Ideal Fluids in Motion — Question 8

Question 8

A bend requires acceleration even at constant speed

A parcel follows a smooth circular arc of radius $R=0.50\,\mathrm{m}$ at constant speed $v=3.0\,\mathrm{m/s}$. The flow is steady, and the arc turns the parcel through $90^\circ$. The fluid density is $1000\,\mathrm{kg/m^3}$. Ignore gravity in the plane of the bend.

1. Find its acceleration magnitude and the time to traverse the arc.

2. What net force per unit volume is needed, and in which direction? Explain why equal inlet and outlet speeds do not imply zero net force on a parcel.

Question 8 - Solution

A bend requires acceleration even at constant speed

(a) Derivation and calculation

$$\boxed{a_n=v^2/R=3.0^2/.500=18.0\,\mathrm{m/s^2}}\quad\text{inward}.$$$$s=R\Delta\theta=R\pi/2\Rightarrow\boxed{t=s/v=\frac{\pi(.500)}{2(3.0)}=.261799\,\mathrm s}.$$

(b) Result and interpretation

$$\boxed{f_{\mathrm{net}}=\rho a_n=(1000)(18.0)=18000\,\mathrm{N/m^3}}\quad\text{inward}.$$$$|\mathbf v_f|=|\mathbf v_i|,\qquad\mathbf v_f\ne\mathbf v_i\Rightarrow\Delta\mathbf p\ne0.$$A transverse pressure gradient can supply the centripetal force. Equal speeds constrain kinetic energy, not momentum direction.

Physical check

The force is perpendicular to the instantaneous velocity and can turn the parcel without changing its speed.