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| In [[fluid dynamics]], a flow with periodic variations is known as '''pulsatile flow'''. The [[cardiovascular]] system of [[Chordate|chordate animals]] is very good example where pulsatile flow is found. Pulsatile flow is also observed in [[engines]] and [[hydraulic system]]s as a result of [[rotating]] [[Mechanism (engineering)|mechanisms]] belonging to them.
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| == Derivation ==
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| To obtain the velocity profile of non-stationary flow, one must solve the [[motion equation|equations of motion]] and [[continuity equation|continuity]]. Depending on the complexity of the [[boundary condition]]s, the problem's analytic solution may be impracticable and thus [[numerical simulation]]s would be necessary. An analytical solution is here given assuming the following hypothesis:
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| *Fluid is [[homogeneous]], [[incompressible]] and [[Newtonian fluid|Newtonian]];
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| *Tube wall is [[Stiffness|rigid]], [[Circle|circular]] and [[cylindrical]];
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| *Motion is [[Laminar flow|laminar]], [[axisymmetric]] and parallel to the tube's axis;
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| *Boundary conditions are axisymmetry at the centre and no-slip condition on the wall;
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| *Pressure gradient drives the fluid;
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| *[[Gravitation]] has no effect on the fluid. <ref>{{cite book
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| |author=Fung, Y. C.
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| |title=Biomechanics – Motion, flow, stress and growth
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| |year=1990
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| |page=569
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| |publisher=Springer-Verlag
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| |place=New York (USA)
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| |url=http://books.google.ie/books?id=33qbOEKAWIwC&printsec=frontcover&dq=Biomechanics+-+Motion,+flow,+stress+and+growth&hl=en&sa=X&ei=bpKiT7jKGMi0hAevo4n-CA&ved=0CDkQ6AEwAA#v=onepage&q=Biomechanics%20-%20Motion%2C%20flow%2C%20stress%20and%20growth&f=false}}</ref><ref>{{cite journal|last=Nield|first=D.A.|coauthors=Kuznetsov, A.V.|title=Forced convection with laminar pulsating flow in a channel or tube|journal=International Journal of Thermal Sciences|year=2007|volume=46|issue=6|pages=551–560|doi=10.1016/j.ijthermalsci.2006.07.011}}</ref>
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| The field equations [[Navier-Stokes equation]] and the equation of continuity are simplified as
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| :<math> \rho \frac{\partial u}{\partial t} = -\frac{\partial p}{\partial x} + \mu \left(\frac{\partial^2 u}{\partial r^2} + \frac{1}{r} \frac{\partial u}{\partial r}\right) \, </math>
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| and
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| :<math> \frac{\partial u}{\partial x} = 0 \, .</math>
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| The pressure gradient is a general periodic function
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| :<math> \frac{\partial p}{\partial x} = \sum^N_{n=0}C_n e^{in\omega t} \, .</math>
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| Velocity profile is driven by the pressure, resulting in
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| :<math> u(r,t) = \sum^N_{n=0}U_n e^{in\omega t} \, .</math>
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| Substituting the pressure gradient and velocity profile in the Navier-Stokes equation gives us
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| :<math> i\rho n\omega U_n = -C_n +\mu \left(\frac{\partial^2 U_n}{\partial r^2} + \frac{1}{r} \frac{\partial U_n}{\partial r}\right) \, .</math>
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| With the boundary conditions satisfied, the general solution is
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| :<math> U_n(r) = A_nJ_0 \left( \alpha \frac{r}{R} n^{1/2}i^{3/2} \right) + B_nY_0 \left( \alpha \frac{r}{R} n^{1/2}i^{3/2} \right) + \frac{iC_n}{\rho n \omega}\, ,</math>
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| where <math>J_0(kr)</math> is the [[Bessel function]] of first kind and order zero, <math>Y_0(kr)</math> is the Bessel function of second kind and order zero, being <math>k</math> a constant. <math>A_n</math> and <math>B_n</math> are arbitrary constants and <math>\alpha</math> is the [[dimensionless]] [[Womersley number]]. In order to determine <math>A_n</math> and <math>B_n</math> the axisymetic boundary condition is used, i.e. <math>\partial U_n/ \partial r = 0</math>, then the derivatives <math>J_0'</math> and <math>Y_0'</math> approaches infinity. Hence <math>B_n</math> must vanish. And the boundary condition at the wall gives us
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| :<math> U_n(R) = 0 = A_nJ_0 \left( \alpha n^{1/2}i^{3/2} \right) + \frac{iC_n}{\rho n \omega}\, .</math> | |
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| Solving this equation for <math>A_n</math>, we obtain the amplitudes of the velocity profile
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| :<math> U_n(r) = \frac{iC_n}{\rho n \omega} \left[ 1 - \frac{J_0(\alpha \frac{r}{R} n^{1/2}i^{3/2})}{J_0(\alpha n^{1/2}i^{3/2})} \right] \, ,</math>
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| which leads to the velocity profile itself
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| :<math> u(r) = \sum^N_{n=0} \frac{iC_n}{\rho n \omega} \left[ 1 - \frac{J_0(\alpha \frac{r}{R} n^{1/2}i^{3/2})}{J_0(\alpha n^{1/2}i^{3/2})} \right] e^{in\omega t} \, .</math> | |
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| The velocity profile depends on Womersley number <math>\alpha</math>.
