Mass Conservation

  Under most circumstances, mass is neither created or destroyed. Therefore, we can say mass is conserved. You are likely already familiar with mass conservation since it's easy to observe day to day. One exception to mass conservation would be nuclear reactions where mass can be transferred into energy. Thankfully most fluid mechanics doesn't rely on nuclear reactions, so we can safely use mass conservation for the problems we'll encounter.

  Imagine a box of arbitrary size and shape such as the tank pictured below. If the amount of fluid in the box is constant in time, the amount of fluid flowing in must be equal to the amount of fluid flowing out. If more fluid is flowing in than is flowing out, the amount of fluid in the tank will increase over time. And vice versa. This form of analysis considering our arbitrary box of fluid is called control volume analysis. It may feel fairly straightforward for mass conservation but control volume analysis is an important tool for fluid mechanics.

  Looking to get a better feel for mass conservation? Try sliding the vertical gates up and down in the animation below to change the flow rates in or out of the tank.

Click and drag the gates below to observe the results.

Mass Flowrate In  

Mass Flowrate Out

Mathematical Statement of Mass Conservation

  Let's transfer this idea of mass conservation to an equation. The simplest expression would be that the time derivative of mass in our control volume is equal to the difference between the mass flow in and out. The dot notation below is used to indicate a time derivative, so \( \dot{m} = \frac{dm}{dt} \).

Accumulation term. The rate at which mass is appearing or disappearing from the control volume. \( {\dot{m}_{CV}} \) \( = \) \( {\dot{m}_{in} - \dot{m}_{out}} \) Flux term. Mass flow in minus mass flow out.
(Simple Form)

   We can rewrite the mass flow rate \( \dot{m} \) in terms of fluid variables if we imagine flow passing through a cross-section A with some velocity v. To go from a volume flow rate to a mass flow rate we can add a factor of density \( \rho \).

\( \dot{m} = \rho A \vec{\mathbf{v}} \)
Mass flow variables labeled in flow through a pipe

  
If the flow properties are different at the inlet and outlet of a given control volume as in the image below, we can query the properties at those locations when evaluating the mass flow. In the example below, the channel widens from left to right. Therefore, either the density or velocity must change between those two points. Typically, the fluid density is a constant (i.e. the fluid is incompressible) so the velocity must have changed. Since the area increased from left to right, the velocity must have decreased from left to right.

Mass flow variables labeled at inlet and outlet of a pipe
Accumulation term. The rate at which mass is appearing or disappearing from the control volume. \( {\dot{m}_{CV}} \) \( 0 \, \) \( = \) \( {{\rho}_{in}{A}_{in}{\vec{\mathbf{v}}}_{in} - {\rho}_{out}{A}_{out}{\vec{\mathbf{v}}}_{out}} \) Flux term. Mass flow in minus mass flow out.
(Simple Form — One Inlet and One Outlet)

  It's worth noting that \( {\dot{m}_{CV}} \) can often be ignored since we are frequently interested in problems where the fluid is incompressible in a confined system such that the amount of mass in the system isn't changing in time. An example, would be the system pictured above. Click the button to activate that assumption and simplify the equation.

Accumulation term. The rate at which mass is appearing or disappearing from the control volume. \( {\dot{m}_{CV}} \) \( 0 \, \) \( = \) \( {\sum\limits_{i=1}^n {\rho}_{i}{A}_{i}{\vec{\mathbf{v}}}_{i} - \sum\limits_{j=1}^m {\rho}_{j}{A}_{j}{\vec{\mathbf{v}}}_{j}} \) Flux term. Mass flow in minus mass flow out.
(Simple Form — Multiple Inlets and Multiple Outlets)

Everyday Applications

One interesting consequence of mass conservation can be seen when constricting and thus accelerating the flow of an incompressible fluid like water. If you ever have put your finger over the end of a hose, you know the water will spray faster and farther the more you cover the end of the hose. If we assume the mass flow rate through the hose is constant, the more you reduce the cross-sectional area A, the greater the velocity v becomes.

