The equation of continuity

In addition to these questions of notation, we must also take into account the fact that what happens in one part of a fluid affects what happens in other parts. For example, we must express in equation form the fact that the net flow across a closed surface in the fluid produces a change in the properties of the fluid inside the surface. Suppose f is some property like entropy density or electric charge density that can be created in the fluid and, once created, can be carried along with the fluid as it moves. For one-dimensional situations, where both f and u depend only on x and t, the flux of  f is the total amount off that passes per second in the positive x direction across a unit cross-sectional area perpendicular to x. If f travels with the fluid, this flux J(x,t) must be equal to the density f times the fluid velocity u at x and t.

Now consider the slice of fluid between a plane surface of unit area, normal to the x axis, at x + dx, and a parallel surface at x. The flux J(x + dx) represents a loss off from the region, and the flux J(x) into the region represents a gain. The difference must equal the rate of loss by outflow off from the region. If, in addition, f is created at a rate Q(x,t) per unit volume of fluid, the net rate of change off within the region is

$$\frac{\partial f}{\partial x}dx=Qdx-J(x+dx)+J(x)=Qdx-\frac{\partial J}{\partial x}dx$$

In previous equation partial derivatives are used because we are talking about a region that is fixed in space.

The equation which states that any increase in f in a region must have been brought there by fluid flow or else by specific creation there, as represented by Q is called the equation of continuity.

In particular let’s say that the function f is the mass density of the fluid ρ and Q is the rate of creation of the fluid then we get

$$Q=\frac{\partial \rho }{\partial t}+\frac{\partial \rho u}{\partial x}$$

If we write the density as ρ+δ(x,t) where ρ is the constant density and δ is the variable part.

$$Q=\frac{\partial \delta }{\partial t}+u\frac{\partial \delta }{\partial x}+\left( \rho +\delta \right)\frac{\partial \delta }{\partial x}=\frac{d\delta }{dt}+\left( \rho +\delta \right)\frac{\partial \delta }{\partial x}$$

Even if Q=0, the density of a given portion of fluid can change with time if the fluid velocity changes with x. This equation is valid only for homogeneous fluids.

It is  not difficult to generalize these equations to three dimensions if we use vector notation. Property f in the Euler description is a function of position, denoted by vector r, with components x,y, and z and fluid velocity at r, t is a vector u with components ux, uy, and uz. The flux of f is vector J=fu.

$$\begin{align} & \frac{df}{dt}dt=f(r+udt,t+dt)-f(r,t) \\ & \frac{df}{dt}dt=\left( \frac{\partial f}{\partial t}+{{u}_{x}}\frac{\partial f}{\partial x}+{{u}_{y}}\frac{\partial f}{\partial y}+{{u}_{z}}\frac{\partial f}{\partial z} \right)dt \\ & \frac{df}{dt}=\frac{\partial f}{\partial t}+\left( u\cdot grad \right)f \\ & u\cdot grad\equiv u\cdot \nabla ={{u}_{x}}\frac{\partial }{\partial x}+{{u}_{y}}\frac{\partial }{\partial y}+{{u}_{z}}\frac{\partial }{\partial z} \\ \end{align}$$

Measures the rate of change of f at r,t caused by the fluid flow, it can be used on either a scalar or a vector.

To obtain the three dimensional equation of continuity we first show that the net flux of volume element dxdydz is

$$\begin{align} & dydz\left( \frac{\partial {{J}_{x}}}{\partial x} \right)dx+dxdz\left( \frac{\partial {{J}_{y}}}{\partial y} \right)dy+dxdy\left( \frac{\partial {{J}_{z}}}{\partial z} \right)dz=divJdxdydz \\ & divJ=\nabla \cdot J=\frac{\partial {{J}_{x}}}{\partial x}+\frac{\partial {{J}_{y}}}{\partial y}+\frac{\partial {{J}_{z}}}{\partial z} \\ & \frac{\partial f}{\partial t}\equiv Q(r,t)-divJ=Q-fdiv\text{ }u-u\cdot grad\text{ }f \\ & \frac{df}{dt}=Q-f\text{ }div\text{ u} \\ \end{align}$$