Question 11.10: Consider a constant density fluid in solid body rotation in t...
Consider a constant density fluid in solid body rotation in the absence of body forces as shown in Figure 11.12. Use the Euler equations in streamline coordinates to analyze the direction of pressure change. Use the Euler equations in cylindrical coordinates to confirm your findings.
The velocity field in solid body rotation is described in cylindrical coordinates by
avr = 0, vθ = rΩ, and vz = 0
where Ω is a constant (see Eq. 10.33).
vr = 0, v_θ = rΩ_0, and vz = 0
By inspection the streamlines are circles. Consider a point on a streamline of radius r. The velocity at this point is vθ = rΩ and is directed along the streamline. Thus the speed is V = rΩ and does not change along the streamline. The radius of curvature at this point is ℜ = r. Applying the Euler equations in streamline coordinates, Eqs. 11.20a and 11.20b, we find
\frac{\partial p}{\partial s}=-\rho V\frac{\partial V}{\partial s}=-\rho (r\Omega )=0
\frac{\partial p}{\partial n}=\rho \left(\frac{V^2}{ℜ} \right) =\rho \left(\frac{(r\Omega )^2}{r} \right) =\rho r\Omega ^2
Thus there is no change in pressure along a streamline, and the pressure increases with increasing radius.
The Euler equations in cylindrical coordinates, Eqs. 11.17a–11.17c, are
\rho\left(\frac{\partial v_r}{\partial t}+v_r \frac{\partial v_r}{\partial r}+\frac{v_\theta}{r} \frac{\partial v_r}{\partial \theta}+v_z \frac{\partial v_r}{\partial z}-\frac{v_\theta^2}{r}\right) =\rho f_r-\frac{\partial p}{\partial r}
\rho\left(\frac{\partial v_\theta}{\partial t}+v_r \frac{\partial v_\theta}{\partial r}+\frac{v_\theta}{r} \frac{\partial v_\theta}{\partial \theta}+v_z \frac{\partial v_\theta}{\partial z}+\frac{v_r v_\theta}{r}\right) =\rho f_\theta-\frac{1}{r} \frac{\partial p}{\partial \theta}
\rho\left(\frac{\partial v_z}{\partial t}+v_r \frac{\partial v_z}{\partial r}+\frac{v_\theta}{r} \frac{\partial v_z}{\partial \theta}+v_z \frac{\partial v_z}{\partial z}\right) =\rho f_z-\frac{\partial p}{\partial z}
Substituting the velocity components for solid body rotation, and noting the absence of body force, we find
\rho \left(-\frac{v^2_{\theta }}{r} \right) =\rho \left(-\frac{(r\Omega )^2}{r} \right) =\rho r\Omega ^2=-\frac{\partial p}{\partial r}, 0=-\frac{1}{r} \frac{\partial p}{\partial \theta }, and 0=-\frac{\partial p}{\partial z}
Thus we have ∂p/∂r = ρrΩ², and since n is equivalent to r in this case, this result can be seen to be consistent with that obtained with the Euler equations in streamline coordinates.