# Continuity equation

When a fluid molecule suffers decrease in area whether smoothly or sharply its velocity vector strikes the wall but both in case of elastic and inelastic collision it looses/changes its velocity direction and magnitude. Then how its velocity increases when it has passed through the orifice/smaller area which is inferred by using the continuity equation $$A_1 v_1 = A_2 v_2$$.

• Do you think that the continuity equation refers to a single molecule? Irrespective of the conservation of momentum and energy, do you think that mass is conserved? – Chet Miller Feb 26 '20 at 13:17
• Continuity equation talks about the flux passing through an area. So to visualize I take an area $A_1$ and consider vector field(arrows) passing through it in a single direction. Then I take an area $A_2$ again with the same vector field but now densely filled than its prior one considering $A1 > A2$. – Vishesh Mangla Feb 27 '20 at 9:25
• I don't understand the second sentence in your comment. – Chet Miller Feb 27 '20 at 12:29
• Is continuity equation derived by probability theory? – Vishesh Mangla Feb 28 '20 at 2:26
• No. It is a mathematical expression based on conservation of mass. – Chet Miller Feb 28 '20 at 3:17

## 2 Answers

In many cases the pressure is much lower beyond an orifice. The molecules are being pushed from behind. With a decrease in area, again where the velocity is higher, the pressure is lower.

• That is true and can be considered. – Vishesh Mangla Mar 11 '20 at 15:37

I think an answer lies in the assumption that velocity of a particle is always along the streamline and we usually we talk about steady state flow in which the streamline has already been bent in that direction of the flow. The $$\nabla \psi$$ where $$\psi$$ is the stream function is the $$\vec{V}$$ or velocity vector. If you consider transition state flow it would definately suffer momentum loss and there would be a mass reflux and continuity equation wouldn't be valid.