What is Equation of continuity: Definition and 16 Discussions

A continuity equation or transport equation is an equation that describes the transport of some quantity. It is particularly simple and powerful when applied to a conserved quantity, but it can be generalized to apply to any extensive quantity. Since mass, energy, momentum, electric charge and other natural quantities are conserved under their respective appropriate conditions, a variety of physical phenomena may be described using continuity equations.
Continuity equations are a stronger, local form of conservation laws. For example, a weak version of the law of conservation of energy states that energy can neither be created nor destroyed—i.e., the total amount of energy in the universe is fixed. This statement does not rule out the possibility that a quantity of energy could disappear from one point while simultaneously appearing at another point. A stronger statement is that energy is locally conserved: energy can neither be created nor destroyed, nor can it "teleport" from one place to another—it can only move by a continuous flow. A continuity equation is the mathematical way to express this kind of statement. For example, the continuity equation for electric charge states that the amount of electric charge in any volume of space can only change by the amount of electric current flowing into or out of that volume through its boundaries.
Continuity equations more generally can include "source" and "sink" terms, which allow them to describe quantities that are often but not always conserved, such as the density of a molecular species which can be created or destroyed by chemical reactions. In an everyday example, there is a continuity equation for the number of people alive; it has a "source term" to account for people being born, and a "sink term" to account for people dying.
Any continuity equation can be expressed in an "integral form" (in terms of a flux integral), which applies to any finite region, or in a "differential form" (in terms of the divergence operator) which applies at a point.
Continuity equations underlie more specific transport equations such as the convection–diffusion equation, Boltzmann transport equation, and Navier–Stokes equations.
Flows governed by continuity equations can be visualized using a Sankey diagram.

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1. Contradicting values of pressure for liquid cross-sections

Consider a fixed horizontal tube of uniform cross section with pressure being 1atm at one of it's end and 5atm at the other (former due to 'open to atmosphere' and latter due to force on a piston), then liquid would flow towards low pressure end. By equation of continuity all cross sections will...
2. Different answers from equation of continuity and Bernoulli

Hi there, So I was doing the dishes this morning using a sink wand hat can toggle between different flow speeds. The way that I've always thought of this working is using the equation of continuity: Volume flow rate: = Area*velocity Pressing a button on the wand decreases the cross-sectional...
3. Fluid Dynamics: Solving the Equation of Continuity

Homework Statement You are watering your lawn with a hose when you put your finger over the hose opening to increase the distance the water reaches. If you are pointing the hose at the same angle, and the distance the water reaches increases by a factor of 4, what fraction of the hose opening...
4. I Equation of continuity - what if more openings are added?

The equation states that A1V1=A2V2. What about in a situation like a showerhead, where it's one long pipe/tube, then opens up to say, 20 holes. Is it now A1V1=A2V2/20 ? why/why not?
5. Equation of Continuity - Flow rate through an artery

I'm really confused by this question, as it is different than the examples we did in class. I've compared my answer with a few classmates and I'm getting a different one, so I'm not sure if I've done it wrong or if they have. I'd really just like to know if I am on the right track! Thank you :)...
6. Bernoulli's Equation & Equation of Continuity

Hi all, I'm considering an fluids example that's giving me an apparent contradiction when I consider it from the perspective of Bernoulli's Equation vs. the Equation of Continuity. What I'm thinking of is the common observation that putting one's thumb over a garden hose results in an increase...
7. What is the equation of continuity?

How does equation of continuity relate to conservation of mass principle? Laymans terms please.
8. Equation of Continuity: Leaking Bucket & Minus Sign

Suppose their is a bucket with water inside it and there is a small hole at the bottom of the bucket such that water leaks from the end. Area of cross section of the bucket is A and area of the small hole is a. The velocity with which water is coming out of the hole is v and the velocity with...
9. Coordinate Transformation of the equation of continuity for a vaporizing droplet

Hey there, I trying to understand the following coordinate transformation of the equation of continuity (spherical coordinates) for a vaporizing liquid droplet\frac{\partial \rho}{\partial t} + \frac{1}{r^2} \frac{\partial}{\partial r} (r^2 \rho v) = 0 into \epsilon \sigma \frac{\partial...
10. Mechanics question on equation of continuity

Homework Statement A large vertical cylindrical rainwater collection tank of cross sectional area A is filled to a depth h. The top of the tank is open and in the centre of the bottom of the tank is a small hole of cross sectional area B (B<<A). Derive expressions for (i) the flow speed...
11. Equation of continuity of water depth

Homework Statement A can of height h and cross-sectional Area Ao is initially full of water. A small hole of area A1<<Ao is cut in the bottom of the can. Find an expression for the time it takes all the water to drain from the Can. Hint: Call the water depth y use the continuity equation to...
12. Equation of Continuity of charge for point charges

Homework Statement I am looking to demonstrate that the expressions for the charge and current density of point charges satisfy the equation of continuity of charge. Intuitively it makes sense to me but I run into trouble with the delta function when I try to prove it mathematically.Homework...
13. Lorenz guage and equation of continuity

π²³ ∞ ° → ~ µ ρ σ τ ω ∑ … √ ∫ ≤ ≥ ± ∃ · θ φ ψ Ω α β γ δ ∂ ∆ ∇ ε λ Λ Γ ô Homework Statement Show that Lorentz' gauge equation ∇.A = -µ(jωε+σe)φ is the equation of continuity ∇ .Ji = (jωε+σe)/ε P(R) Homework Equations The Attempt at a Solution I tried taking curl, div of...
14. How to Calculate the Speed of Water Exiting a Shower Head with 24 Holes?

Homework Statement A water line with an internal radius of 6.1*10^-3 m is connected to a shower head that has 24 holes. The speed of the water in the line is 1.2 m/s. (b) At what speed does the water leave one of the holes (effective radius = 4.6*10^-4 m) in the head...
15. Solve Equation of Continuity Using Schrodinger Equation

Homework Statement Use the Schrodinger Equation to show that \frac{\partial}{\partial t}(\Psi^{*} \Psi) = - \underline{\nabla}. \underline{j} Homework Equations \underline{j} = \frac{-i}{2m} \left[\Psi^{*}(\nabla \Psi) - (\nabla \Psi^{*})\Psi]\right \frac{\partial}{\partial...
16. Bernoulii's Principle and Equation of Continuity help

Homework Statement A liquid with a specific gravity of 0.9 is stored in a pressurized, closed storage tank to a height of 7 m. The pressure in the tank above the liquid is 8700 Pa. What is the intial velocity of the fluid when a 5 cm valve is opened at a point 0.5 m from the bottom of the...