GATE Past Year Questions: Fluid Kinematics | Fluid Mechanics for Mechanical Engineering PDF Download

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 in compressible.

Now magnitude of particleat (1, – 1)

 

on integrating

ψ = xy² + f (x)
= y¹ + f'(x)
f'(x) = 0
⇒ f (x)= constant
so y = xy² + constant

Slope of equipotential Line x slope of stream function = 1 1 They are orthogonal to each other.

Given, u = x²t and v = – 2xyt
Integrating equation (i), we get

ψ =-x²yt + f(y) ...(iii)
Differentiating equation (iii) with respect to y, we get
∂ψ/∂y =–x²t + f(y) ...(v)
Equating the value of ∂ψ/∂y from equations (ii)
and (iv), we get
–x²t = –x²t + f'(y)
Since, f'(y) = 0, thus f(y) = C
(where 'C' is constant of integration)
ψ = -x²yt + C
C is a numerical constant so it can be taken as zero
ψ = -x²yt
For equation of stream lines,
ψ = constant
-x²yt =constant
For a particular instance,
x²y = constant

Given: u_x = ax and u_y = ay
Equation of steam line is,

Integrating both sides, we have
log(ax) = log(ay) + log c
or ax = c×ay
or x = cy
Since the steam line is passed through point (1, 2), therefore
1 = 2c
⇒ c = 1/2
∴ x = y/2
Hence equation of steam line is
2x – y = 0.

(2) 1 –d, 2 – d, 3 - c
(3) Closed contour in a flow field.

x direction scalar of velocity field,
u = dx/dt
u = -kx.e^-kt
y direction scalar of velocity field
v = dy/dt
v = ky₀e^kt

u & v are non zero scalar t ≥ 0 so it is 2D flow. 2D possible flow field

0 + 0 = 0 continuity satisfied.

So, flow is unsteady.

C is the false statement 2D incompressible flow continuity equation.

A – A = 0 it satisfies continuity equation.

As y → ∞.velocity vector field will not be defined along y axis.
So flow will be along x-axis i.e. 1-D flow
⇒ Stream line equation for 2D


In x = – ln y + ln c
ln xy = ln c
xy = c → stream line equation.

Incompressible flow condition

The basic equation of continuity for fluid flow is given by

Now if ρ remains constant, then only we can write

hence incompressible flow.

Clearly zero shear stress and vortex.

hence incompressible.

hence not irrotational.

For continuous and in compressible flow
u_x + u_y = 0
a₁ + b₂ = 0

Convective acceleration is the effect of tim e independent acceleration of fluid with respect to space that means flow is steady non-uniform flow.

Two dimensional velocity field with velocities u, v and along x and y direction.
∴ Acceleration along x direction, a_x = a_convective + a_{temporal or local}

Sicne, ∂u/∂x   = 0 for 2-dimensional field, therefore
Convective acceleration

Radial distance = 120 m


By equating (i) & (ii), we get

By solving above, we get
r = 64 m

For 2D incompressible flow,

Stream function,
ψ = x² - y²