How to derive continuity eqn. in polar form?

[loveblade]

pleasezzzzzzzzzzzzzzzz

 

[jaap de vries]

You know that the continuity equation states out - in + change = 0
Therefore draw a small control colume with dimensions Theta * R, Theta * (R + dR). dR, dZ (looks like piece of pie with the point taken out.
The simply see flow going in on on side out on the other and the difference is the change, dRho/dt * R*dR*dTheta*dz
just to help you out:

 

[ravenprp]

To derive the continuity equation in polar form, we start with the basic definition of continuity, which states that the rate of change of a quantity within a given volume is equal to the net flow of that quantity through the boundaries of the volume. In mathematical terms, this can be written as:

$\frac{\partial \rho}{\partial t} + \nabla \cdot \vec{J} = 0$

where $\rho$ is the density of the quantity, $\vec{J}$ is the flux of the quantity, and $\nabla \cdot$ represents the divergence operator.

In polar coordinates, the divergence operator can be written as:

$\nabla \cdot = \frac{1}{r} \frac{\partial}{\partial r} (r \cdot) + \frac{1}{r \sin \theta} \frac{\partial}{\partial \theta} (\sin \theta \cdot)$

Substituting this into the continuity equation, we get:

$\frac{\partial \rho}{\partial t} + \frac{1}{r} \frac{\partial}{\partial r} (r \vec{J}) + \frac{1}{r \sin \theta} \frac{\partial}{\partial \theta} (\sin \theta \vec{J}) = 0$

Now, we can expand the flux vector $\vec{J}$ in terms of its polar components, $\vec{J}_r$ and $\vec{J}_\theta$, as follows:

$\vec{J} = \vec{J}_r \hat{r} + \vec{J}_\theta \hat{\theta}$

where $\hat{r}$ and $\hat{\theta}$ are the unit vectors in the radial and tangential directions respectively.

Substituting this into the previous equation and simplifying, we get the final form of the continuity equation in polar coordinates:

$\frac{\partial \rho}{\partial t} + \frac{1}{r} \frac{\partial}{\partial r} (r \rho \vec{v}_r) + \frac{1}{r \sin \theta} \frac{\partial}{\partial \theta} (\sin \theta \rho \vec{v}_\theta) = 0$

where $\vec{v}_r$ and $\vec{v}_\theta$ are the radial and tangential components of the