Test 2- Fluid mechanics-ME21101 Indian Institute of Technology Kharagpur-India date: 30/10/2020 • Please workout the questions as a team and submit a single answerscript for a team • Submit the answerscript in .pdf form as a personal message on Microsoft teams as: team00 test2.pdf. You can replace 00 by your team number 01, 02,... 17 etc. • There are 4 questions in Part A and all are compulsory- no choice. • Answers should accompany the question statement, given, find, assumptions made by you and solution in detail • Good hand writing/typesetting is preferred • Just giving some numbers without detailed construction will not be evaluated • Mention your team number and member names on the first page of the answer script • Academic integrity is appreciated. Cross-talking with other teams is not permitted • Answerscripts for evaluation should reach the instructor before 20.00 hrs of 31/10/2020 • Use fundamental principles like the mass continuity, momentum and the energy conservation. You are welcome to use both the integral and differential analysis- no issue. It is a take home examination. • Part B is for experts-which means a bonus grade. Please submit it separately and not as a team, if you wish. Instructor will take a final call in this matter. Part A- 4 questions and all are compulsory- no choice 1. A hemispherical vessel (which holds water) of radius R has a small rounded orifice of area A at the bottom. hShow that the time to lower required i the liquid level from h1 to 3/2 3/2 5/2 5/2 2π 1 2 h2 is given by t = A√2g 3 R h1 − h2 − 5 h1 − h2 . Here, g is the accelaration due to gravity in vertical direction (take the component value). Work it out using the control volume analysis. [10] 2. A tornado can be idealized as a Rankine vortex with a core of diameter 30 m. The gauge pressure at a radius of 15 m is −2000 N/m2 (that is, the absolute pressure is 2000 N/m2 below atmospheric). • Show that the circulation around any circuit surrounding the core is in between 5000 to 6000 m2 /s. • Such a tornado is moving at a linear speed of 25 m/s relative to the ground. Find the time required for the gauge pressure to drop from −500 to −2000 N/m2 . Neglect compressibility effects and assume an air temperature of 25o C, based on that fix physical properties of air. Team number: Please go on to the next page. . . [10] ME21101 Fluid mechanics-ME21101 Page 2 of 2 3. Liquid flowing at a fast speed in a wide, horizontal open channel under some conditions can undergo a hydraulic jump, as shown. For a suitably chosen control volume, the flow which enters and leaves the jump may be considered uniform with hydrostatic pressure distribution. Consider water henters the channel from i D1 = 0.6 m with V1 = 5 p m/s. Show that in general, D2 = D1 1 + 8V12 /gD1 − 1 /2. Evaluate the change in mechanical energy through the hydraulic jump. If heat transfer to the surroundings is neglizible, determine the change in water temperature through the jump. The channel is two-dimensional with unit width normal to the page. [10] 4. A continuous belt, passing upward through a chemical bath at speed U0 , picks up a liquid film of thickness h, density ρ, and viscosity µ. Gravity tends to make the liquid drain down, but the movement of the belt keeps the liquid from running off completely. Assume that the flow is fully developed and laminar with zero pressure gradient, and that the atmosphere produces no shear stress at the outer surface of the film. State clearly the boundary conditions to be satisfied by the velocity at y = 0 and y = h. Obtain an expression for the velocity profile in the fully developed state. [10] Part B- Experts can answer the following question- not mandatory to others 5. Consider a plane flow develops from rest in between two parallel flat plates. The flow is bounded by two rigid boundaries at y = 0 and y = h. The flow motion in x-direction is started from rest by suddenly accelerating the lower plate from rest to a steady velocity U , whereas the upper plate is held stationary. Show that the velocity distribution is ∞ P 2 1 −n2 b t e sin nπy . Here, b = πh2ν , ν is the kinematic given by u(y, t) = U 1 − hy − 2U π n h n=1 viscosity and t is the time. Make suitable assumptions and show the solution step by step. Team number: End of exam [–]
