Conservation of mass
Nothing in this topic is conceptually difficult — mass is not created or destroyed, and a bank balance obeys the same rule. What takes work is turning that plain fact into a statement about a fixed box in space, with fluid streaming through its walls, and then trusting it enough to use it on a duct, a pump or a branching pipe without re-deriving it each time.
Every law in this course eventually gets written for a control volume, because that is the object engineers actually build — a fixed piece of hardware with fluid running through it, not a lump of matter followed around. The Reynolds transport theorem supplies the general machine for making that translation for any quantity at all. Mass is the simplest quantity to run through it, because the answer for a system is not some balance that might tip one way or the other — it is exactly zero, always, by definition. That single fact is the whole of this topic; everything else is applying it carefully.
Mass in, mass out, and storage
Draw a box around anything — a tank, a length of pipe, an entire building's water system — and mass can only change inside that box in one of two ways: some crossed the boundary, or none did and the amount inside is fixed. There is no third mechanism. Write it as a balance:
A tank with a tap running in and a drain running out is the whole idea in miniature. Open the tap alone and the level has nowhere to go but up, because the right-hand side is entirely positive. Open the drain to something slower than the tap and the level still rises, only less steeply, because the two flows partially cancel. Match them exactly and the level stops moving altogether — not because nothing is happening, but because two things are happening at equal and opposite rates. That last case has a name worth fixing early: steady state, and it does not mean the fluid has stopped. It means the amount of it inside the box has stopped changing, which is a much weaker and much more common condition.
The left-hand side is usually the term students under-read. It is not a flow; it is a rate of accumulation, and it can be found from nothing more than the density and the geometry of what is stored: \(m_{cv} = \rho \, V_{cv}\) for a tank of uniform density, so \(dm_{cv}/dt\) is directly the rate at which the liquid level is climbing or falling, scaled by the tank's cross-sectional area. Once that connection is made, the balance becomes something you can watch happen: film a tank filling and draining unevenly, and the picture on the screen is a live plot of \(dm_{cv}/dt\).
The sums on the right are written to admit more than one inlet and more than one outlet on purpose, because almost nothing in practice has exactly one of each. A header feeding three branches, a mixing tank with two supplies and one product line, a reservoir fed by a river and drained by both a spillway and a treatment plant — all of them are this same equation with more terms filled in, not a different equation.
Open the inlet alone and the level has nowhere to go but up.
The continuity equation
The tank argument is intuitive because a tank has an obvious inside and an obvious level to watch. The general statement has to work for a control volume with no free surface at all — a length of pipe, entirely full, with fluid crossing every part of its boundary at once. The route there is the Reynolds transport theorem, with \(b = 1\) and mass on the left, which is fixed for any system whatsoever:
Substituting that into the theorem gives the continuity equation in its full, general form:
The first term is precisely the tank's \(dm_{cv}/dt\) written for a general shape rather than a shape with a convenient liquid level to read off. The second is the net flux of mass through the entire control surface at once, with \(\mathbf{n}\) pointing outward, so the dot product is automatically negative wherever fluid enters and positive wherever it leaves — the sign convention does the bookkeeping the sums did explicitly in the tank version.
Set the flow to steady and the first term is forced to zero by definition, whatever the geometry, leaving
which, for the ordinary case of one inlet and one outlet each crossed by a roughly uniform flow, collapses to \(\dot m_{\text{in}} = \dot m_{\text{out}}\) — exactly the statement that felt obvious for the tank, now derived rather than assumed, and now available in a form that survives an arbitrary number of inlets, outlets, and a control volume of any shape at all.
A differential version exists too, found by shrinking the control volume to a point rather than a finite box:
which for an incompressible fluid — \(\rho\) constant, the working assumption for the rest of this topic and most of liquid flow — reduces further to \(\nabla \cdot \mathbf{V} = 0\). That single line is the constraint every incompressible velocity field in this course has to satisfy, and it is the reason a converging duct necessarily speeds a flow up: there is nowhere else for the divergence to go.
Draw a fixed control volume around any piece of hardware you like.
