Mindlin–Reissner plate theory: Difference between revisions

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In small [[capillary]] [[hemodynamics]], the cell-free layer is a near-wall layer of [[Blood plasma|plasma]] absent of [[red blood cell]]s since they are subject to migration to the [[capillary]] [[Centre (geometry)|center]] in [[Poiseuille flow]].<ref>{{cite journal|title=W. Pan, B. Caswell and G. E. Karniadakis |year=2010|title=A low-dimensional model for the red blood cell|journal= Soft Matter|doi=10.1039/C0SM00183J}}</ref>'''Cell-free marginal layer model''' is a [[mathematical model]] which tries to explain [[Fåhræus–Lindqvist effect]] mathematically.
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==Mathematical modeling==
 
===Governing equations===
Consider [[steady flow]] of [[blood]] through a [[capillary]] of [[radius]] <math>R</math>. The [[capillary]] cross section can be divided into a [[core]] region and cell-free [[Blood plasma|plasma]] region near the wall. The governing equations for both regions can be given by the following equations:<ref>{{cite book|last=Krishnan B. Chandran|first=Alit P. Yoganathan , Ajit P. Yoganathan , Stanley E. Rittgers|title=Biofluid mechanics : the human circulation|year=2007|publisher=CRC/Taylor & Francis|location=Boca Raton|isbn=978-0-8493-7328-2|url=http://www.amazon.com/Biofluid-Mechanics-Circulation-Krishnan-Chandran/dp/084937328X}}</ref>
 
:<math> \frac{ -\Delta P}{ L }=\frac{1}{r}\frac{d}{dr}(\mu_c r \frac{du_c}{dr});</math> <math> 0 \le r\ \le R-\delta\,</math>
:<math> \frac{ -\Delta P}{ L }=\frac{1}{r}\frac{d}{dr}(\mu_p r \frac{du_p}{dr});</math> <math> R-\delta\le r\ \le R\ \,</math>
 
where:
 
:<math>\Delta P </math> is the [[pressure drop]] across the [[capillary]]
:<math>L</math> is the length of capillary
:<math> u_c </math> is [[velocity]] in core region
:<math> u_p </math> is [[velocity]] of plasma in cell-free region
:<math> \mu_{c} </math> is [[viscosity]] in core region
:<math> \mu_{p} </math> is [[viscosity]] of plasma in cell-free region
:<math>\delta</math> is the cell-free [[Blood plasma|plasma]] layer thickness
 
===Boundary conditions===
 
The [[boundary condition]]s to obtain the solution for the two [[differential equation]]s presented above are that the velocity gradient is zero in the tube center, no slip occurs at the tube wall and the [[velocity]] and the [[shear stress]] are continuous at the [[interface (chemistry)|interface]] between the two zones. These [[boundary condition]]s can be expressed mathematically as:
 
*<math>\left. \frac{du_c}{dr}\right|_{r= 0}=0</math>
 
*<math>\left.u_p\right|_{r= R}=0</math>
 
*<math>\left.u_p\right|_{r= R-\delta}=\left.u_c\right|_{r= R-\delta}</math>
 
*<math>\left.\tau_p\right|_{r= R-\delta}=\left.\tau_c\right|_{r= R-\delta}</math>
 
===Velocity profiles===
 
Integrating governing equations with respect to ''r'' and applying the above discussed boundary conditions will result in:
 
:<math> u_c=\frac{ \Delta P R^2}{ 4\mu_p L }[1-(\frac{ R-\delta}{R})^2-\frac{\mu_p}{\mu_c}(\frac{r}{R})^2+\frac{\mu_p}{\mu_c}(\frac{ R-\delta}{R})^2]</math>
 
:<math> u_p=\frac{ \Delta P R^2}{ 4\mu_p L }[1-(\frac{r}{R})^2]</math>
 
===Volumetric flow rate===
 
Total [[volumetric flow rate]] is the algebraic sum of the flow rates in core and plasma region. The expression for the total [[volumetric flow rate]] can be written as:
 
:<math> Q=\frac{ \pi \Delta P R^4}{ 8\mu_p L }[1-(1-\frac{\delta}{R})^4(1-\frac{\mu_p}{\mu_c})]</math>
 
Comparison with the [[viscosity]] which applies in the [[Poiseuille flow]] yields effective [[viscosity]], <math> \mu_{e} </math> as:
 
:<math> \mu_{e}=\frac{\mu_p}{[1-(1-\frac{\delta}{R})^4(1-\frac{\mu_p}{\mu_c})]} </math>
 
It can be realized when the radius of the [[blood vessel]] is very larger than the thickness of the cell-free [[Blood plasma|plasma]] layer, the effective [[viscosity]] is equal to bulk [[blood viscosity]] <math> \mu_c </math> at high shear rates (Newtonian fluid).
 
==See also==
*[[Fåhræus–Lindqvist effect]]
*[[Blood viscosity]]
*[[hemodynamics]]
 
==References==
{{Reflist}}
 
[[Category:Fluid dynamics]]

Latest revision as of 03:33, 9 October 2014

Some of you may well be familiar with the PBS cooking show America's Test Kitchen The same organization publishes the magazines Cook's Illustrated and Cook's Country, which you can consider of as Consumer Reports for cooking in that they do not accept any marketing and are entirely subscriber-supported. In the event you cherished this information as well as you would want to receive more info concerning Very Best Japanese Chef Knife Set [Thebestkitchenknivesreviews.Com] i implore you to check out our internet site. Chicago Cutlery Fusion is the exception to the Japanese Chef Knives Handmade rule due to the fact they are known for producing pretty great knives at very reasonably priced prices. The taper-grind technologies employed to make the knives implies that they remain sharp for longer so cooks don't have to constantly sharpen them. They are known as one of the major knife producers in the planet and have numerous series accessible for significant amateurs and pros.

MY original point is just that a very good sharpening skill and a superior set of whetstones (or a tormek) are both critical if you want to have the finest knife in the planet. I use a mix of Forschner, Wusthof and inexpensive shit in my property kitchen and they perform nicely. I appreciate great tools only since of my exposure to a skilled kitchen. It is sharp, but does not appear sharper than my other knives.

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Ceramic Knives: 1 of the hardest materials identified to exist after diamonds, the blades produced of these ingredient retain their properties such as hardness, sharpness and durability for a quite lengthy time. It is a specialist knife and typically can be located in experienced chef's knife set. There is not 1 knife that can be stated as the fantastic 1, since it all depends on the cook, how to use it. This also keeps the knives in just a single place.

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