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# The tube shown is of a non-uniform cross section. The cross-section area at A is half of the cross-section area at B, C and D. A liquid is flowing through in steady stateThe liquid exerts on the tube:Statement I: A net force towards right Statement II: A net force towards leftStatement III: A net force in some oblique direction Statement IV: Zero net forceStatement V: A net clockwise torqueStatement VI: A net counter clockwise torque Out of these: (A) Only statement I and V are correct(B) Only statement II and VI are correct(C) Only statement IV and VI are correct(D) Only statement III and VI are correct

Last updated date: 03rd Aug 2024
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Hint: We know that the continuity equation reflects the fact that mass is conserved in any non-nuclear continuum mechanics analysis. The equation is developed by adding up the rate at which mass is flowing in and out of a control volume, and setting the net in-flow equal to the rate of change of mass within it. Continuity principle, or continuity equation, Principle of fluid mechanics. Stated simply, what flows into a defined volume in a defined time, minus what flows out of that volume in that time, must accumulate in that volume. The principle is a consequence of the law of conservation of mass. Based on this equation, it is required to solve this problem.

We know that the force has been exerted by liquid on the tube due to change in momentum at the corners i.e., when liquid is taking turn from A to B and from B to C. As cross - section area at A is half of that of B and C, so velocity of liquid flow at B and C is half to that of velocity at A. Let velocity of flow of liquid at A be v and cross section area at A be S, the velocity of flow of liquid at B and C would be $v / 2$ [from continuity equation] and cross section area at B and C would be 2S.
Due to flow of liquid, it is exerting a force per unit time of $\rho \mathrm{Sv}^{2}$ on the tube, where $\rho$ is the density of liquid, S is cross section area and v is velocity of flow of liquid. The force exerted by liquid on the tube is shown in the figure. Which clearly shows that a net force is acting on the tube due to flowing liquid towards right and a clockwise torque sets in.

Therefore, only statements I and V are correct. So, the correct answer is option A.

Note We know that the common applications of continuity equations are used in pipes, tubes and ducts with flowing fluids or gases, rivers, overall procedure as diaries, power plants, roads, logistics in general, computer networks and semiconductor technologies and some other fields. If steady flow exists in a channel and the principle of conservation of mass is applied to the system, there exists a continuity of flow, defined as the mean velocities at all cross sections having equal areas are then equal, and if the areas are not equal, the velocities are inversely proportional to the areas of flow.