Railways Technical Ability Metal Forming, Casting and Cutting Question Bank Metal Forming

  • question_answer An invicid irrotational flow field of free vortex motion has a circulation constant \[\Omega \,.\] the tangential velocity at any point in the flow field is given by \[\text{ }\!\!\Omega\!\!\text{ /r}\] where, r, is the radial distance from the centre. At the centre, there is a mathematical singularity which can be phiysically substituted by a forced vortex motion \[(r={{r}_{c}}),\] the angular velocity E is given by:

    A) \[\text{ }\!\!\Omega\!\!\text{ /(}{{\text{r}}_{c}}{{)}^{2}}\]                    

    B) \[\text{ }\!\!\Omega\!\!\text{ /}{{\text{r}}_{c}}\]

    C) \[\text{ }\!\!\Omega\!\!\text{ }\,{{\text{r}}_{c}}\]                        

    D) \[\text{ }\!\!\Omega\!\!\text{ }\,\text{r}_{c}^{2}\]

    Correct Answer: A

    Solution :

    At the interface of free and forced vortex, tangential velocities are equal.    \[\frac{\Gamma }{2\pi {{r}_{c}}}=\omega {{r}_{c}}\] Or         \[\omega =\frac{\Gamma }{2\pi }\times \frac{1}{r_{c}^{2}}=\frac{\Omega }{r_{c}^{2}}\] Where \[\omega =\frac{\Gamma }{2\pi }\]

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