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Mechanical Properties of Fluids

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Question : 26 of 31
Marks: +1, -0
(a) What is the largest average velocity of blood flow in an artery of radius 2×10−32 \times 10^{-3} m if the flow must remain laminar?
(b) What is the corresponding flow rate? (Take viscosity of blood to be 2.084×10−3Pa s2.084 \times 10^{-3} \text{Pa s}).
Density of blood is 1.06×103kg/m31.06 \times 103 \text{kg/m}^3
Solution:  
Here, ρ=2×10−3m;\rho = 2 \times 10^{-3} \text{m} ;
D=2r=2×2×10−3D = 2 r = 2 \times 2 \times 10^{-3}
=4×10−3m= 4 \times 10^{-3} \text{m}
η=2.084×10−3Pa s\eta = 2.084 \times 10^{-3} \text{Pa s}
For flow to be laminar, NR=2000N_{R}=2000
(a) $ $ Now, vc=NRηρDv_{c} = \frac{N_{R} \eta}{\rho D}
=2000×(2.084×10−3)(1.06×103)×(4×10−3)= \frac{2000 \times (2.084 \times 10^{-3})}{(1.06 \times 10^{3}) \times (4 \times 10^{-3})}
=0.98ms−1= 0.98 \text{ms}^{-1}
(b) Volume flowing per second
=πr2vc= \pi r^{2} v_{c}
=227×(2×10−3)2×0.98= \frac{22}{7} \times (2 \times 10^{-3})^{2} \times 0.98
=1.23×10−5m3s−1= 1.23 \times 10^{-5} \text{m}^{3} \text{s}^{-1}.
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