81. Which combination of the following statements about steady incompressible forced vortex flow is correct?
    P: Shear stress is zero at all points in the flow.
    Q: Vorticity is zero at all points in the flow.
    R: Velocity is directly proportional to the radius from the centre of the vortex.
    S: Total mechanical energy per unit mass is constant in the entire flow field.
    Select the correct answer using the codes given below:

1. 1. P and Q

   
   
   

   2. R and S

   
   
   

   3. P and R

   
   
   

   4. P and S

   
   
   

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1. View Hint View Answer Discuss in Forum

   Clearly zero shear stress and vortex.

   Correct Option: B

   Clearly zero shear stress and vortex.

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82. The 2-D flow with, velocity
    V̄ = (x + 2y + 2) î + (4 – y) ĵ is

1. 1. Compressible and irrotational

   
   
   

   2. Compressible and not irrotational

   
   
   

   3. In compressible and irrotational

   
   
   

   4. In compressible and not irrotational

   
   
   

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1. View Hint View Answer Discuss in Forum

   v̄ = (x + 2y +2)î + (4 - y)ĵ
   u =x + 2y + 2, v = 4 – y
   

   ∴	δv	= 1,	δv	= - 1
   	δx		δy

   
   

   ∴	δv	+ 1,	δv	= 0
   	δx		δy

   
   hence in compressible.
   

   Again, ω =	1			δv	-	δv
   	2			δx		δy

   
   

   =	1	(0 - 2) = - 1.
   	2

   
   hence not irrotational.

   Correct Option: D

   v̄ = (x + 2y +2)î + (4 - y)ĵ
   u =x + 2y + 2, v = 4 – y
   

   ∴	δv	= 1,	δv	= - 1
   	δx		δy

   
   

   ∴	δv	+ 1,	δv	= 0
   	δx		δy

   
   hence in compressible.
   

   Again, ω =	1			δv	-	δv
   	2			δx		δy

   
   

   =	1	(0 - 2) = - 1.
   	2

   
   hence not irrotational.

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83. Existence of velocity potential implies that

1. 1. Fluid is in continuum

   
   
   

   2. Fluid is irrotational

   
   
   

   3. Fluid is ideal

   
   
   

   4. Fluid is compressible

   
   
   

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1. View Hint View Answer Discuss in Forum

   Fluid is irrotational

   Correct Option: B

   Fluid is irrotational

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84. A closed cylinder having a radius R and height H is filled with oil of density ρ. If the cylinder is rotated about its axis at an angular velocity of ω, the thrust at the bottom of the cylinder is

1. 1. πR²ρ.gH

   
   
   

   2. πR2	ρ ω2 R2
      	4

   
   
   

   3. πR²(ρ ω²R² + ρ gH)

   
   
   

   4. πR2		ρ ω2R2	+ ρ gH
      		4

   
   
   

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1. View Hint View Answer Discuss in Forum

   We know that
   

   	δp	=	ρv2	=	ρ.ω2r2	= δω2r
   	δr		r		r

   
   [∵ v = ω × r]
   

   	p	δp =		r	ρω2r dr
   	0			0

   
   

   p =	ρ	ω2r2
   	2

   
   Area of circular ring = 2πrdr
   Force on elementry ring
   = Intensity of pressure × Area of ring
   

   =	ρ	ω2r22πr dr
   	2

   
   ∴ Total force on the top of the cylinder
   
   

   =		R	ρ	ρω2r22π r dr
   		0	2

   
   

   =	ρ	ρω2r22π		R	r3 dr
   	2			0

   
   

   	ρ	. ω22π	R2	=	ρ	ω2 × πR4
   	2		4		4

   
   Thrust at the bottom of the cylinder
   = Weight of water in cylinder + Total force on the top of cylinder
   

   = ρg × πR2 × H +	ρ	ω2 × πR4
   	4

   
   

   = πR2		ρω2R2	+ ρgH
   		4

   Correct Option: D

   We know that
   

   	δp	=	ρv2	=	ρ.ω2r2	= δω2r
   	δr		r		r

   
   [∵ v = ω × r]
   

   	p	δp =		r	ρω2r dr
   	0			0

   
   

   p =	ρ	ω2r2
   	2

   
   Area of circular ring = 2πrdr
   Force on elementry ring
   = Intensity of pressure × Area of ring
   

   =	ρ	ω2r22πr dr
   	2

   
   ∴ Total force on the top of the cylinder
   
   

   =		R	ρ	ρω2r22π r dr
   		0	2

   
   

   =	ρ	ρω2r22π		R	r3 dr
   	2			0

   
   

   	ρ	. ω22π	R2	=	ρ	ω2 × πR4
   	2		4		4

   
   Thrust at the bottom of the cylinder
   = Weight of water in cylinder + Total force on the top of cylinder
   

   = ρg × πR2 × H +	ρ	ω2 × πR4
   	4

   
   

   = πR2		ρω2R2	+ ρgH
   		4

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85. The horizontal and vertical hydro static forces F_x and F_y on the semicircular gate, having a width w into the plane of figure, are
    

1. 1. F_x = ρghrw and F_y = 0

   
   
   

   2. F_x = 2ρghrw and F_y = 0

   
   
   

   3. F_x = ρghrw and F_y = ρgwr²

   
   
   

   4. F_x = 2ρghrw and F_y = πρgwr²/2

   
   
   

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1. View Hint View Answer Discuss in Forum

   F_x = 2ρghrw
   

   Fy = w2 - w1	πr2	. ρg × w =	πr2ρgw
   	2		2

   Correct Option: D

   F_x = 2ρghrw
   

   Fy = w2 - w1	πr2	. ρg × w =	πr2ρgw
   	2		2

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