Physics

The rate of flow of a liquid through a capillary tube of length $$L$$ and radius $$r$$ under a pressure difference $$p$$ is given by the poiseuille's formula $$V = \pi \,p\,{r^4}/8\eta L$$. Determine the dimensions of the coefficient of viscosity $$\eta $$ of the liquid.


SOLUTION

It is given that

$$ V=\dfrac{\pi {{\Pr }^{4}}}{8\eta L} $$

$$ \eta =\dfrac{\pi {{\Pr }^{4}}}{8VL} $$

Where V is volume rate $$\left[ {{L}^{3}}{{T}^{-1}} \right]$$

P is pressure $$\left[ M{{L}^{-1}}{{T}^{-2}} \right]$$

$$\eta $$  is viscosity

L is length $$\left[ L \right]$$

So the dimensional formula of viscosity is

$$ \dfrac{\left[ M{{L}^{-1}}{{T}^{-2}} \right]\left[ {{L}^{4}} \right]}{\left[ {{L}^{3}}{{T}^{-1}} \right]\left[ L \right]} $$

$$ \left[ M{{L}^{-1}}{{T}^{-1}} \right] $$


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