The relation between tangential velocity (v) and radius (r) is given by:
Vortex Motion in Fluid Dynamics
Vortex motion is a fundamental concept in fluid dynamics, describing the rotational flow of a fluid. This type of flow can be broadly categorized into two main types: forced vortex and free vortex. Understanding the relationship between tangential velocity (\(v\)) and radius (\(r\)) is crucial for differentiating between these flow patterns.
Forced Vortex Flow Analysis
A forced vortex, also known as a solid body rotation, occurs when an external torque is continuously applied to the fluid. In this type of flow, the entire mass of fluid rotates as if it were a rigid body.
- External Torque: An external force causes the fluid to rotate.
- Constant Angular Velocity: Every fluid particle within the forced vortex rotates with the same constant angular velocity (\(\omega\)). This is similar to a spinning wheel.
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Tangential Velocity Relation: The tangential velocity (\(v\)) of any fluid particle is directly proportional to its radial distance (\(r\)) from the center of rotation.
Mathematically, this relationship is expressed as:
\(\mathrm{v} = \omega \mathrm{r}\)
Since \(\omega\) is constant for a forced vortex, dividing both sides by \(\mathrm{r}\) gives:
\(\frac{\mathrm{v}}{\mathrm{r}} = \omega = \text{constant}\) - Irrotationality: A forced vortex is a rotational flow, meaning the curl of the velocity field is non-zero.
Free Vortex Flow Analysis
A free vortex, also known as an irrotational vortex, occurs when there is no external torque applied to the fluid. This type of flow is governed by the conservation of angular momentum. Examples include the flow in a whirlpool or the water draining from a sink.
- No External Torque: The fluid flows freely without any external forces driving its rotation.
- Conservation of Angular Momentum: For ideal, inviscid flow, the angular momentum per unit mass (\(v \times r\)) is conserved for each fluid particle.
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Tangential Velocity Relation: The tangential velocity (\(v\)) of a fluid particle is inversely proportional to its radial distance (\(r\)) from the center.
Mathematically, this relationship is expressed as:
\(\mathrm{v} \times \mathrm{r} = \text{constant}\) (often denoted as \(C\))
So, \(\mathrm{v} = \frac{C}{\mathrm{r}}\) - Irrotationality: A free vortex is an irrotational flow (except possibly at the singular point at the center where velocity approaches infinity in ideal flow), meaning the curl of the velocity field is zero.
Vortex Relation Options Explained
Let's analyze each given option based on the definitions of forced and free vortices:
| Option | Relation | Analysis | Correctness |
|---|---|---|---|
| 1 | \(\rm \frac{v}{r}\) = constant for forced vortex | As derived for a forced vortex, \(\mathrm{v} = \omega \mathrm{r}\), which implies \(\frac{\mathrm{v}}{\mathrm{r}} = \omega\) (a constant angular velocity). This relation accurately describes a forced vortex. | Correct |
| 2 | \(\rm v \times r\) = constant for forced vortex | For a forced vortex, \(\mathrm{v} = \omega \mathrm{r}\). Therefore, \(\mathrm{v} \times \mathrm{r} = (\omega \mathrm{r}) \times \mathrm{r} = \omega \mathrm{r}^2\). This is not constant but varies with \(\mathrm{r}^2\). This relation is characteristic of a free vortex, not a forced vortex. | Incorrect |
| 3 | \(\rm \frac{v}{r}\) = constant for free vortex | For a free vortex, \(\mathrm{v} \times \mathrm{r} = \text{constant}\). This means \(\mathrm{v} = \frac{\text{constant}}{\mathrm{r}}\). Therefore, \(\frac{\mathrm{v}}{\mathrm{r}} = \frac{\text{constant}}{\mathrm{r}^2}\). This is not constant but varies with \(\frac{1}{\mathrm{r}^2}\). | Incorrect |
| 4 | \(\rm v \times r^2\) = constant for free vortex | For a free vortex, \(\mathrm{v} \times \mathrm{r} = \text{constant}\). Therefore, \(\mathrm{v} \times \mathrm{r}^2 = (\frac{\text{constant}}{\mathrm{r}}) \times \mathrm{r}^2 = \text{constant} \times \mathrm{r}\). This is not constant but varies with \(\mathrm{r}\). | Incorrect |
Based on the detailed analysis, the relation \(\rm \frac{v}{r}\) = constant accurately describes the behavior of a forced vortex, where \(v\) is the tangential velocity and \(r\) is the radius.
Important Questions from Fluid Dynamics
The coefficient of contraction Cc for an orifice can be determined using other coefficients; discharge and velocity Cv by the relation.
The ratio of the actual discharge from an orifice to the theoretical discharge from the orifice is defined as:
The equation of continuity of flow is applicable when the-
The science which deals with the action of forces on bodies such that the bodies are at rest is called-
The coefficient of velocity is defined as the ratio of the-
