Speed Of Outer Cylinder Given Velocity Gradient Calculator

Angular Speed Formula:

\[ \Omega = \frac{VG}{\frac{\pi \cdot r_2}{30 \cdot (r_2 - r_1)}} \]

Velocity Gradient (VG):

m/s

Radius of Outer Cylinder (r₂):

Radius of Inner Cylinder (r₁):

Angular Speed (Ω):

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1. What is the Angular Speed Formula?

The Angular Speed formula calculates the rotational speed of the outer cylinder based on velocity gradient and cylinder dimensions. This is particularly useful in fluid mechanics for viscosity measurements using coaxial cylinder viscometers.

2. How Does the Calculator Work?

The calculator uses the Angular Speed formula:

\[ \Omega = \frac{VG}{\frac{\pi \cdot r_2}{30 \cdot (r_2 - r_1)}} \]

Where:

\( \Omega \) — Angular Speed (rad/s)
\( VG \) — Velocity Gradient (m/s)
\( r_2 \) — Radius of Outer Cylinder (m)
\( r_1 \) — Radius of Inner Cylinder (m)
\( \pi \) — Mathematical constant (≈3.14159)

Explanation: The formula relates the velocity gradient between coaxial cylinders to the angular speed of the outer cylinder, accounting for the geometric configuration of the system.

3. Importance of Angular Speed Calculation

Details: Accurate angular speed calculation is essential for determining fluid viscosity, analyzing shear rates in rotational viscometers, and understanding fluid behavior in industrial processes and research applications.

4. Using the Calculator

Tips: Enter velocity gradient in m/s, both radii in meters. Ensure the outer cylinder radius is greater than the inner cylinder radius. All values must be positive.

5. Frequently Asked Questions (FAQ)

Q1: What is velocity gradient in this context?
A: Velocity gradient refers to the difference in velocity between adjacent fluid layers in the annular space between the coaxial cylinders.

Q2: Why is the 30 factor included in the formula?
A: The factor 30 is used to convert between different units and ensure dimensional consistency in the calculation.

Q3: What are typical applications of this calculation?
A: This calculation is primarily used in rheology for viscosity measurements, quality control in manufacturing, and research involving non-Newtonian fluids.

Q4: Are there limitations to this formula?
A: The formula assumes laminar flow, Newtonian fluid behavior, and perfect coaxial alignment of the cylinders. Results may vary with turbulent flow or non-Newtonian fluids.

Q5: How does cylinder radius affect angular speed?
A: Larger radius differences generally result in lower angular speeds for the same velocity gradient, while smaller gaps require higher angular speeds to maintain the same gradient.