If dimensions of critical velocity, v _ c of a liquid flowing...

Dimensions of velocity, coefficient of viscosity and density are,

$\left[v\right] = \left[{LT}^{- 1}\right]$

\left[η\right] = \dfrac{\left[F\right]}{\left[A\dfrac{d v}{d x}\right]} = \dfrac{\left[{MLT}^{- 2}\right]}{\left[{L}^{2}\right]\left[{T}^{- 1}\right]} = \left[{ML}^{- 1}{T}^{- 1}\right]$</p></blockquote><blockquote><p>\left[ρ\right] = \left[{ML}^{- 3}\right]$

Now, \left[v\right] = {\left[η\right]}^{x}{\left[ρ\right]}^{y}{\left[r\right]}^{z}$</p><p>$⇒ \left[{LT}^{- 1}\right] = {\left[{ML}^{- 1}{T}^{- 1}\right]}^{x}{\left[{ML}^{- 3}\right]}^{y}{\left[L\right]}^{z}$</p><p>Now, by using dimensional homogeneity, we will get,</p><p>${M}^{0} = {M}^{\left(x + y\right)}$</p><p>$⇒ y = - x$</p><p>For$T$, </p><p>$x = 1$</p><p>$⇒ y =- x =- 1$</p><p>For$L$,</p><p>$1 = - x - 3y + z$</p><p>$⇒ z = - 1