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Origin: sally.csie.ntu.edu.tw

※ 引述《gratain (恩典湧泉)》之銘言：
: 假設血管壁上流速=0, blood is a Newtonian fluid,已知 blood density,
: blood viscosity, 入口流速/壓力,出口壓力後,希望分析wall shear stress
: 使用Navier-Stokes equation還有Newtonian fluid的假設這兩個方程式要如何做呢？
: 謝謝
一邊看論文一邊打  所以以下都用英文
Following computation is just for reference.

First to establish corresponding coordinate, is cylindrical coord (r,θ,z),
then Navier-Stokes eqn of motion holds followings,

 1 @   @u     1  @^2u      1 dp
--*--(r--) + ---*------ = --*---  ... (1)         @=partial differential
 r @r  @r    r^2 @θ^2    μ dz

where u=velocity along axis direction, r=radii, p=pressure, z=cylindrical z.

to find eqn (1), consider Navier-Stokes equations in tensor form,

   @V                                       1
ρ(-- + (V。▽)V) = ρf - ▽p + μ(▽^2V + ---▽(▽。V))  ... (2)
   @t                                       3

where the Einstein summation convention have been used.
      and ρ,V,f,p,μ are mass density, velocity,
      body force, pressure, and viscosity, respectively.

blood flow is mostly assumed to be steady-state flow, @V/@t=0,
no body forces, ρf=0
imcompressured flow and laminar flow, ▽。V=0
finally, eqn (2) can be transformed into (1) by coordinate transformation.

For geometric nature, it's assumed the flow is symmetric so that u=u(r),
that is @/@θ=0, then equation (1) reduces,

 1   d   du     1 dp
---*---(r--) = --*--- ...(3)
 r  dr   dr    μ dz

general solution for (3) is found to be,

  r^2   dp
u=----*--- + Aln(r) + B ... (4)
  4μ   dz

where A,B are constants to be determined.

in your case, dp/dz=pressure drop between entrance and exit, I denote by P.
also,I THINK your information knowing the velocity distribution is no use
for calculation since it is dependent on pressure gradient.

Now, apply boundary conditions on (4),
no slip condition, u=0 at r=a if your blood vessel radius is a.
symmetric nature on centre yields du/dr=0 at r=0.

Finally, the solution for velocity distribution is,

      1
u= - ---(a^2-r^2)*P ... (5)
     4μ

This is an outstanding velocity profile, called Hagen-Poiseuille flow.

To analysis shear stress τ at the tube wall, just consider,

        @u
τ= -μ*---  at r=a ... (6)
        @r

apply (5) you can find τ from (6).

For more analysis about other assumptions, refer to your handbook or prof.
It's not easy to handle other assumptions since Navier-Stokes equation
is nonlinear in PDE system.

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◆ From: 59.117.132.101
※ 編輯: harry901        來自: 59.117.132.101       (02/12 22:55)