How to derive continuity eqn. in polar form?

In summary, the continuity equation in polar form is a mathematical expression that describes the conservation of mass for a fluid in polar coordinates. It can be derived by applying the principle of conservation of mass to a small control volume and has several assumptions, including the fluid being incompressible and the flow being steady. It is a useful tool in fluid dynamics for analyzing the behavior of fluids and is also used in the development of other important equations. However, it has limitations such as assuming a steady, incompressible, and two-dimensional flow, and not taking into account certain factors such as turbulence or compressibility.
  • #1
loveblade
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  • #2
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:
 
  • #3


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
 

1. What is the continuity equation in polar form?

The continuity equation in polar form is a mathematical expression that describes the conservation of mass for a fluid in polar coordinates. It states that the rate of change of mass within a certain region must equal the net flow of mass into or out of that region.

2. How is the continuity equation derived in polar form?

The continuity equation in polar form can be derived by applying the principle of conservation of mass to a small control volume in polar coordinates. This involves using the divergence theorem and simplifying the resulting equations to arrive at the final form of the continuity equation.

3. What are the assumptions made when deriving the continuity equation in polar form?

The main assumptions made when deriving the continuity equation in polar form include the fluid being incompressible, the flow being steady, and there being no external sources or sinks of mass within the control volume. Additionally, the flow is assumed to be two-dimensional and axisymmetric.

4. How is the continuity equation used in fluid dynamics?

The continuity equation is an important tool in fluid dynamics as it allows us to analyze the behavior of fluids in different situations. By using the continuity equation, we can determine how a fluid's velocity and density change in response to changes in the flow or other external factors. It is also used in the development of other important equations in fluid dynamics, such as the Navier-Stokes equations.

5. Are there any limitations to the continuity equation in polar form?

While the continuity equation in polar form is a useful tool in fluid dynamics, it does have some limitations. It assumes a steady, incompressible, and two-dimensional flow, which may not always be the case in real-world scenarios. Additionally, it does not take into account certain factors such as turbulence or compressibility, which can affect the accuracy of the results obtained from using this equation.

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