Continuity equation for a Stream Filament

For a Stream element

$$\text{Flow rate}=\text{Rate of Mass entering - Rate of mass leaving}$$ $$\frac{dm}{dt}={\dot{m}}_1-{\dot{m}}_2\text{\hspace{1em}(1)}$$

Stream element extends from $s$ to $s+ds$ and the location is given by $z(s)$ Naturally all the flow properties $(v,p,\rho )$ are functions of $(s,t)$

Mass flux through left face

$${\dot{m}}_1=\rho \cdot A\cdot v\text{\hspace{1em}(2)}$$

Using the The Taylor Expansion $(x=ds,a=0)$ Mass flux through right face can be approximated to.

$$\begin{array}{rl}{\dot{m}}_2& =\left(\rho +\frac{\partial \rho }{\partial s}ds\right)\cdot \left(A+\frac{\partial A}{\partial s}ds\right)\cdot \left(v+\frac{\partial v}{\partial s}ds\right)\end{array}\text{\hspace{1em}(3)}$$

Ignoring higher order terms ($ds\cdot ds$ terms are ~ $0$)

$$\begin{array}{rl}& =\rho Av+\left(Av\frac{\partial \rho }{\partial s}+\rho A\frac{\partial v}{\partial s}+\rho v\frac{\partial A}{\partial s}\right)ds+...\end{array}\text{\hspace{1em}(4)}$$

Applying reverse chain rule

$$\begin{array}{rl}& =\rho Av+\frac{\partial (\rho Av)}{\partial s}ds\end{array}\text{\hspace{1em}(5)}$$

Flow rate $\dot{m}$ can be expressed as

$$\frac{\partial m}{\partial t}=\frac{\partial (\rho \cdot dv)}{\partial t}=\frac{\partial \rho }{\partial t}\cdot A\cdot ds\text{\hspace{1em}(6)}$$

Substituting values of $\dot{m},m_1$ & $m_2$ in $(1)$

$$\frac{\partial \rho }{\partial t}\cdot A\cdot ds=\rho Av-\left(\rho Av+\frac{\partial (\rho Av)}{\partial s}ds\right)\text{\hspace{1em}(7)}$$

Subsequently

$$\boxed{\frac{\partial \rho }{\partial t}+\frac{1}{A}\cdot \frac{\partial (\rho Av)}{\partial s}=0}\text{\hspace{1em}(8)}$$

Gives the Continuity equation for a 1 Dimensional Compressible flow in a Stream Filament.

Application

1. Compressible Steady Flow

$$\boxed{\rho Av=\text{const}}$$

2. Unsteady Incompressible Flow

$$\boxed{Av=\text{const}=f(t)}$$

3. Steady Incompressible Flow

$$\boxed{Av=\text{const}}$$