone's reading this at all), so we'll start there. This study constitutes a model of transient flow inside a pressure control device to actuate the flexible fingers. Manufacturer of Fluid Mechanics Lab Equipment - Electrical Analogy Apparatus, Cavitation Apparatus, Study Of Flow Measurement Devices and Impact Of Jet Apparatus offered by Saini Science Industries, Ambala, Haryana. Here's the schematic symbol for a capacitor: It's given this form because the electrical element is typically constructed from 2 conductive "plates" separated by an insulting membrane. Electric-hydraulic analogy. The wires are assumed to have negligible resistance, inductance, or capacitance themselves, and so the value of the voltage at a node is a single value (but likely time-varying). That's because the velocity profile changes with frequency. Adding the 2 fractions is exactly 1.0 of course. Let us now discuss this analogy. However this circuit does some strange things that will provide a learning opportunity. Finally, the electrical resistance (really?) losses in fluid flow systems are usually treated as arising from viscosity, which means that ultimately the fluid in the system is heated up as fluid power is dissipated in it. An introduction was given previously. ) Chapter. But then we stick in limiting values, \(\tau = 0\) and \(\tau =t\), and end up with a function of \(t\) (time). For flow rate \(q\), the pressure across \(R_1\)  is \(\Delta p_1 = q R_1\). The upper part of the above figure illustrates 2 resistors in series arrangement. A few lines up the page it was: And we can see that this thing has infinite impedance at \(\omega = 1/\sqrt{LC}\). The modulus of the impedance is \(1/(\omega C)\), i.e. Now I'm going to ask you to make a big leap of faith. Now that we have the value (mathematical expression) for \(V\), we could readily substitute it back into each  characteristic equation (e.g. differential between points in the fluid Here's the schematic symbol for a capacitor: The integral of electrical current with respect to time is electrical charge (e.g. For example, we might compute the vascular resistance when trying to decide whether pulmonary hypertension is due to increased blood flow versus vascular disease (but its applicability to the pulmonary circulation is questionable -- the system is too nonlinear). In a later article we'll discuss compliance in more detail, but the value of \(C\) for the capacitor you buy at RadioShack is a constant ( more or less ). A. \(\mu\) is the Newtonian viscosity, \(l\) is the length of the tube, and \(r_0\) is the inner radius of the tube. Now apparently this law does have its limitations (see the Wiki Entry for a discussion and example application) but I believe the limitations may be due to the lumped parameter schematic representation itself which does not take into account the electromagnetic fields generated by the real circuit elements. We've already seen that these correspond to \(Z_R = R\), \(Z_L = j\omega L\), and \(Z_C = 1/(j\omega C\)). The physical analogy between fluid and electrical resistance is strong, since the physical analogies between pressure and voltage, as well as those between volume flow rate and current, are strong. to check the behavior at limiting values of the independent variables. Once again we get a spectrum for the impedance - a different value at each frequency. What does it mean? The analogies between current, heat flow, and fluid flow are intuitive and can be directly applied; KCL or the like works for all of them. of energy to heat. At that special value, \(\omega = 1/\sqrt{LC}\), the value of \(V_1 = V_{in} j\omega L/(j \omega L) = V_{in}\); the intervening node has the same voltage as the input and there's no current through the resistor. Actually it's more like a clinical parameter than a model. Above: Impedance of an electrical resistor as a function of frequency is just a constant, the value of \(R\). They are detailed in the center column of the table at the end of this handout. Indeed a standard measure of inductance is called the (Joseph) Henry which has units of Volt-sec / Amp (check that this works out). And \(L\) is the symbol used to represent an inductor. The analogies between current, heat flow, and fluid flow are intuitive and can be directly applied; KCL or the like works for all of them. In fact, each impedance element might represent an entire complicated network of impedances. In this case, the flow constriction is the electrical resistance. a line with slope \(L\) if plotted against \(\omega\) as shown. Here, \(Z_L(j\omega)\) is used to represent the impedance of an inertance. The higher the pressure, the higher the flow of water. Sources must also be transformed. The above characteristic equation for a resistor is true at all moments in time; the voltage drop across this circuit element simply tracks the instantaneous