Question

A ship is to be operated in a fluid medium with kinematic viscosity $0.032 \times 10^{-3} \, \text{m}^2/\text{s}$. A one-tenth scale model of the ship is built for testing. Consider, inertia, viscous, and gravity forces are dominant for the ship and its model during the operation. The required kinematic viscosity of the liquid for testing the model is $P \times 10^{-6} \, \text{m}^2/\text{s}$. Find $P$ (rounded off to 2 decimal places).

Hint:

For ship modeling, ensure both Froude and Reynolds similarity are satisfied: scale velocity using Froude law, then adjust model fluid viscosity to satisfy Reynolds similarity.

Answer 1

Correct Answer: 0.95

Step 1: Condition for similarity.
Since inertia, viscous, and gravity are important → Froude and Reynolds similarity both must hold. - Froude similarity: \[ \frac{V_m}{\sqrt{g L_m}} = \frac{V_s}{\sqrt{g L_s}} \Rightarrow V_m = V_s \sqrt{\frac{L_m}{L_s}}. \] - Reynolds similarity: \[ \frac{V_m L_m}{\nu_m} = \frac{V_s L_s}{\nu_s}. \]

Step 2: Model scaling.
$L_m/L_s = 1/10$. So, \[ V_m = V_s \sqrt{\frac{1}{10}} = \frac{V_s}{\sqrt{10}}. \]

Step 3: Apply Reynolds similarity.
\[ \frac{V_m L_m}{\nu_m} = \frac{V_s L_s}{\nu_s}. \] \[ \nu_m = \frac{V_m L_m \nu_s}{V_s L_s}. \] Substitute $V_m = \frac{V_s}{\sqrt{10}}, \; L_m = \frac{L_s}{10}$. \[ \nu_m = \frac{\left(\frac{V_s}{\sqrt{10}}\right)\left(\frac{L_s}{10}\right)\nu_s}{V_s L_s}. \] \[ \nu_m = \frac{1}{10\sqrt{10}} \nu_s. \]

Step 4: Substitute values.
\[ \nu_s = 0.032 \times 10^{-3} = 32 \times 10^{-6} \, \text{m}^2/\text{s}. \] \[ \nu_m = \frac{1}{10\sqrt{10}} (32 \times 10^{-6}) = \frac{32}{31.62 \times 10} \times 10^{-6}. \] \[ \nu_m = 1.0117 \times 10^{-4} \,\text{m}^2/\text{s} = 101.17 \times 10^{-6}. \] Final Answer:
\[ \boxed{P = 101.17} \]