$10$A$\mathrm{.}7.$ Viscous heating in a ball$-$point pen. You are asked to decide whether the apparent decrease in viscosity in ball$-$point pen inks during writing results from "shear thinning" $($decrease in viscosity because of non$-$Newtonian effects$)$ or "temperature thinning" $($decrease in viscosity because of temperature rise caused by viscous heating$).$ If the temperature rise is less than $1K,$ then "temperature thinning" will not be important. Estimate the temperature rise using Eq$.10\mathrm{.}4-9$ and the following estimated data:

$\backslash$table$[[$Clearance between ball and holding cavity,$5{10}^{-5}in.],[$Diameter of ball,$1mm$