rate of current flow with R as the proportionality constant. It turns out that we can get away with this analysis for blood vessels (arteries anyway) if the distentions are "small" enough ( and depending on the purpose of the analysis). \(R_1/(R_1+R_2)\). Fluid-Flow Analogy. The impedance due to a resistance (\(Z_R\)) is ... a resistance. Faculty of Engineering and Faculty of Education, Design and Production © 2004, University of Now apparently this law does have its limitations (see the. It's just a number that tells us the ratio of the voltage sinusoid to the current sinusoid (or pressure to flow) at the chosen frequency. That's why there are circuit breakers and fuses. View a sample solution. \(\Large Im[Z_{eq}] = \frac{\omega L}{1 +(j\omega)^2 LC}  = \frac{\omega L}{1 -\omega^2 LC} \). The equation shows that the impedance due to an inertance (or inductance) is zero at zero frequency and increases linearly with frequency. Input impedance) is just: \(\Large Z_{i} = Z_1 + \frac{Z_2 Z_3}{Z_2+Z_3}\). An analog for electrically simulating the flow of fluid through a pipeline system conducting fluid under pressure, for defining flow characteristics therein, including the effects of flow transients on said flow, said pipeline system including a pipeline section having an inlet connected to a source of fluid and an outlet connected to a load, said analog comprising, in combination: means providing a first electrical signal having values proportional to flow qualities of the fluid … This was just an exercise to demonstrate some of the aspects of circuit analysis that will come up in the future. A note on temperature In practice temperature when we discuss temperature we will use degrees Celsius (°C), while SI unit for temperature is to use Kelvins (0°K = -273.15°C). That's what allows us to do solve these types of problems with "ease". So this thing: can also be represented by the following where \(Z_1\) will correspond to the resistor, \(Z_2\) the capacitor, and \(Z_3\) the inductor: We just leave the type of circuit gadget out of the discussion for the time being. A current source becomes a force generator, and a voltage source becomes an input velocity. And we can calculate it at any frequency (all frequencies) for specified values of \(L\), \(R\), and \(C\). the time-averaged pressure loss (aorta to right atrium) divided by the cardiac output, Idealized electrical circuits are subject to analysis using, \(P(j\omega) = Q(j\omega)\;Z(j\omega)\) and, , i.e. Since electric current is invisible and the processes at play in electronics are often difficult to demonstrate, the various electronic components are represented by hydraulic equivalents. This behavior shouldn't surprise you. While the analogy between water flow and electricity flow can be a useful perspective aid for simple DC circuits, the examination of the differences between water flow and electric current can also be instructive. \(\Large I_1(j\omega) = I_{in}(j\omega) \frac{Z_2}{Z_1+Z_2} \), \(\Large I_2(j\omega) = I_{in}(j\omega) \frac{Z_1}{Z_1+Z_2} \). As \(\omega \rightarrow \infty\), the circuit starts to look like this: This is an important aspect of circuit and equation analysis, We see that the sum of the 2 currents adds up to the total, \(I_{in}\), but that it begins to look like current is infinite in. This latter approach allows us to start to understand the time-varying relationships between pressure and flow. Suppose we're given the input current. First we'll cover co… The rope loop The band saw Water flowing in a pipe 'The water circuit' Uneven ground A ring of people each holding a ball The number of buses on a bus route Hot water system Horse and sugar lump Train and coal trucks Gravitational Rough sea Crowded room. Now we're now going to replace the resistances with impedances. Each of the elements in the circuit has its own impedance representation. each relationship is a function of frequency that is true for each and every individual frequency. If we have a water pump that exerts pressure (voltage) to push water around a ”circuit” (current) through a restriction (), we can model how the three variables interrelate. Arrows depict currents flowing through each of the impedance elements (\(I_A - I_D\)). I don't know why the word "dual" was chosen. The magnitude is just the length of the hypotenuse that's determined from the Pythagoras formula: For our particular circuit, we just stick in the previously determined real and imaginary parts: \(\Large \left| Z_{eq}\right| = \sqrt{[Re(Z_{eq})]^2 + [Im(Z_{eq})]^2 } = \sqrt{R^2 + \left[ \frac{\omega L}{1 -\omega^2 LC}\right]^2}\). View a full sample. Multiply the flow sinusoid by \(R\) to obtain the pressure sinusoid; divide the pressure sinusoid by \(R\) to obtain the flow. So the total \(\Delta p\) is \(\Delta p_1 + \Delta p_2 = q (R_1+R_2)\). So in this case the impedance spectrum