Question

Find the dimensional formula for the coefficient of viscosity $\eta$.

Answer 1

Hint: Dimensional formula is representation of a formula in dimensional form, at first we take the coefficient of viscosity formula, and then we compare it with the dimensional analysis figures that is, M for mass, L for length, T for time.

Complete step by step answer:
We know the formula for coefficient of viscosity is,
$\eta =\dfrac{F}{A(\dfrac{dv}{dx})}$ ………….eq1,
F=force, A= area, dv/dx= velocity gradient.
In dimensional analysis,
M=MASS
L=LENGTH
T=TIME
Force=MA=$mass\times acceleration$
In dimensional analysis,
Force=$M\times [L{{T}^{-2}}]$…………eq2
A=${{L}^{2}}$ …………….eq3
Velocity gradient=${{T}^{-1}}$……………eq4
Therefore the dimensional formula for viscosity is;
On substituting the value of eq2, eq3, eq4 onto eq1, we get
The dimensional value of viscosity as,
$[M{{L}^{-1}}{{T}^{-1}}]$ .

Additional Information:
The viscosity is calculated in terms of the coefficient of viscosity. It is constant for a liquid and depends on it’s liquid’s nature. The Poiseuille’s method is formally used to estimate the coefficient of viscosity, in which the liquid flows through a tube at the different level of pressures.
The coefficient of viscosity of fluids will be decreased as the temperature increases, while it is inverse in the case of gases. While the coefficient of viscosity of gases will increase with the increase in temperature. The increase in temperature for the fluid deliberate the bonds between molecules. These bonds are directly associated with the viscosity and finally, the coefficient is decreased.

Note: During conversion it is required to be precise in basic knowledge of the formulas, it is good to break one formula into as many parts as possible. It will help to get to the result more precisely without any silly mistake.