Fluid Kinematics · Fluid Mechanics · GATE ME
Marks 1
The velocity field of a two-dimensional, incompressible flow is given by $\overrightarrow{V} = \ 2sin{h}\,x\,\hat{i} + v(x,y)\hat{j}$ where $ \hat{i}$ and $\underset{\dot{}}{j}$ denote the unit vectors in x and y directions, respectively. If $v(x, 0) = cosh\ x$, then $v(0,-1)$ is
$$\overrightarrow V = \left( {5 + {a_1}x + {b_1}y} \right)\widehat i + \left( {4 + {a_2}x + {b_2}y} \right)\widehat j,$$
where $${a_1},\,\,{b_1},\,\,{a_2}$$ and $${b^2}$$ are constants. Which one of the following conditions needs to be satisfied for the flow to be incompressible?
Assume both inlet and outlet to be at the same elevation.

Marks 2
A steady two-dimensional flow field is specified by the stream function
ψ = kx3y,
where x and y are in meters and the constant k = 1 m-2s-1. The magnitude of acceleration at a point (x, y) = (1 m, 1 m) is ________ m/s2 (round off to 2 decimal places).
Identify the CORRECT statements from below.
(1) The flow is incompressible
(2) The flow is unsteady
(3) $$y$$-component of acceleration, $${a_y} = {{ - y} \over {{{\left( {{x^2} + {y^2}} \right)}^2}}}$$
(4) $$x$$-component of acceleration , $${a_x} = {{ - \left( {x + y} \right)} \over {{{\left( {{x^2} + {y^2}} \right)}^2}}}$$

$$V = \left( {{a_1}x + {a_2}y + {a_3}z} \right)i + \left( {{b_1}x + {b_2}y + {b_3}z} \right)j$$ $$$ + \left( {{c_1}x + {c_2}y + {c_3}z} \right)k,$$$
Where $${a_1} = 2$$ and $${c_3} = - 4.$$ The value of $${b_2}$$ is _____________.
(i) It is a continuous line such that the tangent at any point on it shows the velocity vector at that point
(ii) There is no flow across streamlines
(iii) $${{dx} \over u} = {{dy} \over v} = {{dz} \over w}$$ is the differential equation of a streamline, where $$u,v$$ and $$w$$ are velocities in directions $$x,y$$ and $$z,$$ respectively
In an unsteady flow, the path of a particle is a streamline
Which one of the following combinations of the statements is true?
$$P:$$ $$u = 2y,\,\,\,v = - 3x$$
$$Q:$$ $$u=3xy,$$ $$\,\,\,\,$$$$v=0$$
$$R:$$ $$u=-2x,$$ $$\,\,\,\,$$$$v=2y$$
Which flows should be recommended when the application requires the flow to be incompressible and irrotational?

The radial velocity $${V_r},$$ at any radius $$r$$, when the gap width is $$h,$$ is

The radial component of the fluid acceleration at $$r=R$$ is
P: Shear stress is zero at all points in the flow.
Q: Velocity is directly proportional to the radius from the centre of the vortex.
R: Total mechanical energy per unit mass is constant in the entire flow field.
S: Total mechanical energy per unit mass is constant in the entire flow field.
$${V_r} = - \left( {{{60 \times {{10}^3}} \over {2\pi r}}} \right)m/s$$
and $${V_\theta } = - \left( {{{300 \times {{10}^3}} \over {2\pi r}}} \right)m/s.$$
Where $$r$$ (in meters) is the distance from the centre of the whirlpool . What will be the distance of the leaf from the centre when it has moved through half a revolution?
Where $$x,y,z$$ are in $$m$$ and $$V$$ $$m/s.$$ Determine if
(i) It represents an incompressible flow
(ii) The flow is irrotational
(iii) The flow is steady .
The magnitude of the velocity at point $$(1,1)$$ is
The rate of shear deformation $${\varepsilon _{yz}}$$ at the point $$x=-2, y=-1$$ and $$z=2$$ for the given flow is :