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Chapter 6: Problem 95

Blood flows at volume rate \(Q\) in a circular tube of radius \(R\). The blood cells concentrate and flow near the center of the tube. while the cell-free fluid (plasma) flows in the outer region. The center core of radius \(R_{c}\) has a viscosity \(\mu_{c}\) and the plasma has a viscosity \(\mu_{p}\). Assume laminar, fully developed flow for both the core and plasma flows and show that an "apparent" viscosity is defined by $$\mu_{\mathrm{app}}=\frac{\pi R^{4} \Delta p}{8 L Q}$$ is given by $$\mu_{\mathrm{app}}=\frac{\mu_{p}}{1-\left(R_{c} / R\right)^{4}\left(1-\mu_{p} / \mu_{c}\right)}$$

Short Answer

Expert verified
The apparent viscosity \(\mu_{\mathrm{app}}\) which accounts for a core of blood cells with different viscosity within the plasma is given by \( \mu_{\mathrm{app}}=\frac{\mu_{p}}{1-\left(R_{c} / R\right)^{4}\left(1-\mu_{p} / \mu_{c}\right)} \).

Step by step solution

01

Formulate Flow Rate Equation

First, consider the volume flow rates \(Q_{c}\) for the core and \(Q_{p}\) for the plasma; they add up to give the total flow rate Q. Formulate the equations: \( Q_{c} = \int_0^{Rc} 2 \pi r u dr \) and \( Q_{p} = \int_{Rc}^{R} 2 \pi r u dr \)
02

Substitute Velocity and perform integration

Now, substitute the laminar flow profile velocity \(u = \frac{1}{4 \mu} {(\Delta p / L)}(R^2 - r^2)\) into the integral where \(\mu\) is the viscosity. For the core region \(\mu = \mu_c\) and for plasma \(\mu = \mu_p \). Perform the integration to obtain \(Q_c\) and \(Q_p\).
03

Derive the apparent viscosity equation

The total flow rate \(Q\) which equals \(Q_c + Q_p\) can also be defined as \(Q = \frac{\pi R^4 \Delta p}{8 L \mu_{\mathrm{app}}}\). Substitute the expressions of \(Q_c\) and \(Q_p\) obtained from Step 2 into the equation and simplify to get the expression for \( \mu_{\mathrm{app}}\).
04

Simplify Expression

The last step involves simplifying the derived expression for \( \mu_{\mathrm{app}}\) to obtain: \( \mu_{\mathrm{app}}=\frac{\mu_{p}}{1-\left(R_{c} / R\right)^{4}\left(1-\mu_{p} / \mu_{c}\right)} \).

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