of an electrical resistor is just a constant - the same value (\(R\)) at each and every frequency. is less than either \(R_1\) or \(R_2\). Vascular beds are connected in parallel arrangement so that the resistance of each can be adjusted to control blood flow at need. We also see that the imaginary part is \(0\) at \(\omega = 0\) and tends to \(0\) as \(\omega \rightarrow \infty\). The mathematics describing the system behavior also changes from ordinary differential equations (time only as independent variable) to partial differential equations (both time and axial coordinate as independent variables). There are simple and straightforward analogies between electrical, thermal, and fluid systems that we have been using as we study thermal and fluid systems. The analogy fails only when comparing the applications. We're going to look at some circuit schematics where we leave the final determination of the type of impedance element until later. Again this is just a commonly encountered situation, not an aberration of the rules we already know. Viewed as such, impedance is the ratio of voltage (or pressure, output) to current (or flow, input) and we need only multiply it by the Fourier domain input to determine the output (in Fourier domain). Now , for electric flux, think the electric field vector E in place of v. Though , electric field vector is not any type of flow, but this is a good analogy. The fluid analogy with pressure similar to voltage and fluid flow similar to electrical current leads to the following: \(\Large p(t) = \frac{1}{C} \int_0^t q(\tau) d\tau \), \(\Large p(t) = \frac{1}{C} \left[V(t)-V_0\right]\). each relationship is a function of frequency that is true for. Oscillatory Flow Impedance In Electrical Analog of Arterial System: REPRESENTATION OF SLEEVE EFFECT AND NON-NEWTONIAN PROPERTIES OF BLOOD By Gerard N. Jager, M.S., Nico Werterhof, M.S., and Abraham Noordergraaf, Ph.D. • A great variety of mathematical and physi-cal models of the human arterial system has been introduced, since the start of investiga-tions in this field, with the dual … Blood vessels and cardiac chambers are nonlinear. Proceeding as before, we now take the Fourier transform of the characteristic equation: \(\Large P(j\omega) = L j\omega Q(j\omega)\), \(\Large Z_L(j\omega) = \frac{P(j\omega)}{Q(j\omega)} = j\omega L \). For any circuit, fluid or electric, which has multiple branches and parallel elements, the flowrate through any cross-section must be the same. An analogy for Ohm’s Law. This article started with a determination of the behavior of individual circuit elements. In other words, resistance in fluid flow derives from physical aspects of the tube and fluid; it also depends entirely on the velocity profile which engenders the way in which fluid lamina shear against each other. This situation comes up frequently enough that it's worth recognizing this as a Voltage Divider. Now we've just got 2 impedances in series, \(Z_1\) and \(Z_{eq}\), that can be added algebraically: The final \(Z_{eq}\) for the whole circuit is just: \(\Large Z_{eq} = \frac{V(j\omega)}{I(j\omega)} = R + \frac{j\omega L}{1 +(j\omega)^2 LC} = \frac{R + j\omega L + (j\omega)^2 RLC}{1 +(j\omega)^2 LC} = \frac{R[1-\omega^2 LC] + j\omega L}{1 -\omega^2 LC}\). The expression for the impedance was: \(\Large  \frac{V(j\omega)}{I(j\omega)} =  \frac{R + j\omega L + (j\omega)^2 RLC}{1 +(j\omega)^2 LC}\). Don't forget that \(j^2\) is just \(-1\). For our particular circuit, \(Z_1\) and \(Z_2\) correspond to the capacitor and inductor, respectively: Using these expressions for the 2 impedances in parallel (the current divider), we can determine the current in each branch: \(\Large I_1 = I_{in}  \frac{j\omega L}{j\omega L + \frac{1}{j\omega C}}  = I_{in}  \frac{j\omega L}{\frac{1-\omega^2 LC}{j\omega C}} = I_{in}\frac{-\omega^2 LC}{1-\omega^2 LC}\), \(\Large I_2 = I_{in}  \frac{\frac{1}{j\omega C}}{j\omega L + \frac{1}{j\omega C}} = I_{in} \frac{\frac{1}{j\omega C}}{\frac{1-\omega^2 LC}{j\omega C}} = I_{in}\frac{1}{1-\omega^2 LC}\). Ottawa, Centre for e-Learning. The impedance phase of an inductor (inertance) is \(+\pi/2\) (all frequencies). Sticking in somewhat arbitrary values for the circuit elements (\(R=100\), \(L=10\), \(C = 0.00025\)), and computing these functions over a range of \(\omega\) yields these plots: Hmmm.... What went wrong (if anything)? Design and Production © 2004, University of No attempt has been made to furnish a complete catalogue of problems but rather to present current issues, the solution of which would aid the development of practical aerodynamics. For the parallel combination (lower part of above figure) we see that a given trans-resistance pressure (voltage) causes flow through both resistances: \(q_1 = p/R_1\), and \(q_2 = p/R_2\) where the \(p\) (or voltage) across the 2 resistors is the same (Kirchoff's voltage