Question

The velocity v of blood that flows in a blood vessel with radius r and length ℓ at a distance r from the central axis is v(r) = p 4ηℓ (r2 − r2) where p is the pressure difference between the ends of the vessel and η is the viscosity of the blood. find the average velocity vave (with respect to r) over the interval 0 ≤ r ≤ r. (use eta for η. use capital p for pressure.)

Answer 1

$v(r)=4P\eta\ell(R^2-r^2)$

(assuming $R$ is the radius of the vessel, and that $0\le r\le R$)

The average velocity of the blood is given by

$\displaystyle\frac1{R-0}\int_{r=0}^{r=R}v(r)\,\mathrm dr$
$=\displaystyle\frac{4P\eta\ell}R\int_{r=0}^{r=R}(R^2-r^2)\,\mathrm dr$
$=\dfrac{4P\eta\ell}R\left(\dfrac23R^3\right)$
$=\dfrac{8P\eta\ell}3R^2$

Answer 2

The average velocity in the interval, 0 ≤ r ≤ R, describes the velocity of

blood flow that can be expected in the interval.

• $\displaystyle \mathrm{The \ average \ velocity \ \underline{v(r)_{ave} = \frac{P\cdot R^2}{6 \cdot \eta \cdot l}}}$

Reasons:

The given function for velocity of blood flow is presented as follows;

$\displaystyle v(r) = \frac{P}{4 \cdot \eta \cdot l} \cdot \left (R^2 - r^2 \right)$

Required:

The average velocity with respect to r.

Solution:

The average rate of change of velocity is given by the sum of all the

velocity divided by the size of the set of data;

$\displaystyle The \ dataset \ sum = \frac{P}{4 \cdot \eta \cdot l} \cdot\int\limits^R_0 {\displaystyle \left (R^2 - r^2 \right)} \, dr = \mathbf{ \frac{P}{4 \cdot \eta \cdot l} \cdot \left [R^2\cdot r - \frac{r^3}{3} \right]^R_0}$

$\displaystyle \frac{P}{4 \cdot \eta \cdot l} \cdot \left [R^2\cdot r - \frac{r^3}{3} \right]^R_0= \frac{P}{4 \cdot \eta \cdot l} \cdot\frac{2\cdot R^3}{3} = \mathbf{\frac{P\cdot R^3}{6 \cdot \eta \cdot l}}$

The dataset size = 0 ≤ r ≤ R = R - 0 = R

The average velocity, $v(r)_{ave}$, is therefore;

$\displaystyle Average \ velocity, \, v(r)_{ave} = \frac{Dataset \ sum }{Dataset \ size} = \frac{ \frac{P\cdot R^3}{6 \cdot \eta \cdot l}}{R} = \frac{P\cdot R^2}{6 \cdot \eta \cdot l}$

$\displaystyle \underline{The \ average \ velocity, v(r)_{ave} = \frac{P\cdot R^2}{6 \cdot \eta \cdot l}}$

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