Integral Forms

Accumulation term. The rate at which mass is appearing or disappearing from the control volume. \( {\displaystyle \iiint_V \dfrac{\partial\rho}{\partial{t}} \;dV} \) \( 0 \, \) \( = \) \( {-\displaystyle \iint_S \rho (\vec{\mathbf{v}}\cdot\hat{n}) \;dS} \) Flux term. Mass flow in minus mass flow out.
(General Integral Form)

  Where \( \rho \) is the density of the fluid (which potentially varies in our control volume), \( \vec{v} \) is the fluid velocity, and \( \hat{n} \) is the normal vector of the control surface (i.e. surface of control volume)—pointing outwards. Check for yourself that the dot product results in mass flow in being positive and mass flow out being negative.

Differential Forms

  Although the integral form of mass conservation may be more intuitive, it is sometimes preferable to write mass conservation in differential form for an infinitesimal volume of fluid.

Material derivative of density (i.e. how does density change in space and time). \( \dfrac{D\rho}{D{t}} \) \( + \) \( (\nabla\cdot\vec{\mathbf{v}})\rho \) \( = 0 \) The compressibility of the fluid (i.e how the velocity field contracts or expands the fluid locally).
(General Differential Form)

  Expanding the material/lagrangian derivative of density into its time component and space component, we can rewrite the equation as follows.

Time derivative of density. How the fluid density changes along a streamline in the flow. \( \dfrac{\partial\rho}{\partial{t}} \) \( + \) \( (\vec{\mathbf{v}}\cdot\nabla)\rho \) \( + \) \( (\nabla\cdot\vec{\mathbf{v}})\rho \) \( = 0 \) The compressibility of the fluid (i.e how the velocity field contracts or expands the fluid locally).
(General Differential Form)

  If we assume that the fluid is incompressible (i.e. the fluid density is constant in space and time), then the material derivative of density is zero. This results in the simple expression that for an incompressible fluid the divergence of the velocity field is zero.

\( \nabla\cdot\vec{\mathbf{v}} \) \( = 0 \)
(Incompressible Differential Form)

  In Cartesian coordinates, the divergence of the velocity field would be written as follows.

\( \dfrac{\partial{v_x}}{\partial{x}} + \dfrac{\partial{v_y}}{\partial{y}} + \dfrac{\partial{v_z}}{\partial{z}} \)
(Incompressible Differential Form)

Derivation of the Differential Form

Accumulation term. The rate at which mass is appearing or disappearing from the control volume. \( {\displaystyle \iiint_V \dfrac{\partial\rho}{\partial{t}} \;dV} \) \( + \) \( {\displaystyle \iint_S \rho (\vec{\mathbf{v}}\cdot\hat{n}) \;dS} \) \( = 0 \) Flux term. Mass flow in minus mass flow out.

Rewriting, using Gauss's Divergence theorem:

Accumulation term. The rate at which mass is appearing or disappearing from the control volume. \( {\displaystyle \iiint_V \dfrac{\partial\rho}{\partial{t}} \;dV} \) \( + \) \( {\displaystyle \iiint_V \nabla\cdot (\rho\vec{\mathbf{v}}) \;dV} \) \( = 0 \) Flux term. Mass flow in minus mass flow out.
Accumulation term. The rate at which mass is appearing or disappearing from the control volume. \( \displaystyle \iiint_V \) \( \dfrac{\partial\rho}{\partial{t}} \) \( + \) \( \nabla\cdot (\rho\vec{\mathbf{v}}) \) \( \;dV = 0 \) Flux term. Mass flow in minus mass flow out.

Since this integral must equal zero for any arbitrary control volume, we know the integrand must also be zero:

Accumulation term. The rate at which mass is appearing or disappearing from the control volume. \( \dfrac{\partial\rho}{\partial{t}} \) \( + \) \( \nabla\cdot (\rho\vec{\mathbf{v}}) \) \( = 0 \) Flux term. Mass flow in minus mass flow out.