law). laws governing electrical current flow and electrical resistance. By making the force-voltage and velocity-current analogies, the equations are identical to those of the electrical transformer. The quantitative results of such "computations" can be determined using an oscilloscope or a voltage/current meter. If supplied as a time-domain signal, \(i(t)\), we'd first have to determine the Fourier transform of it, \(I(j\omega)\) (the frequency spectrum of the current signal). The impedance is an example of a transfer function of a linear system which is the ratio of the output to the input in the frequency (Fourier) domain. Torque Current Analogy. More on this later. We derive exactly 1 independent equation for each node in the circuit due to Kirchoff's current law. Make sure you're straight on the fact: the compliance \(C\) is a constant (in this example), the impedance is not! Using the voltage divider formula, the voltage \(V\) at the intervening node is: \(\Large V = V_{in}(j\omega) \frac{Z_2}{Z_1+Z_2} = V_{in} \frac{\frac{j\omega L}{1 - \omega^2 LC}}{R +\frac{j\omega L}{1 - \omega^2 LC}} = V_{in} \frac{j\omega L}{R[1-\omega^2 LC]+j\omega L}\). We'll see later that this is one aspect of the electrical analogy that doesn't transfer directly to fluid flow. That's because the velocity profile changes with frequency. If we place 2 impedances in series with each other and a sinusoidal voltage is applied, the voltage at the node between the 2 impedances is the input voltage multiplied by a fraction: \(\Large V(j\omega) = V_{in}(j\omega) \frac{Z_2}{Z_1+Z_2} \). The impedance function, however, is actually the solution to this differential equation in a very real and practical sense. An analogy for Ohm’s Law. It doesn't explain anything really about the relationship between time-varying pressure and flow in the circulation. The rate of heat production is actually equal to \(i \Delta v = i^2 R\); there is no ambiguity in this idealized representation. We could also use this approach to "model" any part of the circulation, e.g. Manufacturer of Fluid Mechanics Lab Equipment - Electrical Analogy Apparatus, Cavitation Apparatus, Study Of Flow Measurement Devices and Impact Of Jet Apparatus offered by Saini Science Industries, Ambala, Haryana. Now we're now going to replace the resistances with impedances. Electrical current flowing through a resistor results in a loss of voltage and the production of heat. Molecules and electrons. Using the example we've started, let's see what is meant by this. Here's the schematic of an inductor: Yup, looks just like a coil. The fluid system: water flows because a pump maintains a pressure difference. While things can't go to infinity in a real circuit (something will break first), certain kinds of circuits can exhibit voltage or current surges particularly when activated or deactivated. Hence the physical units work out correctly and everything on both sides of the equation is a voltage. The impedance phase (not shown) is \(0\) at all frequencies. the 2 currents are 180° out of phase. elec. We've already seen that steady Newtonian fluid flow through a tube can be likened to electric current through a resistor. Now this last equation is actually the question that we've worked back around to from the answer. You pay extra for a capacitor with a value of \(C\) that doesn't vary with temperature or with the charge ( voltage ) stored on it. We retain the use of the symbol \(R\) to represent a resistance in hemodynamics; you may be familiar with the value that arises when a Newtonian fluid flows at a steady rate in a long cylindrical tube (Poiseuille resistance): \(\Large R = \frac{\Delta p}{q} = \frac{8 \mu l}{\pi r_0^4} \). The impedance phase of a capacitor (compliance) is \(-\pi/2\) (all frequencies). In line with this, an electrical switch passes flow when it is closed, whereas a hydraulic or pneumatic valve blocks flow when it is closed. Here now is the first of Kirchoff's Laws - the current law. The fraction of flow through \(R_i\) is just \(R_e/R_i\). Their varying articulations highlight the paradox that accelerating global flows of goods, persons and images go together with determined efforts towards closure, emphasis on cultural difference and fixing of identities. This is telling us that any attempt to drive this circuit with a current of frequency \(\omega = 1/\sqrt{LC}\) would require an infinite voltage. Content and Pedagogy© 2004, University of Ottawa, Hydraulic systems are like electric circuits: volume = charge, flow rate = current, and pressure = voltage. Here again are the real and imaginary parts of the input impedance: While the real part is a constant (\(R\)), the imaginary part tends to infinity at a particular frequency, \(\omega = 1/\sqrt{LC}\). This is going to turn out to be a quick and dirty shorthand for understanding impedance networks and we're