Stream Function For Uniform Incompressible Flow In Polar Coordinates Calculator

Stream Function Formula:

\[ \psi = V_{\infty} \times r \times \sin(\theta) \]

Stream Function (ψ):

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1. What is the Stream Function for Uniform Flow?

The Stream Function for uniform incompressible flow in polar coordinates describes the flow pattern where fluid moves with constant velocity in a specific direction. It is a scalar function whose contours represent streamlines of the flow field.

2. How Does the Calculator Work?

The calculator uses the Stream Function formula:

\[ \psi = V_{\infty} \times r \times \sin(\theta) \]

Where:

\( V_{\infty} \) — Freestream velocity (m/s)
\( r \) — Radial coordinate (m)
\( \theta \) — Polar angle (radians)

Explanation: The stream function represents the volume flow rate between the streamline and a reference streamline. For uniform flow, it varies linearly with the radial coordinate and sinusoidally with the polar angle.

3. Importance of Stream Function Calculation

Details: Stream functions are fundamental in fluid dynamics for visualizing flow patterns, analyzing potential flows, and solving various boundary value problems in aerodynamics and hydrodynamics.

4. Using the Calculator

Tips: Enter freestream velocity in m/s, radial coordinate in meters, and polar angle in radians. All values must be positive and valid for physical flow conditions.

5. Frequently Asked Questions (FAQ)

Q1: What is the physical significance of the stream function?
A: The stream function represents the volume flow rate between streamlines. Constant values of ψ define streamlines, and the difference between two stream functions gives the flow rate between them.

Q2: Why use polar coordinates for uniform flow?
A: Polar coordinates are particularly useful for analyzing flows around circular objects or in configurations with radial symmetry, making mathematical treatment more convenient.

Q3: What are the units of stream function?
A: In SI units, stream function has units of square meters per second (m²/s), representing volumetric flow rate per unit depth.

Q4: Can this formula be used for compressible flow?
A: No, this specific formula applies only to incompressible flow where density remains constant throughout the flow field.

Q5: How does uniform flow relate to potential flow theory?
A: Uniform flow is one of the fundamental potential flows and serves as a building block for more complex flow patterns through superposition with other elementary flows.