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| == Cardiovascular flow ==
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| These pulsating characteristics have been shown to be a result of two [[pumps]]. As the primary pump, the [[heart]] causes the [[blood]] flow and [[velocity]] to [[oscillate]] from zero to very high rates as the [[valves]] at the entrances and exits to the [[Ventricle (heart)|ventricles]] intermittently close and open with each beat of the heart. The second pump is a result of the [[respiratory]] and [[skeletal muscle|skeletal systems]], which exert their greatest action on venous flow.<ref>{{cite book
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| |author=Lee, B. Y., and Trainor, F. S.
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| |title=Peripheral Vascular Surgery: Hemodynamics of Arterial Pulsatile Blood Flow
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| |year=1973
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| |publisher=Appleton-Century-Crofts
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| |location=New York
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| |page=270
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| |url=http://books.google.ie/books/about/Peripheral_vascular_surgery_hemodynamics.html?id=wIFsAAAAMAAJ&redir_esc=y}}</ref> Specifically pulsation that result from the release of blood from the left ventricle show that they exhibit [[non-linear]], [[transient]] pulsations in [[pressure]] and flow. These create complex pulse patterns which are further propagated through the rest of the network. This results in variations in the applied shear stress to the layer of [[endothelium|endothelial cells]] covering the vessel wall. Depending upon the amount of [[Stress (mechanics)|stress]], the endothelial cells will react releasing chemicals that either induce [[vasodilation|dilation]] or [[vasoconstriction|constriction]] of the smooth muscle surrounding the vessel.
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| It is nearly impossible to mathematically model such a flow using the standard [[Navier-Stokes equations]]. Rather than give an equation that can model the flow, which has proven to be near impossible; the [[Womersley number]] is used. This [[dimensionless number]] has been developed to give a measure of the frequency and magnitude of pulsations rather than a model of the actual flow.
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| {{Center|<math>\alpha = R \left( \frac{\omega}{\nu} \right)^{1/2} \ = R \left( \frac{\omega \rho}{\mu} \right)^{1/2} \, </math>}}
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| As you can see, the equation can take on two forms by substituting ''mu''/''rho'' for ''nu''. It can also be shown that Wormesley number is primarily influenced by the size of the vessel which can be shown in the table below. Since the density of blood and blood viscosity remain fairly constant (with slight variations throughout) the value of the square root will be similar for all, thus vessel size is most important.
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| {| class="wikitable"
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| ! Section !! Radius (cm) !! alpha
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| | Ascending Aorta || 0.75|| 14.628
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| |-
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| | Descending Aorta || 0.65 || 12.677
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| | Abdominal Aorta || 0.45 || 8.777
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| | Femoral Artery || 0.2 || 3.901
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| | Cartoid Artery || 0.25 || 4.876
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| | Arteriole || 0.0025|| 0.049
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| | Capillary || 0.0003 || 0.006
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| | Venule || 0.002 || 0.039
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| | Inferior Vena Cava|| 0.5 || 9.752
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| | Main Pulmonary artery || 0.85|| 16.578
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| |}
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| These values were calculated using a cardiac frequency of 2 Hz, a blood density of 1060 kg/m^3 at 37 C, and a dynamic viscosity of 0.035 Pa-s
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| == References ==
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| {{Reflist}}
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| [[Category:Biological engineering]]
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| [[Category:Biology]]
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| [[Category:Fluid dynamics]]
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Office Manager Palmer Dorothy from Sainte-Croix, spends time with hobbies for example jogging, como ganhar dinheiro na internet and collecting music albums. Loves to visit unknown destinations like Verla Groundwood and Board Mill.
My web site :: ganhando dinheiro na internet