Practice Questions

The continuity equation in fluid flow states that the net rate of inflow into a small volume must be zero. It represents the law of conservation of mass and applies to steady flow where density is constant

States that energy is constant along a streamline

States that energy is constant everywhere in the fluid

States that the net rate of inflow into small volume must be zero

Stream function is a scalar function that defines only for two-dimensional flow. It represents the streamline and is constant along the streamline. The partial derivative of stream function with respect to any direction gives the velocity component perpendicular to that direction.

Streamline, pathline and streakline are different in a flow field. Streamline is an imaginary line that represents the direction of the instantaneous velocity vector at any point.

The ideal flow of any fluid must satisfy the continuity equation, which represents the law of conservation of mass. 
Pascal's law states that the pressure at a point in a fluid at rest is the same in all directions. Viscosity is a property of a fluid by virtue of which it offers resistance to flow, and 
Newton's law of viscosity relates shear stress to shear strain. 
The boundary layer theory describes the thin layer close to the wall in a flow field where viscous stresses are prominent.

Steady uniform flow is a type of flow in which the conditions of flow do not alter with regard to the time at any given point on the passage.It is represented by flow through a long pipe at constant rate. The velocity and hydrodynamic parameters remain constant throughout the flow field.

flow through an expanding tube at an increasing rate

flow through an expanding tube at constant rate

flow through a long pipe at decreasing rate

flow through a long pipe at constant rate

Incompressible flow means the density of the fluid is constant at every point in the flow field.

The stream function in a two-dimensional flow automatically satisfies the continuity equation

A streamline is an imaginary line in a fluid, the tangent which gives the direction of the flow velocity at that position. In other words, it's a line that is everywhere tangent to the velocity vector. It shows the pathway a fluid element will follow in a steady flow. So, the correct answer is option 3

Uniform flow is the type of flow in which the velocity at any given time does not change with respect to space. 
Laminar flow is the flow in which fluid particles flow in layers. 
Turbulent flow is the flow in which fluid particles flow in a random fashion.

Gradually varied flow is steady non-uniform, where the velocity of water remains constant at a specified point but changes from one point to another.

Smoke emitting from a lighted cigarette represents a Streakline, which is the locus of points of all fluid particles that have passed continuously through a particular spatial point in Streamlines are an in space such that the tangent drawn at any point to it will give the direction of velocity.

During the opening of a valve in a pipeline, the flow is unsteady as the hydrodynamic parameters and fluid properties change with time.

Streamlines, streaklines, and pathlines are different ways of representing the movement of fluid particles. Streamlines show the direction of a fluid element at any point, streaklines show the path of all fluid particles that have passed through a specific point, and pathlines show the trajectory of an individual fluid particle. They all coincide in steady flow.

Flow velocities do not change steadily with time

In irrotational flow, fluid particles do not rotate about their own axis while flowing along streamlines. Viscous forces are not present in the irrotational flow.

One-dimensional flow involves zero transverse components of flow and can be expressed as functions of time and one space coordinate. Example: the flow in a pipe is considered one-dimensional when variations of pressure and velocity occur along the length of the pipe, but any variation over the cross-section is assumed negligible.

Involves zero transverse components of flow

The discharge of an incompressible fluid is usually expressed as the volume of flowing fluid per unit of time. This is because the units of discharge for liquids are m3/s or litres/s.

Vortex flow is the motion of a fluid in a curved path. Forced vortex is rotational, while free vortex is irrotational. Irrotational flow is the one where no external torque is required to rotate the fluid mass. Couette flow is the laminar flow of a viscous fluid between two parallel plates, where the flow is driven by viscous drag force. The moment of momentum is constant in a free vortex, and it is applicable as

A forced vortex is a rotational flow influenced by an external torque. The linear velocity of the forced vortex is directly proportional to the radius,

Turns in an opposite direction to a free vortex

Always occurs in conjunction with a free vortex

Has the linear velocity directly proportional to the radius

Has the linear velocity inversely proportional to the radius

Streakline is the locus of points of fluid particles that have passed through a specific point in the past. Streamlines are curves that show the direction of a fluid element at any given time. Pathlines are trajectories of individual fluid particles. For steady flow, all three coincide. Fluid particles cannot cross streamlines. Streamlines converge when accelerated and diverge when retarded.

Vorticity of a fluid is the local rotation of a fluid particle and is defined as the ratio of circulation to the area of a closed contour. It is twice the rotational component and equal to the curl of the velocity field.

A velocity potential function exists only when the flow is irrotational, as it is defined as a scalar function of space and time whose negative derivative with respect to any direction gives the velocity in that direction.

The most relevant statement in fluid kinematics is option 2, which states that the Lagrangian approach considers a fluid particle of fixed mass. This approach tracks individual fluid particles and their properties over time, while the Eulerian approach defines a control volume and describes fluid properties as fields within that volume. Pressure is a scalar quantity, not a vector field. Fluid kinematics deals with the motion of fluids without reference to forces.

In kinematics of fluid flow, pressure is a vector field.

In the Lagrangian approach, a fluid particle of fixed mass is considered.

Fluid kinematics deals with the action of the forces in producing or changing motion of fluid.

In the Eulerian method, control volume approach in not used.

Both velocity potential and stream function exist in both rotational and irrotational flows, but vorticity only exists in rotational flows.

In a free vortex, the flow is irrotational as there is no external torque required to rotate the fluid mass. The momentof momentum is constant, i.e. vr = constant. 
Examples
draining a bathtub or the flow of liquid through a hole in a container.

A static fluid cannot sustain shear stress as it lacks the rigidity to resist deformation under shear forces. Therefore, options involving non-zero shear stress are incorrect. Fluids at rest can only exert normal stresses, which are pressure forces acting perpendicular to any surface. Hence, the correct scenario involves positive normal stress and zero shear stress.

Positive normal stress and zero shear stress

Zero normal stress and non-zero shear stress

Non-zero normal stress and non-zero shear strees

Positive normal stress and non-zero shear stress

When a circular cylinder partially filled with a liquid is rotated about its axis at w rad/s, the centrifugal force causes the liquid to rise along the walls of the cylinder. The height of the liquid rise can be determined by balancing the centrifugal force and gravitational force acting on a liquid element. By doing this, we find that the rise of the liquid surface at the walls is given by w² r²/4g

Rotational Flow	Velocity is zero or infinite
Irrotational flow	Inversely proportional to velocity
Singularities	Vorticity exists
Stream line spacing	Vorticity is zero

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