Question

What are the units or dimensions of the shear rate dv/dy (English units)? Then, what are the dimensions of the shear stress τ= μ*dV/dy? Then, by dimensional analysis, show that the shear stress has the same units as momentum divided by (area*time).What are the unit or dimensions of viscosity?

Answer 1

1) Dimensions of shear rate is
`[T^(-1)]` .

2)Dimensions of shear stress are
`[ML^(-1)T^(-2)]`

Step-by-step explanation:

Since the dimensions of velocity 'v' are
`[LT^(-1)]` and the dimensions of distance 'y' are
`[L]` , thus the dimensions of
`(dv)/(dy)` become

`([LT^(-1)])/([L])=[T^(-1)]` and hence the units become
`s^(-1)`.

Now we know that the dimensions of coefficient of dynamic viscosity
`\mu` are
`[ML^(-1)T^(-1)]` thus the dimensions of shear stress can be obtained from the given formula as

`[\tau ]=[ML^(-1)T^(-1)]* [T^(-1)]\\\\[\tau ]=[ML^(-1)T^(-2)]`

Now we know that dimensions of momentum are
`[MLT^(-1)]`

The dimensions of
`Area* time` are
`[L^(2)T]`

Thus the dimensions of
`(Moumentum)/(Area* time)=(MLT^(-1))/(L^(2)T)=[MLT^(-2)]`

Which is same as that of shear stress. Hence proved.