## Funnel: how long does the fluid take to go through?

Posted by Diego Assencio on 2013.12.24 under Physics (Fluid dynamics)

In this post I will compute, under some simple assumptions, how much time a given amount of fluid (e.g. water) takes to go through a funnel. The result presented here is by no means a general formula but will be a good estimate in certain regimes (namely, if the stem radius is much smaller than the mouth radius and if the stem is short compared to the funnel height).

The funnel is shaped like an "upside down" cone whose tip has been cut (see figure 1). It is initially completely filled with fluid. The volume of the neck itself is assumed to be negligible with respect to the total volume of the funnel.

The fluid is assumed to be incompressible and the flow is assumed to be inviscid (meaning we treat the fluid as having zero viscosity).

 Fig. 1: A funnel with conical shape. The funnel has mouth radius $b$ and stem radius $a$. The initial fluid height is $h_0$, which is the height of the funnel itself. At time $t$, the height of the fluid is $h(t)$ and the radius of the fluid's surface is $r(t)$.

It will be assumed here that the top surface of the fluid moves down very slowly. This assumption will be good as long as $h(t)$ is not too small, but might break as $h(t) \rightarrow 0$. With this assumption, the problem becomes similar to the "orifice in a tank" problem, meaning we can treat the flow as steady. The advantage this brings is we can use Bernoulli's equation. To be more precise, consider points $A$ and $B$ on figure 1. We can assume both points are on the same flow streamline (the dashed line), meaning: $$\displaystyle\frac{v_A^2(t)}{2} + \frac{p_A}{\rho} + gh_A(t) = \frac{v_B^2(t)}{2} + \frac{p_B}{\rho} + gh_B(t) \label{post_62f54ab6114d473b6933ad5bf5a5fc88_eq_bern}$$ where $\rho$ is the density of the fluid, $v_A(t)$ is the fluid speed at $A$, $p_A$ is the pressure at $A$ and $h_A(t)$ is the height of point $A$ (all quantities referring to $B$ are equivalently defined). Since both points $A$ and $B$ are in direct contact with the surrounding air, we have $p_A = p_B = p_0$, where $p_0$ is the atmospheric pressure. Also, since we assume the surface of the fluid moves down slowly, we can take $v_A(t) \approx 0$. Equation \eqref{post_62f54ab6114d473b6933ad5bf5a5fc88_eq_bern} then becomes: $$\frac{p_0}{\rho} + gh_A(t) = \frac{v_B^2(t)}{2} + \frac{p_0}{\rho} + gh_B(t)$$ and therefore: $$v_B(t) = \sqrt{2g(h_A(t) - h_B(t))} = \sqrt{2gh(t)} \label{post_62f54ab6114d473b6933ad5bf5a5fc88_eq_vB}$$ so the fluid comes out of the neck's orifice with speed $v_B(t) = v(t) = \sqrt{2gh(t)}$.

Before we proceed, let's obtain a relation between $r(t)$ and $h(t)$. Since we assume $a$ to be small ($a \ll r(t)$ and $a \ll b$), the cross section of the cone shown on figure 1 is close to a triangle, so: $$\displaystyle\frac{r(t)}{h(t)} = \frac{b}{h_0} \Longrightarrow r(t) = h(t)\frac{b}{h_0}$$

The total volume of fluid contained in the funnel at time $t$ is given by (as a reminder, the volume of the funnel's neck is assumed to be negligible): $$V(t) = \displaystyle\frac{1}{3}\pi r^2(t) h(t) = \frac{\pi}{3}\left(\frac{b}{h_0}\right)^2h^3(t) \label{post_62f54ab6114d473b6933ad5bf5a5fc88_eq_volume}$$ This approximation will be good as long as $r(t) \gg a$, meaning the volume of the "removed tip" of the cone is negligible with respect to the fluid volume $V(t)$. As $h(t) \rightarrow 0$, $r(t) \rightarrow a$ so this approximation becomes poor, but by that time the small amount of fluid left should not be large enough to break the time estimate.

