31. For a steady flow, the velocity field is V = (–x² + 3y) i + (2xy)j. The magnitude of the acceleration of a particle at (1, - 1) is

1. 1. 2

   
   
   

   2. 1

   
   
   

   3. 2√5

   
   
   

   4. 0

   
   
   

---

1. View Hint View Answer Discuss in Forum

   NA

   Correct Option: C

   NA

---

32. For a two-dimensional flow, the velocity field is
    

    u =		x	i +	y	j
    		x² + y²		x² + y²

    
    where i and j are the basis vectors in the x – y Cartesian coordinate system. Identify the CORRECT statements from below.
    1. The flow is incompressible
    2. The flow is unsteady
    3. y-component of acceleration,
    

    ay =	- y
    	(x² + y²)²

    
    4. x-component of acceleration,
    

    ax =	-(x + y)
    	(x² + y²)²

1. 1. 2 and 3

   
   
   

   2. 1 and 3

   
   
   

   3. 1 and 2

   
   
   

   4. 3 and 4

   
   
   

---

1. View Hint View Answer Discuss in Forum

   axu		δu	+ v	δu
   		δx		δy

   
   
   

   =		x(x² + y² -2x²)-2xy²	=	- x3 - xy²
   		(x² + y²)(x² + y²)		(x² + y²)

   
   

   ∴ ax =	x
   	(x² + y²)²

   
   

   ay = u		δv	+ v	δv
   		δx		δy

   
   
   

   =		-x² + yx² - y3	=	- y
   		(x² + y²)3		(x² + y²)²

   
   The velocity components are not functions of time, so flow is steady according to continuity equation,
   
   Since it satisfies the above continuity equation for 2D and incompressible flow.
   ∴ The flow is incompressible.

   Correct Option: B

   axu		δu	+ v	δu
   		δx		δy

   
   
   

   =		x(x² + y² -2x²)-2xy²	=	- x3 - xy²
   		(x² + y²)(x² + y²)		(x² + y²)

   
   

   ∴ ax =	x
   	(x² + y²)²

   
   

   ay = u		δv	+ v	δv
   		δx		δy

   
   
   

   =		-x² + yx² - y3	=	- y
   		(x² + y²)3		(x² + y²)²

   
   The velocity components are not functions of time, so flow is steady according to continuity equation,
   
   Since it satisfies the above continuity equation for 2D and incompressible flow.
   ∴ The flow is incompressible.

---

33. For a certain two-dimensional incompressible flow, velocity field is given by 2xyi – y² j. The streamlines for this flow are given by the family of curves

1. 1. x²y² = constant

   
   
   

   2. xy² = constant

   
   
   

   3. 2xy – y² = constant

   
   
   

   4. xy – constant

   
   
   

---

1. View Hint View Answer Discuss in Forum

   v = 2xy i – y²j
   

   u =		δx	, v = -	δx
   		δy		δx

   
   

   2xy =	δx	= 2
   	δy

   
   2xyδy = δψ
   on integrating
   ψ = xy² + f(x)
   = y¹ + f'(x)
   f'(x) = 0
   ⇒ f(x)= constant
   so ψ= xy² + constant

   Correct Option: B

   v = 2xy i – y²j
   

   u =		δx	, v = -	δx
   		δy		δx

   
   

   2xy =	δx	= 2
   	δy

   
   2xyδy = δψ
   on integrating
   ψ = xy² + f(x)
   = y¹ + f'(x)
   f'(x) = 0
   ⇒ f(x)= constant
   so ψ= xy² + constant

---

34. The volumetric flow rate (per unit depth) between two streamlines having stream function ψ1 and ψ2 is

1. 1. |ψ₁ + ψ₂|

   
   
   

   2. ψ₁ ψ₃

   
   
   

   3. ψ₁ /ψ₂

   
   
   

   4. |ψ₁ – ψ₂|

   
   
   

---

1. View Hint View Answer Discuss in Forum

   NA

   Correct Option: D

   NA

---

35. The velocity field of an incompressible flow is given by V = (a₁x + a₂y + a₃ z)i + (b₁x + b₂y + b₃ z)j + (c₁x + c₂y + c₃ z)k, where a₁ = 2 and c₃ = –4. The value of b₂ is _________.

1. 1. 2

   
   
   

   2. 1

   
   
   

   3. 3

   
   
   

   4. 0

   
   
   

---

1. View Hint View Answer Discuss in Forum

   		δu	+	δv	+	δw	= 0
   		δx		δy		δz

   
   a₁ + b₂ + c₃ = 0
   2 – 4 + b₂ = 0
   b₂ = 2
   

   Correct Option: A

   		δu	+	δv	+	δw	= 0
   		δx		δy		δz

   
   a₁ + b₂ + c₃ = 0
   2 – 4 + b₂ = 0
   b₂ = 2
   

---