A train with cross-sectional area $$S_t$$ is moving with speed $$v_t$$ inside a long tunnel of cross-sectional area $$S_0(S_0 = 4S_t)$$. Assume that almost all the air (density $$\rho$$)in front of the train flows back between its sides and the walls of the tunnel. Also, the air flow with respect to the train is steady and laminar. Take the ambient pressure and that inside the train to be $$p_0$$. If the pressure in the region between the sides of the train and the tunnel walls is 𝑝, then $$p_0 - p = \frac{7}{2N}\rho v_t^2$$. The value of N is _____________.

Let the train move with a constant speed $$v_t$$ through still air of density $$\rho$$. Work in the train’s rest frame; this frame is inertial because the train travels at constant velocity.

1. Velocity of air far ahead of the train (point A)
In the train’s frame the air far in front approaches the train with speed $$v_t$$ through the full tunnel cross-section $$S_0$$. Pressure at point A is the ambient value $$p_0$$, and its speed is $$v_t$$.

2. Velocity of air in the annular gap (point B)
The train blocks the central area $$S_t$$, so air can pass only through the remaining annular area $$S_0 - S_t = S_0 - \tfrac{S_0}{4} = \tfrac{3S_0}{4}.$$

Continuity (volume conservation) in the train’s frame:

$$v_t\,S_0 = u\,\bigl(S_0 - S_t\bigr)$$ $$\Longrightarrow\; u = v_t\,\frac{S_0}{S_0 - S_t} = v_t\,\frac{4S_t}{3S_t} = \frac{4}{3}\,v_t.$$

Thus the air in the gap (point B) moves backwards past the train with speed $$u = \tfrac{4}{3}v_t$$.

3. Pressure difference using Bernoulli’s theorem
For steady, incompressible, laminar flow, Bernoulli’s equation between points A and B (same height) is

$$p_0 + \frac{1}{2}\,\rho\,v_t^{\,2} = p + \frac{1}{2}\,\rho\,u^{2}\;.$$

Therefore

$$p_0 - p = \frac{1}{2}\,\rho\,(u^{2} - v_t^{\,2}).$$

Insert $$u = \tfrac{4}{3}v_t$$:

$$u^{2} - v_t^{\,2} = \left(\frac{4}{3}v_t\right)^{\!2} - v_t^{\,2} = \frac{16}{9}v_t^{\,2} - \frac{9}{9}v_t^{\,2} = \frac{7}{9}v_t^{\,2}.$$

Hence

$$p_0 - p = \frac{1}{2}\,\rho\,\frac{7}{9}v_t^{\,2} = \frac{7}{18}\,\rho\,v_t^{\,2}.$$

4. Compare with the given expression
The problem states $$p_0 - p = \frac{7}{2N}\,\rho\,v_t^{\,2}.$$

Equate the two formulas:

$$\frac{7}{2N} = \frac{7}{18} \;\;\Longrightarrow\;\; \frac{1}{2N} = \frac{1}{18} \;\;\Longrightarrow\;\; 2N = 18 \;\;\Longrightarrow\;\; N = 9.$$

Therefore, the required value of $$N$$ is 9.