Question 11.1: Are the flows represented by the following velocity and densi...
Are the flows represented by the following velocity and density functions physically possible?
A. u = Axy², v = −Ax²y, w = 0, and ρ = Bxy, where A and B are constants
B. vr = U∞(1− R²/r²) cos θ, vθ = −U∞(1+ R²/r²) sin θ, vz = 0, and ρ = Cz + ρ0, where U∞, R, C, and ρ0 are constants
To be physically possible, the continuity equation must be satisfied. For case A, we use Eq. 11.2a, [∂ρ/∂t + u(∂ρ/∂x) + v(∂ρ/∂y) + w(∂ρ/∂z)] + ρ(∂u/∂x + ∂v/∂y + ∂w/∂z) = 0, and substitute the given functions for u, v, w, and ρ. Since this a steady flow, the time derivative is zero. Also, there is no z dependence in the density or velocity, and the z velocity component is zero. Thus, after substituting we find:
(Axy^2)\frac{\partial (Bxy)}{\partial x}+(-Ax^2y)\frac{\partial (Bxy)}{\partial y}(Bxy)\left(\frac{\partial (Axy^2)}{\partial x}+\frac{\partial (-Ax^2y)}{\partial y}\right)=0
Simplifying, we obtain:
(ABxy³) + (−ABx³y) + (ABxy³) + (−ABx³y) = (xy³ −x³y) ≠ 0
Since this flow does not satisfy the continuity equation, it cannot be a physically possible flow.
For case B we use Eq. 11.2b:
\left(\frac{\partial \rho }{\partial t}+v_r\frac{\partial \rho }{\partial r}+\frac{v_{\theta }}{r}\frac{\partial \rho }{\partial \theta }+v_z\frac{\partial \rho }{\partial z}\right)+\rho \left(\frac{1}{r}\frac{\partial (rv_r)}{\partial r}+\frac{1}{r}\frac{\partial v_{\theta } }{\partial \theta }+\frac{\partial v_z}{\partial z}\right)=0
and substitute the given functions, noting that this is a steady flow with vz = 0. Ignoring terms that are zero, the result is
(Cz+\rho _0)\left(\frac{1}{r}\frac{\partial \left\{r[U_∞(1− R^2/r^2)\cos θ]\right\}}{\partial r}+\frac{1}{r}\frac{\partial ∂[−U_∞(1+ R^2/r^2)\sin θ]}{\partial \theta }+\frac{\partial (0)}{\partial z}\right)=0Evaluating the derivatives we find
(Cz+\rho _0)\left[\frac{U_∞ \cos θ }{r} \left(1-\frac{R^2}{r^2}-1-\frac{R^2}{r^2}+\frac{2R^2}{r^2}\right) \right]=0
Since the continuity equation is satisfied, the flow described in case B is a physically possible flow. In fact it describes inviscid flow over a cylinder.