Radius of Inner Cylinder given Velocity Gradient Calculator

Formula Used:

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

Radius of Inner Cylinder (r₁):

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1. What is the Radius of Inner Cylinder given Velocity Gradient?

The Radius of Inner Cylinder given Velocity Gradient refers to the calculation of the inner cylinder's radius in a rotational viscometer setup, based on the velocity gradient, outer cylinder radius, and angular speed. This is crucial for viscosity measurement in fluid dynamics.

2. How Does the Calculator Work?

The calculator uses the formula:

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

Where:

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

Explanation: The formula calculates the inner cylinder radius by considering the velocity gradient, outer cylinder dimensions, and rotational speed in a coaxial cylinder viscometer setup.

3. Importance of Radius Calculation

Details: Accurate calculation of the inner cylinder radius is essential for proper viscometer calibration, ensuring precise viscosity measurements in various industrial and research applications involving fluid flow characterization.

4. Using the Calculator

Tips: Enter velocity gradient in m/s, outer cylinder radius in meters, and angular speed in rad/s. All values must be positive (velocity gradient and outer cylinder radius > 0, angular speed ≥ 0).

5. Frequently Asked Questions (FAQ)

Q1: What is a velocity gradient in fluid mechanics?
A: Velocity gradient refers to the rate of change of velocity with respect to distance perpendicular to the flow direction, indicating the shear rate in the fluid.

Q2: Why is the inner cylinder radius important in viscosity measurement?
A: The inner cylinder radius determines the gap spacing between cylinders, which directly affects the shear rate and torque measurements used to calculate viscosity.

Q3: What types of fluids can this calculation be applied to?
A: This calculation is primarily used for Newtonian fluids in rotational viscometers with coaxial cylinder geometry.

Q4: Are there limitations to this formula?
A: The formula assumes ideal conditions and may need adjustments for non-Newtonian fluids, end effects, or turbulent flow conditions.

Q5: How does angular speed affect the inner radius calculation?
A: Higher angular speeds generally result in smaller calculated inner radii for a given velocity gradient and outer radius, following the inverse relationship in the formula.