Outflow velocity from a funnel over time for water without the simplifications made in Torricelli's law I am trying to calculate the outflow velocity $v_2(t)$ from the orifice of a funnel for water. I am aware of Torricelli's law derived from the Bernoulli equation assuming $v_1(t)\approx 0$ resulting in $v_2(t)=\sqrt{2gh_1(t)}$. Since $A_2\ll A_1$ does not apply in my case this equation is not precise enough for my calculations, especially for $h_1\to h_2$.
Is there a more exact model describing the outflow velocity $v_2(t)$?

 A: For the continuity of the flow, we have that:
$$v_1(t)A_1 = v_2(t) A_2.$$
The previous equation reads as follows: the quantity of water entering in the funnel is equal to water exiting from it.
Now, you can use the Bernoulli equation together to the previous one in order to find both $v_1(t)$ and $v_2(t)$.

As a side remark, notice that if $A_1 = A_2$, then the first equation reduces to:
$$v_1(t) = v_2(t)$$ 
as expected. Moreover, in this very case, you also get that $h_1 = h_2$. Of course, we are assuming that the pressure on the top and on the bottom of the funnel is the same.
A: Because this is an unsteady state problem, the usual Bernoulli equation (which applies strictly speaking to steady state flows) is not a good approximation for the flow you are considering.  However, I have presented a method in this link 
Time taken to reach efflux velocity
(2nd answer) that accurately describes the unsteady state flow of an inviscid fluid draining from a vessel.  I suggest you consider applying this method to your draining cone problem.
The basics of this method are presented in Transport Phenomena by Bird, Stewart, and Lightfoot where they present an application of the method to a similar problem.