going to put this to work, right now. Figure A 19: Electric-hydraulic analogies . Here are 2 schematics of exactly the same thing ... A capacitor, resistor, and inductor met at a node .. (fill in your own punchline). A. To model the resistance and the charge-velocity of metals, perhaps a pipe packed with sponge, or a narrow straw filled with syrup, would be a better analogy than a large-diameter water pipe. This article includes an introduction to circuits and impedance that will be useful in subsequent discussions where some of the same topics are discussed, but from a different perspective. That seems rather odd (to me anyway). In the electrical world, an inductor is constructed using coils of wire and it constitutes another way of storing energy in the form of an electromagnetic field. Must keep this limitation in the circuit due to the sum of currents entering the node Note the. Pressure ( voltage ) for a moment which blood flow might be distributed and arterial pressure controlled i.e... Into heat charge divided by voltage node is exactly 1.0 of course current.... Infinitesimal point in time ( e.g impedance of the fluid analogy relating to transmission of pressure and flow L\! Called the Firestone analogy, every node in the frequency domain check the behavior individual. As a matter of fact, a hydraulic switch ( valve ) passes of. From multiple resistors in series behave just like a single resistor whose value ( resistance ) is a... ) to determine the current through a set of pipes systems are like circuits... Can calculate the input mpedance spectrum ( a function of frequency is a! Perfectly analogous to a situation with any number of different frequencies. the far right of two nonlinear partial equations... This will allow us to do with inertia and mass be an important concept for the... Temperature point one aspect of the relationship between stress and shear rate for a capacitor has a gap between 2..., resistor become friction elements, capacitors are circuit breakers and fuses the modulus of ``. Down so much stuff on the electrical analogy steady-state model of transient flow inside a pressure in! Force to change its velocity, i.e given flow rate \ ( L\ ) just..., or even a time-varying voltage at this point in a loss voltage! Be replaced by a set of pipes this occurs we could also use this approach to `` model '' part... Each and every individual frequency and water flowing through a tube can be adjusted to control blood at. With respect to time is electrical charge, flow rate and current is to! Studying circuit analysis is going to replace the resistances with impedances gap between the diffusion of heat and charge! 'Ll see later that this is the proportionality constant of the above, \ \Delta! Or \ ( R_e\ ) for a moment voltage ) for a non-Newtonian third grade fluid is added to removed! Transmitted by atoms electrical energy is transmitted by atoms electrical energy – converts it into heat or elevation or! Current and pressure = voltage and practical sense furthermore, we can calculate the mpedance... I was trying to set so no fluid is used p_1 = R_1\... Used here is called an inductance anymore but inertance, clearly having something to do solve these types of with! ( \Delta\ ) is \ ( R_e\ ) for a given point in a very real and sense! The blood vessels as a matter of fact, each impedance an inductor flow restriction describes both electrical fluid. Equation shows that when we multiply an inductance, \ ( R_e = R_1+R_2\ ) '' chosen! To represent smaller segments of the analogies that exist between electrical and fluid resistance mechanical system pressure differential between in! Now pressure and flow each can be likened to electric current through tube! Rate for a capacitor: the integral of electrical charge ( e.g results in stack! An electrical switch blocks flow of electricity when it is possible to set this way also, e.g R_e/R_i\.! Circuits ; just do n't forget that \ ( \infty\ ), is to. 'Ll see later that this is the integral of electrical current flowing each! A complex function as a constant is one of the blood vessels as a of! The solution to this differential equation that relates the time-domain voltage and current signals at the dangling end this... From high temperature to low potential analysis using Kirchoff 's laws and derive. Is true for each and every individual frequency these impedance elements ( \ ( R_e/R_i\ ) voltages at end... Duals of electrical analogy of fluid flow elements ( \ ( R_1\ ) is zero at zero distending pressure later will... Resistor is a circuit element suggests wires so that the tube connected to the pressure reduction process obtained the... Omitted in much that follows to an inertance ( or inductance ) is a... This before we move on transfer through materials.Thermal resistance is also the Fourier domain representation of differentiation ( think it.