Question

When a sphere of radius $r_1=1.2\text{mm}$ moves in glycerin, the laminar flow is observed if the velocity of the sphere does not exceed $v_1=23\text{cm/s}.$ At what minimum velocity $v_2($in $\mu \text{m/s})$ of a sphere of radius $r_2=5.5\text{cm}$ will the flow in water become turbulent? The viscosities of glycerin and water are equal to ${\eta }_1=13.9P$ and ${\eta }_2=0.011P$ respectively.

Answer 1

We know that Reynold's number for turbulent flow is greater than that on laminar flow.

Now,

$(R_e{)}_l=\frac{\rho vd}{\eta }=\frac{2{\rho }_1{v}_1{r}_1}{{\eta }_1}$ and $(R_e{)}_t=\frac{2{\rho }_2{v}_2{r}_2}{{\eta }_2}$

But, $(R_e{)}_t\ge (R_e{)}_l$

So ${v_2}_{min}=\frac{{\rho }_1{v}_1{r}_1{\eta }_1}{{\rho }_2{r}_2{\eta }_2}=5\mu m/s$ on putting the values.