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.
A civil rights organization on Thursday demanded a meeting with the CEO of Barneys New York and [https://Www.Google.com/search?hl=en&gl=us&tbm=nws&q=threatened&btnI=lucky threatened] to picket the luxury department store in Manhattan after two black customers said they were stopped by police after making expensive purchases.<br><br>Trayon Christian, 19, of Queens said after he bought a $349 Ferragamo belt on April 29 he was handcuffed and detained for two hours before being released with no charges. He filed a discrimination lawsuit against Barneys and the New York City Police Department on Monday.<br><br>Undercover police swarmed Kayla Phillips, 21, of Brooklyn at a subway station and demanded to see her credit card after she left Barneys with her purchase of a $2,500 [http://www.pcs-systems.co.uk/Images/celinebag.aspx Celine Bag Sale] purse on Feb. 28, said her lawyer Kareem Vessup.<br>Phillips, a nursing student, had received a tax return and decided to splurge on the designer purse, Vessup said.<br><br>She filed notice of an upcoming lawsuit against the NYPD and plans to sue Barneys, he said.<br>In his lawsuit, Christian, a mechanical engineering student, said when he bought the belt, Barneys telephoned police to report a criminal act. When he stepped out of the store with his shopping bag, he was handcuffed on the sidewalk and brought to the 19th Precinct, the lawsuit said.<br><br>Police interrogated him "as to how a young black man such as himself could afford to purchase such an expensive belt and that the debit card he had in his possession had to be fake," the lawsuit said.<br>Michael Palillo, a lawyer representing Christian, said his client had saved up earnings from a work-study program at New York City College of Technology and was excited to treat himself to a trendy belt he'd admired on some of the rappers he follows on television.<br><br>"He enjoys fashion," Palillo said.<br>The racial profiling incident violated his civil rights, said the lawsuit, which was filed in state Supreme Court in Manhattan and seeks unspecified damages.<br>Barneys on its Facebook page said "no employee of Barneys" was involved and it "has zero tolerance for any form of discrimination and we stand by our long history in support of all human rights."<br><br>The NYPD did not immediately respond to a request for comment.<br>National Action Network, a civil rights group led by Reverend Al Sharpton, said it was mobilizing activists to take direct action against the store and the NYPD.<br>"National Action Network will immediately demand a meeting with the Barneys New York CEO," NAN spokeswoman Rachel Noerdlinger said in a statement.<br><br>The group is "planning to picket the store if the pattern of racial profiling is not immediately rectified after numerous incidents of discrimination have come to light," Noerdlinger said.<br>The incidents came to light little more than two months after a judge ruled the NYPD discriminated against black and Hispanic youths in its controversial stop-and-frisk crime fighting tactic.<br><br>On its website, Barneys describes itself as "... a mecca for discerning fashionistas and clothing connoisseurs since 1923" and quotes "Sex and the City" actress Sarah Jessica Parker as telling Vanity Fair, "If you're a nice person and you work hard, you get to go shopping at Barneys. It's the decadent reward."<br><br>(Additional reporting by Luke Swiderski; editing by Gunna Dickson)
 
==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]]

Revision as of 22:20, 20 February 2014

A civil rights organization on Thursday demanded a meeting with the CEO of Barneys New York and threatened to picket the luxury department store in Manhattan after two black customers said they were stopped by police after making expensive purchases.

Trayon Christian, 19, of Queens said after he bought a $349 Ferragamo belt on April 29 he was handcuffed and detained for two hours before being released with no charges. He filed a discrimination lawsuit against Barneys and the New York City Police Department on Monday.

Undercover police swarmed Kayla Phillips, 21, of Brooklyn at a subway station and demanded to see her credit card after she left Barneys with her purchase of a $2,500 Celine Bag Sale purse on Feb. 28, said her lawyer Kareem Vessup.
Phillips, a nursing student, had received a tax return and decided to splurge on the designer purse, Vessup said.

She filed notice of an upcoming lawsuit against the NYPD and plans to sue Barneys, he said.
In his lawsuit, Christian, a mechanical engineering student, said when he bought the belt, Barneys telephoned police to report a criminal act. When he stepped out of the store with his shopping bag, he was handcuffed on the sidewalk and brought to the 19th Precinct, the lawsuit said.

Police interrogated him "as to how a young black man such as himself could afford to purchase such an expensive belt and that the debit card he had in his possession had to be fake," the lawsuit said.
Michael Palillo, a lawyer representing Christian, said his client had saved up earnings from a work-study program at New York City College of Technology and was excited to treat himself to a trendy belt he'd admired on some of the rappers he follows on television.

"He enjoys fashion," Palillo said.
The racial profiling incident violated his civil rights, said the lawsuit, which was filed in state Supreme Court in Manhattan and seeks unspecified damages.
Barneys on its Facebook page said "no employee of Barneys" was involved and it "has zero tolerance for any form of discrimination and we stand by our long history in support of all human rights."

The NYPD did not immediately respond to a request for comment.
National Action Network, a civil rights group led by Reverend Al Sharpton, said it was mobilizing activists to take direct action against the store and the NYPD.
"National Action Network will immediately demand a meeting with the Barneys New York CEO," NAN spokeswoman Rachel Noerdlinger said in a statement.

The group is "planning to picket the store if the pattern of racial profiling is not immediately rectified after numerous incidents of discrimination have come to light," Noerdlinger said.
The incidents came to light little more than two months after a judge ruled the NYPD discriminated against black and Hispanic youths in its controversial stop-and-frisk crime fighting tactic.

On its website, Barneys describes itself as "... a mecca for discerning fashionistas and clothing connoisseurs since 1923" and quotes "Sex and the City" actress Sarah Jessica Parker as telling Vanity Fair, "If you're a nice person and you work hard, you get to go shopping at Barneys. It's the decadent reward."

(Additional reporting by Luke Swiderski; editing by Gunna Dickson)

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