Test: Fluid Flow Kinematics - 1 - Civil Engineering (CE) MCQ

The correct answer is:

B: these lines intersect each other orthogonally forming curvilinear squares

Explanation:
A flownet is a graphical representation used in fluid mechanics and groundwater flow analysis. It consists of two sets of lines:
1. Streamlines: Represent the path followed by fluid particles.
2. Equipotential lines: Represent lines of constant potential (e.g., hydraulic head in groundwater flow).

These lines intersect each other orthogonally (at right angles) and form a grid of curvilinear squares. This property is a key characteristic of a flownet and is used to analyze flow patterns and calculate quantities like flow rate or pressure distribution. 

The other options are incorrect:
- A: Streamlines indicate the direction of flow, but not the magnitude of the velocity vector directly.
- C: The lines do not intersect at various angles; they intersect orthogonally.
- D: The velocity potential does not necessarily increase in the direction of flow; it depends on the specific flow conditions.

The correct answer is:

B: steady flow

Explanation:
In fluid mechanics, streamlines, streak lines, and path lines are concepts used to describe the motion of fluid particles. These lines are defined as follows:

Streamlines: Lines that are tangent to the velocity vector of the fluid at every point at a given instant in time. They represent the instantaneous direction of flow.

Streak Lines: Lines formed by all the fluid particles that have passed through a specific point in the flow field over a period of time.

Path Lines: The actual trajectories followed by individual fluid particles over time.

In steady flow, the velocity field does not change with time. As a result:

Streamlines remain fixed in space.

Path lines (trajectories of particles) coincide with streamlines because the flow direction does not change over time.

Streak lines (traces of particles passing through a point) also coincide with streamlines and path lines because the flow pattern is constant.

Other options can't be right as:

A: Uniform flow: Uniform flow means the velocity is the same at every point in the flow field, but it does not guarantee that the flow is steady (time-independent). If the flow is unsteady, the lines will not coincide.

C: Unsteady flow: In unsteady flow, the velocity field changes with time, so streamlines, streak lines, and path lines will generally differ.

D: Non-uniform flow: Non-uniform flow means the velocity varies in space, but this does not imply steadiness. If the flow is unsteady, the lines will not coincide.

A is correct because a velocity potential exists in irrotational flow.

B, C, and D are incorrect because they do not necessarily apply to irrotational flow of an ideal fluid.

Integrating (i), we get,

Differentiating (iii) w.r.t. y, we get,

Equating (ii) and (iv), we get,
f'(y) = 0    ....... (v)
Integrating (v), we get,
f(y) = C
where C is a numerical constant which can be treated as zero.
From (iii),. we get,
 

For two dimensional incompressible flow, :
(i) the continuity equation is' 
(ii) the equation of a stream line is vdx - udy = 0

A path-line may be defined as the line traced by a single fluid particle as it moves over a period of time, Thus a path-line will show the direction of velocity of the same fluid particle at successive instants of time. As indicated earlier a streamline on the other hand shows the direction of velocity of a number of fluid particles at the same instant of time. A fluid particle always moves tangentially to the streamline, and in the case of steady flow since there is no change in direction of the velocity vector at any point with time, the streamline is fixed in space. Therefore in steady flow the pathlines and streamlines are identical. However, in unsteady flow since the direction of velocity vector at any point may change with time, streamline may shift in space from instant to instant. A particle then follows one streamline at one instant and another at the next instant and so on, so that the path of the particle may have no resemblance to any given instantaneous streamline. In other words, in unsteady flow path-lines and streamlines are different.

at (1, 1) V_x = -2 m/s = V_y