From equation \eqref{post_62f54ab6114d473b6933ad5bf5a5fc88_eq_vB}, the volumetric rate at which the fluid goes through the orifice of the tunnel is equal to: $$\Phi(t) = A v(t) = (\pi a^2) v(t) = \pi a^2 \sqrt{2gh(t)} \label{post_62f54ab6114d473b6933ad5bf5a5fc88_eq_flow_rate}$$ where $A = \pi a^2$ is the cross sectional area of the funnel's stem. Since this flow rate is the rate at which the fluid leaves the funnel, we have (below the notation $\dot{q}$ is used to represent $dq/dt$): $$\dot{V}(t) = -\Phi(t) \label{post_62f54ab6114d473b6933ad5bf5a5fc88_eq_v_phi}$$ But from equation \eqref{post_62f54ab6114d473b6933ad5bf5a5fc88_eq_volume} we have: $$\dot{V}(t) = \frac{\pi}{3}\left(\frac{b}{h_0}\right)^2 3 h^2(t) \dot{h}(t) = \pi\left(\frac{b}{h_0}\right)^2 h^2(t) \dot{h}(t)$$ Inserting this on equation \eqref{post_62f54ab6114d473b6933ad5bf5a5fc88_eq_v_phi} and using equation \eqref{post_62f54ab6114d473b6933ad5bf5a5fc88_eq_flow_rate}, we get: $$\pi\left(\frac{b}{h_0}\right)^2 h^2(t) \dot{h}(t) = -\pi a^2 \sqrt{2gh(t)}$$ Cancelling common terms on both sides of the equation above yields: $$\left(\frac{b}{h_0}\right)^2 h^{3/2}(t) \dot{h}(t) = -a^2 \sqrt{2g} \label{post_62f54ab6114d473b6933ad5bf5a5fc88_eq_hr}$$

Since: $$h^{3/2}(t) \dot{h}(t) = \displaystyle\frac{2}{5}\frac{d}{dt}h^{5/2}(t)$$ then equation \eqref{post_62f54ab6114d473b6933ad5bf5a5fc88_eq_hr} can be rewritten as: $$\displaystyle\frac{d}{dt}h^{5/2}(t) = -\frac{5}{2} \left(\frac{h_0}{b}\right)^2 a^2 \sqrt{2g}$$ Integrating both sides with respect to time yields: $$\begin{eqnarray} \int_{t'=0}^{t'=t} \displaystyle\frac{d}{dt'}h^{5/2}(t')dt' &=& -\int_{t'=0}^{t'=t} \frac{5}{2} \left(\frac{h_0}{b}\right)^2 a^2 \sqrt{2g} dt' \nonumber\\[5pt] h^{5/2}(t)\bigg|_{t'=0}^{t'=t} &=& -\frac{5}{2} \left(\frac{h_0}{b}\right)^2 a^2 \sqrt{2g} t \nonumber\\[5pt] h^{5/2}(t) - h_0^{5/2} &=& -\frac{5}{2} \left(\frac{h_0}{b}\right)^2 a^2 \sqrt{2g} t \end{eqnarray}$$ Therefore: $$\begin{eqnarray} \displaystyle h(t) &=& \left[ h_0^{5/2} - \frac{5}{2} \left(\frac{h_0}{b}\right)^2 a^2 \sqrt{2g} t \right]^{2/5} \\[5pt] &=& h_0 \left[ 1 - \frac{5}{2} \frac{1}{\sqrt{h_0}b^2} a^2 \sqrt{2g} t \right]^{2/5} \\[5pt] \end{eqnarray}$$ A bit more algebraic manipulation yields: $$\boxed{ \displaystyle h(t) = h_0\left[ 1 - \frac{5}{2} \left(\frac{a}{b}\right)^2 \sqrt{\frac{2g}{h_0}} t \right]^{2/5} } \label{post_62f54ab6114d473b6933ad5bf5a5fc88_eq_h_vs_t}$$

The time $T$ it takes for the fluid to go through the funnel is such that $h(T) = 0$, which means: $$1 - \displaystyle \frac{5}{2} \left(\frac{a}{b}\right)^2 \sqrt{\frac{2g}{h_0}} T = 0 \Longrightarrow \boxed{ \displaystyle T = \frac{2}{5} \left(\frac{b}{a}\right)^2 \sqrt{\frac{h_0}{2g}} }$$ Equation \eqref{post_62f54ab6114d473b6933ad5bf5a5fc88_eq_h_vs_t} can then be written in a simpler form: $$\displaystyle h(t) = h_0\left( 1 - \frac{t}{T} \right)^{2/5}$$ Figure 2 shows a graph of $h(t)/h_0$ versus $t / T$.

 Fig. 2: Fluid height $h(t)/h_0$ vs. $t/T$.

As the graph on figure 2 shows, the fluid's surface moves down slowly until about $t = 0.95T$. After that point, $h(t)$ starts decreasing very quickly, meaning some of the approximations we made will fail to be appropriate. But as initially expected, this should not badly break our computed estimate of $T$.

 Diego Assencio on Mar 12, 2016: @Cielo: That equation has not been derived here. It just uses the fact (from Calculus) that for a given function $f(t)$, we have: $\displaystyle \frac{d}{dt}f^\alpha(t) = \alpha f^{\alpha-1}(t) \frac{d f(t)}{dt} = \alpha f^{\alpha-1}(t) \dot{f}(t)$ If $\alpha \neq 0$, then we can also write the equation above as: $\displaystyle \frac{1}{\alpha}\frac{d}{dt}f^\alpha(t) = f^{\alpha-1}(t) \dot{f}(t)$ This is exactly equation (11) with $f(t) = h(t)$ and $\alpha = 5/2$. I hope that helps. Let me know if something is still unclear :-)