Maharashtra BoardEnglish MediumSTD 12 SciencePhysicsMechanical Properties of Fluids4 Marks
Question
State Stokes' law. Derive Stokes' law using dimensional analysis.
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Answer
Stokes' law : If a fluid flows past a sphere or a sphere moves through a fluid, for small enough $\therefore$ Viscous force $=$ gravitational force-buoyant force $ =\mathrm{mg}-\mathrm{m}_{\llcorner\mathrm{g}} $ where $\mathrm{m}=$ mass of the sphere $=\frac{4}{3} \pi r^3 \rho$ and $\mathrm{m}_{\mathrm{L}</ \mathrm{sub}}=$ mass of the liquid displaced $=\frac{4}{3} \pi r^3 \rho_{\mathrm{L}}$. At its terminal speed $v_{t^{\prime}}$, the magnitude of the viscous force by Stokes' law is $ \begin{aligned} & f=6 \pi \eta r v_{\mathrm{t}} \\ & \therefore 6 \pi \eta r v_{\mathrm{t}}=\frac{4}{3} \pi r^3\left(\rho-\rho_{\mathrm{L}}\right) g \\ & \therefore v_{\mathrm{t}}=\frac{2}{9} \frac{r^2\left(\rho-\rho_{\mathrm{L}}\right) g}{\eta} \end{aligned} $ [Note: Theoretically, $v \rightarrow v_t$ as time $t \rightarrow \infty$. In practice, if $\eta$ is appreciable, then $v$ tends to $v_t$ in a very small time interval.] [Data: $\mathrm{g}=9.8 \mathrm{~m} / \mathrm{s}^2$ ]
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