Force by vane on steady stream

A steady stream is a continuous flow in which the same mass of particles passes through any section per second, so the momentum of the particles inside the control region (between the entry section A and the exit section B) does not change with time.

     entry A          vane           exit B
  v_A -->  ======== curved ========  --> v_B
           (mass flow m_dot in = out)

Derivation

Consider the particles in the portion of the stream between A and B at time $t$, plus the mass $\Delta m$ that enters at A during $\Delta t$.

• Momentum at $t$: $\vec H_{AB}+\Delta m\,\vec v_A$, where $\vec H_{AB}$ is the momentum of the particles between A and B.
• Momentum at $t+\Delta t$: $\vec H_{AB}+\Delta m\,\vec v_B$ (the mass $\Delta m$ has left at B; the contents between A and B are unchanged in steady flow).

The impulse-momentum principle: momentum at $t$ + impulse of the external forces = momentum at $t+\Delta t$:

$$\vec H_{AB}+\Delta m\,\vec v_A+\vec F\,\Delta t=\vec H_{AB}+\Delta m\,\vec v_B$$ $$\vec F\,\Delta t=\Delta m(\vec v_B-\vec v_A)$$

Dividing by $\Delta t$ and using $\dot m=\dfrac{dm}{dt}=\rho Av$:

$$\boxed{\vec F=\dot m(\vec v_B-\vec v_A)=\rho Q(\vec v_B-\vec v_A)}$$

where $\vec F$ is the resultant force exerted by the vane on the stream (and $-\vec F$ is the force of the stream on the vane) and $Q=Av$ is the discharge.

• Components: $F_x=\dot m(v_{Bx}-v_{Ax})$, $F_y=\dot m(v_{By}-v_{Ay})$.
• For a vane moving with speed $V$, the same result holds with $\vec v_A,\vec v_B$ taken relative to the vane and $\dot m=\rho A(v_A-V)$ for a single moving vane.