Steady, incompressible flow through a duct
Apply the integral form to the single case that shows up more than any other: steady flow of a liquid through a duct whose cross-sectional area changes along its length. With one inlet, one outlet, and density constant and equal at both, \(\dot m_{\text{in}} = \dot m_{\text{out}}\) becomes
Density cancels outright, which is the whole benefit of restricting to an incompressible liquid — the balance becomes a statement about area and velocity alone, with nothing to look up in a table. The product \(AV\) is the volume flow rate, \(Q\), and the equation says it is the same at every station along the duct: whatever volume per second is swept through the wide section must be swept through the narrow one too, because none of it is being created, destroyed, or squeezed into storage along a duct with rigid walls and steady flow.
The consequence that catches people off guard the first time is how little say the fluid has in the matter. Narrow the duct to a quarter of its original area and the velocity does not adjust by some complicated amount depending on the fluid's properties — it rises by a factor of exactly four, because that is the only way \(A_1 V_1 = A_2 V_2\) can still hold. Viscosity, density, even the fluid itself, are absent from the equation entirely. This is also the one piece of physics that makes a venturi meter, a nozzle and a nozzle-shaped section of aorta all work by the same geometric argument, well before any pressure has been mentioned.
It is worth being precise about what is being compared at each station. \(V\) here is not the peak velocity on the centreline — real flow through a duct has a full profile, fastest at the middle and zero at the wall — it is the single average velocity that reproduces the correct volume flow rate for that section, which is the subject of the next section in its own right.
Density is fixed, so it drops out of the balance entirely.
Mass flow rate and volume flow rate
Two different quantities get called "the flow rate" depending on who is asking, and using the wrong one is an easy way to be out by a factor of the fluid's density. The volume flow rate is a rate of swept volume:
an integral of the true velocity profile \(u(y)\) across the whole cross-section, since the fluid at the centreline and the fluid near the wall are not moving at the same speed and each patch of area only contributes what actually passes through it. The mass flow rate is the same idea with density folded in:
and for a single fluid of uniform density this is simply \(\dot m = \rho \dot V\). Neither integral is something you evaluate by hand in routine work, because the whole point of the average velocity is to make it unnecessary:
\(V_{\text{avg}}\) is defined precisely so that a single uniform value, spread across the entire area, sweeps out exactly the same volume per second as the real, non-uniform profile — a rectangle of the same area as the curve. Every \(V\) appearing in the duct equation of the previous section is this average, silently, and the shorthand is nearly universal because writing the profile integral out every time would bury the physics in notation for no benefit — the balance only ever needs the total, never the shape.
\(\dot m = \rho A V_{\text{avg}}\) is worth keeping as the one line to reach for first, because it is the form continuity, momentum and energy balances all quote directly, and because it makes the units discipline explicit: get \(\rho\) in kg/m³, \(A\) in m² and \(V\) in m/s, and the answer is unavoidably in kg/s.
The true speed is not one value — it falls to zero at both walls.
Continuity in a pipe network
Nothing new is needed to move from a single duct to a network of pipes that split and rejoin — the same balance is simply applied at every junction in turn, treating each one as its own tiny control volume with no storage of its own, since a junction holds no appreciable volume of fluid to accumulate in:
A single pipe carrying 12 kg/s into a tee that splits into two branches must see those branches carry away exactly 12 kg/s between them, in whatever proportion the resistance of each branch happens to demand — 5 and 7, or 6 and 6, or anything else that sums correctly. What decides the split is a question about pressure drop and resistance, well outside this topic; what continuity guarantees, without needing to know any of that, is the total.
This is precisely the same structure as Kirchhoff's current law in a circuit, and for the same underlying reason: both are conservation statements applied at a node with negligible storage, one for mass and one for charge. Anyone who has balanced a circuit node already has the right instinct for a pipe network — sum what arrives, sum what leaves, and set them equal, however many pipes happen to meet at the point in question.
Scale that idea up to a network with dozens of junctions and it becomes the backbone of how a water distribution system is checked: write one balance equation per node, assign a flow direction and magnitude to every pipe, and the whole set of unknowns is constrained before a single pressure equation has been written down. Continuity supplies the equations that make the network solvable in principle; getting numbers out of it in practice is a question of how much head each branch costs, which is exactly where the energy equation takes over.