A non-viscous liquid of constant density $1000\,\text{kg/m}^{3}$ flows in a streamline motion along a tube of variable cross section. The tube is kept inclined in the vertical plane as shown in Figure. The area of cross section of the tube two points $P$ and $Q$ at heights of $2$ metres and $5$ metres are respectively $4\times10^{-3}\,\text{m}^{2}$ and $8\times10^{-3}\,\text{m}^{2}$. The velocity of the liquid at point $P$ is $1\,\text{m/s}$. Find the work done per unit volume by the pressure and the gravity forces as the fluid flows from point $P$ to $Q$.

Step 1: Identify the given parameters.

The given values for the fluid flow are:

$$\rho = 1000\,\text{kg/m}^{3}$$ $$A_P = 4\times10^{-3}\,\text{m}^{2}$$ $$A_Q = 8\times10^{-3}\,\text{m}^{2}$$ $$h_P = 2\,\text{m}$$ $$h_Q = 5\,\text{m}$$ $$v_P = 1\,\text{m/s}$$ $$g = 9.8\,\text{m/s}^{2}$$

Step 2: Calculate the velocity of the liquid at point $Q$.

Apply the equation of continuity.

$$A_P v_P = A_Q v_Q$$ $$v_Q = \frac{A_P v_P}{A_Q}$$ $$v_Q = \frac{4\times10^{-3} \times 1}{8\times10^{-3}}$$ $$v_Q = 0.5\,\text{m/s}$$

Step 3: Calculate the work done per unit volume by gravity.

The work done per unit volume by gravity is the negative change in potential energy per unit volume.

$$W_{\text{gravity}} = -\rho g (h_Q - h_P)$$ $$W_{\text{gravity}} = -1000 \times 9.8 \times (5 - 2)$$ $$W_{\text{gravity}} = -2.94\times10^{4}\,\text{J/m}^{3}$$

Step 4: Calculate the work done per unit volume by pressure.

Apply Bernoulli's equation along the streamline from $P$ to $Q$.

$$P_P + \frac{1}{2}\rho v_P^2 + \rho g h_P = P_Q + \frac{1}{2}\rho v_Q^2 + \rho g h_Q$$

The work done per unit volume by pressure forces is equal to the pressure difference.

$$W_{\text{pressure}} = P_P - P_Q$$ $$W_{\text{pressure}} = \rho g (h_Q - h_P) + \frac{1}{2}\rho (v_Q^2 - v_P^2)$$ $$W_{\text{pressure}} = 1000 \times 9.8 \times (5 - 2) + \frac{1}{2} \times 1000 \times (0.5^2 - 1^2)$$ $$W_{\text{pressure}} = 29400 - 375$$ $$W_{\text{pressure}} = 2.9025\times10^{4}\,\text{J/m}^{3}$$

Answer: The work done per unit volume by gravity is $-2.94\times10^{4}\,\text{J/m}^{3}$ and by pressure is $2.9025\times10^{4}\,\text{J/m}^{3}$.