Chapter 4.5, Problem 65E

Solution

To calculate : The contact angle between the capillary vessel and the blood surface if the film of liquid a thing tube has surface tension $\gamma$ modelled by $\gamma =\frac{1}{2}h\rho gr\sec \phi$ .

The contact angle between the capillary vessel and the blood surface if the film of liquid a thing tube has surface tension $\gamma$ is $0.8952\text{ rad}$ .

Given information:

The function $\gamma =\frac{1}{2}h\rho gr\sec \phi$ .

Formula:

Solve for the value of $\phi$ using trigonometric values.

Calculation:

Consider the function $\gamma =\frac{1}{2}h\rho gr\sec \phi$ ,

If the blood has surface tension of $0.058\text{ N/m}$ and density is $1050{\text{ kg/m}}^3$ and blood rises to the height of $1.5\text{ m}$ in a capillary vessel of radius $4.7\cdot {10}^{-6}$ .

Substitute all the values in the formula $\gamma =\frac{1}{2}h\rho gr\sec \phi$ .

  $\begin{array}{c}0.058=\frac{1}{2}\left(1.5\right)\left(1050\right)\left(9.8\right)\left(4.7\cdot {10}^{-6}\right)\sec \phi \\ 0.058=36272.25\sec \phi \\ \sec \phi =1.6\\ \phi \approx 0.8952\text{ rad}\end{array}$

Thus, the contact angle between the capillary vessel and the blood surface if the film of liquid a thing tube has surface tension $\gamma$ is $0.8952\text{ rad}$ .