Glycerin of density $$1.25 \times 10^3$$ kg m$$^{-3}$$ is flowing through the conical section of pipe. The area of cross-section of the pipe at its ends are $$10$$ cm$$^2$$ and $$5$$ cm$$^2$$ and pressure drop across its length is $$3$$ N m$$^{-2}$$. The rate of flow of glycerine through the pipe is $$x \times 10^{-5}$$ m$$^3$$ s$$^{-1}$$. The value of $$x$$ is _____.

$$\text{By continuity equation: } A_1 v_1 = A_2 v_2 \implies 10 \cdot v_1 = 5 \cdot v_2 \implies v_2 = 2v_1$$

$$\text{By Bernoulli's equation: } P_1 + \frac{1}{2}\rho v_1^2 = P_2 + \frac{1}{2}\rho v_2^2 \implies P_1 - P_2 = \frac{1}{2}\rho (v_2^2 - v_1^2)$$

$$3 = \frac{1}{2}(1.25 \times 10^3)[(2v_1)^2 - v_1^2] \implies 3 = \frac{1}{2}(1250)(3v_1^2) \implies v_1^2 = \frac{2}{1250} = \frac{4}{2500}$$

$$v_1 = \frac{2}{50} = 0.04\text{ m s}^{-1}$$

$$\text{Volume flow rate: } Q = A_1 v_1 = (10 \times 10^{-4}\text{ m}^2)(0.04\text{ m s}^{-1}) = 4 \times 10^{-5}\text{ m}^3\text{ s}^{-1}$$