Nabla operations, vector calculus problem

[Karl Karlsson]

Homework Statement

A liquid flows in a cylindrical tube with the z-axis as the axis of symmetry and radius R. The flow rate of the liquid is given by $$ \vec v = \frac {p_0} {4lμ} (R^2-ρ^2)$$
where $\frac {p_0} {l}$ is the pressure drop in the pipe per unit length, $μ$ is the viscosity of the liquid and $ρ$ is the radial cylinder coordinate. Calculate $∇\cdot\vec v$ and $∇^2\cdot\vec v$

Relevant Equations

$$ \vec v = \frac {p_0} {4lμ} (R^2-ρ^2)$$

Here is how my teacher solved this:

I understand what the nabla operator does, $∇\cdot\vec v$ means that I am supposed to calculate $\sum_{n=1}^3\frac {d\vec v} {dx_n}$ where $x_n$ are cylindrical coordinates and $\vec e_3 = \vec e_z$. I understand why $∇\cdot\vec v = 0$, I would get the same answer as my teacher if I used $\sum_{n=1}^3\frac {d\vec v} {dx_n}$ to calculate it. However when I calculate $∇^2\cdot\vec v$ I don't get the same answer. I would get $∇^2\cdot\vec v = (∇\cdot (∇v_1),∇\cdot (∇v_2),∇\cdot (∇v_3)) = \frac {p_0} {4lμ}\cdot (0,0, ∇\cdot (∇(R^2-ρ^2))) =\frac {p_0} {4lμ}\cdot (0,0, \frac {\partial^2 (R^2-ρ^2)} {\partial ρ^2})$. Which is not equal to what my teacher got. Did I miss something? Also could someone explain the steps he took to arrive at his answer?

Thanks in advance!

 

[Mark44]

Homework Statement:: A liquid flows in a cylindrical tube with the z-axis as the axis of symmetry and radius R. The flow rate of the liquid is given by $$ \vec v = \frac {p_0} {4lμ} (R^2-ρ^2)$$
where $\frac {p_0} {l}$ is the pressure drop in the pipe per unit length, $μ$ is the viscosity of the liquid and $ρ$ is the radial cylinder coordinate. Calculate $∇\cdot\vec v$ and $∇^2\cdot\vec v$
Relevant Equations:: $$ \vec v = \frac {p_0} {4lμ} (R^2-ρ^2)$$

Here is how my teacher solved this:
View attachment 268604
I understand what the nabla operator does, $∇\cdot\vec v$ means that I am supposed to calculate $\sum_{n=1}^3\frac {d\vec v} {dx_n}$ where $x_n$ are cylindrical coordinates and $\vec e_3 = \vec e_z$. I understand why $∇\cdot\vec v = 0$, I would get the same answer as my teacher if I used $\sum_{n=1}^3\frac {d\vec v} {dx_n}$ to calculate it. However when I calculate $∇^2\cdot\vec v$ I don't get the same answer. I would get ##∇^2\cdot\vec v = (∇\cdot (∇v_1),∇\cdot (∇v_2),∇\cdot (∇v_3))

The last expression above doesn't seem right to me, but I am somewhat rusty with this stuff. The part that bothers me is where you're taking the gradient of the individual components $v_1, v_2, v_3$.

= \frac {p_0} {4lμ}\cdot (0,0, ∇\cdot (∇(R^2-ρ^2))) =\frac {p_0} {4lμ}\cdot (0,0, \frac {\partial^2 (R^2-ρ^2)} {\partial ρ^2})##. Which is not equal to what my teacher got. Did I miss something? Also could someone explain the steps he took to arrive at his answer?

Thanks in advance!

 

[vela]

I understand what the nabla operator does, $∇\cdot\vec v$ means that I am supposed to calculate $\sum_{n=1}^3\frac {d\vec v} {dx_n}$ where $x_n$ are cylindrical coordinates and $\vec e_3 = \vec e_z$.

Not exactly. If you're using cartesian coordinates, then
$$\nabla \cdot \vec v = \frac{\partial v_x}{\partial x} + \frac{\partial v_y}{\partial y} +\frac{\partial v_z}{\partial z}.$$ If you're using cylindrical coordinates, however, the divergence takes on a more complicated form. See, for example, https://en.wikipedia.org/wiki/Del_in_cylindrical_and_spherical_coordinates.

I understand why $∇\cdot\vec v = 0$, I would get the same answer as my teacher if I used $\sum_{n=1}^3\frac {d\vec v} {dx_n}$ to calculate it. However when I calculate $∇^2\cdot\vec v$ I don't get the same answer. I would get $∇^2\cdot\vec v = (∇\cdot (∇v_1),∇\cdot (∇v_2),∇\cdot (∇v_3)) = \frac {p_0} {4lμ}\cdot (0,0, ∇\cdot (∇(R^2-ρ^2))) =\frac {p_0} {4lμ}\cdot (0,0, \frac {\partial^2 (R^2-ρ^2)} {\partial ρ^2})$. Which is not equal to what my teacher got. Did I miss something? Also could someone explain the steps he took to arrive at his answer?

I think if you visit that Wikipedia page and see what the Laplacian is in cylindrical coordinates, it'll answer your question. Your notation is a bit off too. Note that your professor wrote $\nabla^2 \vec v$, not $\nabla^2 \cdot \vec v$.

 

[pasmith]

This is another consquence of $\mathbf{e}_\rho$ and $\mathbf{e}_\phi$ not being constant.

In this case it's not too difficult to calculate $\nabla \cdot \mathbf{v}$ and $\nabla^2 \mathbf{v}$ in cartesian coordinates using $$\nabla (\rho^2) = 2x\mathbf{e}_x + 2y\mathbf{e}_y$$ and $$\nabla \cdot (f\mathbf{g}) = f\nabla \cdot \mathbf{g} + \mathbf{g} \cdot \nabla f.$$