Suppose that mass is continuously added to the continuum by some external source

Question

Suppose that mass is continuously added to the continuum by some external source at rate r(x,l), as shown below. (For example, a jet engine where jet fuel is continuously added to the air flow.) Derive the continuity equation for this case using a similar reasoning that we used in deriving the continuity equation where the mass was conserved. 1. q(a,t) oatt x=a 2. Suppose that an observers who measures fish concentration in a river travels with a motor boat upstream and downstream with velocity u(xt). All the time the observer is recording the fish concentration as a function of time (t) and the current boat location (x). Calculate the time rate of change of the observed fish concentration.| This rate of change is often referred to as the total time derivative (d/dt)

Explanation / Answer

1. given

mass flow at time t into the control volume = r(x,t)dt

mass flow into the contropl volume due to boundary changes = rho(q(a,t) - q(b,t))dt

hence

integral(rho*dx) from a to b = r*(x,t)dt + rho(dq/dx)(b - a)dt

for constant density,

rho = integral(rdt)/(b - a) + rho*integral(dq/dx)dt

hence this is the continuity equation for this case

integral(rdt)/(b - a) = rho(1 - integral(dq/dx)dt)

2. motor boat speed = u(x,t)

fish concentration as a funciton of time and position = c(x,t)

total time derivative

Dc/dt = u(x,t)dc/dx + dc/dt

hence

Dc/Dt = u(x,t)dc(x,t)/dx + dc(x,t)/dt

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