Indian Institute of Technology Kharagpur (IIT-K) 2008 B.Tech Mechanical Engineering Fluid mechanics - Question Paper
Fluid Mechanics, ME 21101, Mid Semester Examination (Full Marks 60, time 2 hours)
Ql.
(a) Consider the two dimensional semicircular gate AB of radius 1 m, as shown in the figure. The gate is hinged at the point B. Weight of the gate may be neglected.
(i) Determine the horizontal and vertical components of the net force exerted on the gate by the water (density = 1000 kg/m3) and indicate the appropriate senses of these force components in a sketch (free body diagram of the gate).
(ii) Can the gate remain in equilibrium under this forcing condition?
(iii) Whal is the principal stress at the point O as shown in the figure (center of the circle of which AB is an arc)? What is the orientation of the plane at which this state of stress occurs?
(b)A rectangular tank of dimensions 1 m*2 m><3m is filled with water to one fourth of its height and is closed at a!! sides. !f the tank accelerates towards the right, then what is the minimum value of acceleration for which there is no net force exerted on the top face? Is it possible to have any acceleration greater than that limiting value for which still the force on that face is zero? Give reasons through calculations.
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(c) Consider a tube occupied with immiscible liquid and vapor phases, in which the equilibrium configurations of the interface are shown in the figures below, for two different cases. In each case, pressure at either end of the tube is identical to the atmospheric pressure (pa). Qualitatively sketch how the pressure varies along the axis of the tube (p versus x plot) for the two cases, and justify the nature of your plots. Assume the pressure to be linearly varying with axial position in each individual phase.
Q2 j
+ 6 + 3 = 15 marks]
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(a) A tank, weighing 150 N when empty, contains water with a density of 1000 kg/m . The tank is of cylindrical shape with a diameter of 1 m, and is kept on a smooth slab of ice. Coefficient of static friction between the tank and the ice slab is 0.01 and the coefficient of kinetic friction between the two is 0.001. Assume Torricellis idealization of efflux from a hole in the side of the
tank as V = yj2gh , The hole diameter is 9 cm.
r
A
L
Water - L is
(i) What are the major assumptions under which the Torricellis idealization is valid?
(ii) Does the tank move from its initial state of rest towards the right, if h =0.6 m?
(iii) Derive a differential equation of motion describing the velocity of the tank as a function of time, assuming an initial value of h as ho at t ==0. Express the equation in terms of the instantaneous level of water in the tank (which in turn should also be expressed as a function of time). Solution of the equation of motion is not necessary.
(b) Consider a pipe wall in which a pressure tapping is made, as shown in the figure. Viscous effects in the flow may be neglected. Is it possible that = p2 + pghl If so, under what special circumstances?
Free water surface
[2 +3+ 8+2= 15 marks]
V
Q3.
(a) The velocity components in an inviscid, constant density (= 1000 kg/ m ), steady flow field A A
are given as follows: w = (jc + ,y + z), v = + + z), w = -A(x + y + z), where A is a
dimensional constant, with a numerical value of ! unit. Consider a directed line segment in the How Held, connecting the points Pi (0, 0, 0) and P2 (-3, 3, 0). The pressure is given as zero gauge at the origin.
(i) Is the line P1P2 a streamline? Justify with calculations.
(ii)Can the Bernoullis equation be applied to find the change in pressure experienced on moving from the point P| to the point P2 along the direction P1P2? Justify with calculations.
(iii) What is the pressure at P2 (derive from first principles, starting from Eulers equation of motion in differential form)? What is the stagnation pressure at the same point?
__(A) Ruler's equation of motion in rectangular coordinates is given as
- + h ~(y V) K, where h is the body force per unit mass. (B) You may use the following P ' '
vector identity for your analysis: VA + A'j xVx ,
(b) A designer designs a siphon, as schematically shown in the figure. What is the maximum elevation upto which the water may rise in the siphon, if the atmospheric pressure is pa and the vapor pressure at the prevailing temperature is pv?
(c) A constant density fluid is flowing through a variable area duct with negligible viscous effects. If the flow is unsteady, is it possible to apply the expression = A2V2 for relating the velocities at two points 1 and 2 located on the centerline of the duct at any instant of time, where A| and A2 are the areas of cross section containing the points 1 and 2? Does the same equation apply for a viscous flow? Justify.
[10 + 2 + 3 = 15 marks]
Q4.
(a) The density in a two-dimensional flow-field varies with position as p = k}xy + clt where kj and c, are dimensional constants. The viscosity also varies with position as p. = k2xy + c2, where k: and Cj are dimensional constants. Velocity components describing the flow-field are as: u-kix, v = -kty, where is a dimensional constant and t is the time. A swimmer swims in the
flow-field with a velocity relative to the flow given as yswimmerfflmt ~ -vj
(i) What is the acceleration that the swimmer experiences at a given point (x/, >>/)?
(ii) Consider a two dimensional rectangular fluid element in the above-mentioned flow-field, which originally had two of its adjacent sides oriented along the x and y axis, respectively. Represent the non-zero stress tensor components (with appropriate subscripts or indices) acting on the faces of this identified element. Justify your sketch with reasons. Also sketch a possible deformed configuration of the same fluid element, superimposed on its undeformed configuration.
(iii) For the above flow-field, streamlines, streaklines, and pathlines are not identical, because of the reason that it is an unsteady flow- is this statement correct? Justify your answer.
(iv) Is the flow incompressible or compressible? Justify with calculations, if necessary.
(b) Is it possible that the flow in a converging section of a vertical venturimeter takes place in a direction from lower pressure to higher pressure, if (i) the flow is in the direction of gravity, and
(ii) the flow is opposite to the direction of gravity? Give reasons. Neglect viscous effects.
(c) Why is the angle of the converging cone in a venturimeter steeper than the diffuser angle?
[11 +2 + 2=15 marks]
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