Consider the following statements regarding dynamics of inviscid flows : 1. Euler's equation of motion describes the dynamics of inviscid flows. 2. Flows having only tangential velocities with streamlines as concentric circles are known as plane circular vortex flows. 3. A free vortex flow is a rotational vortex flow where the tangential velocity is directly proportional to the radius of curvature. Which of the above statements is/are correct?
Analyzing Statements on Inviscid Flow Dynamics
We will evaluate each statement regarding the dynamics of inviscid flows.
Statement 1: Euler's Equation and Inviscid Flows
Euler's equation is a fundamental equation in fluid dynamics. It represents the conservation of momentum for an inviscid (zero viscosity) fluid. It is derived directly from Newton's second law applied to a fluid particle, neglecting viscous forces. Therefore, Euler's equation accurately describes the dynamics of inviscid flows.
Conclusion: Statement 1 is correct.
Statement 2: Plane Circular Vortex Flows
A plane circular vortex flow is defined by fluid motion where the velocity vector is purely tangential (circumferential) and lies in planes perpendicular to the axis of rotation. The streamlines for such a flow are concentric circles. This description matches the given conditions.
Conclusion: Statement 2 is correct.
Statement 3: Free Vortex Flow Characteristics
A free vortex flow is a type of vortex flow. In an idealized free vortex, the tangential velocity ($v$) is inversely proportional to the radius ($r$) from the center of the vortex, meaning $v \propto \frac{1}{r}$. The statement claims the tangential velocity is directly proportional to the radius of curvature ($v \propto r$), which contradicts the definition of a free vortex.
Conclusion: Statement 3 is incorrect.
Final Conclusion
Based on the analysis, statements 1 and 2 are correct, while statement 3 is incorrect. Therefore, the combination of correct statements is 1 and 2 only.
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