1. The problem asks to find the washout function for laminar flow in a pipe and explain it. 2. In fluid mechanics, laminar flow in a pipe is characterized by a parabolic velocity profile given by the Hagen-Poiseuille equation: $$u(r) = U_{max} \left(1 - \frac{r^2}{R^2}\right)$$ where $u(r)$ is the velocity at radius $r$, $U_{max}$ is the maximum velocity at the centerline ($r=0$), and $R$ is the pipe radius. 3. The washout function describes how a scalar quantity (like a contaminant or dye) is transported and diluted by the flow over time. 4. For laminar flow, the washout function $W(t)$ can be derived from the velocity profile and the residence time distribution. It often involves integrating the velocity profile to find the distribution of transit times. 5. The residence time $\tau$ for fluid at radius $r$ is: $$\tau(r) = \frac{L}{u(r)} = \frac{L}{U_{max} \left(1 - \frac{r^2}{R^2}\right)}$$ where $L$ is the length of the pipe segment. 6. The washout function $W(t)$ is the fraction of fluid that has exited the pipe by time $t$, which can be expressed as: $$W(t) = \frac{\text{Volume of fluid with } \tau \leq t}{\text{Total volume}}$$ 7. Using the velocity profile, the volume fraction with residence time less than $t$ corresponds to fluid at radius less than $r_t$ where: $$t = \tau(r_t) = \frac{L}{U_{max} \left(1 - \frac{r_t^2}{R^2}\right)}$$ Solving for $r_t$: $$1 - \frac{r_t^2}{R^2} = \frac{L}{U_{max} t} \implies r_t = R \sqrt{1 - \frac{L}{U_{max} t}}$$ 8. The volume fraction is proportional to the cross-sectional area up to $r_t$: $$W(t) = \frac{\pi r_t^2}{\pi R^2} = \frac{r_t^2}{R^2} = 1 - \frac{L}{U_{max} t}$$ 9. The washout function is thus: $$W(t) = 0 \quad \text{for } t < \frac{L}{U_{max}}$$ $$W(t) = 1 - \frac{L}{U_{max} t} \quad \text{for } t \geq \frac{L}{U_{max}}$$ 10. This means no fluid exits before the minimum transit time $\frac{L}{U_{max}}$, and after that, the fraction exiting increases asymptotically to 1. 11. The corresponding figure would show a parabolic velocity profile and a plot of $W(t)$ versus $t$ starting at zero and increasing as above. This washout function captures the effect of velocity distribution on the exit time of fluid elements in laminar pipe flow.