Fluid Kinematics MCQ Quiz - Objective Question with Answer for Fluid Kinematics - Download Free PDF

Explanation:

Two-Dimensional Flow of Ideal Fluids

• In fluid mechanics, two-dimensional flow refers to a flow scenario where the velocity components are functions of only two spatial coordinates, typically x and y, and the flow parameters do not vary in the third coordinate (z). For an ideal fluid, which is incompressible and inviscid, the flow can be analyzed using mathematical tools such as the stream function and potential function.

Option 3: "Stream function will exist but potential function may or may not exist."

This statement is correct because, in the case of two-dimensional flow of ideal fluids:

1. Stream Function: The stream function (ψ) always exists for two-dimensional, incompressible flows. The stream function is a scalar function whose contours represent streamlines. It helps visualize the flow pattern and satisfies the condition for continuity in two-dimensional flow. For incompressible flow, the continuity equation ensures the existence of the stream function.
2. Potential Function: The potential function (φ) exists only if the flow is irrotational. For a flow to be irrotational, the vorticity (the curl of the velocity field) must be zero. Hence, while the stream function exists for all two-dimensional incompressible flows, the potential function may or may not exist depending on whether the flow is irrotational or not.

Explanation:

Navier-Stokes Equation:

• The Navier-Stokes equation is a set of partial differential equations that describe the motion of fluid substances. These equations are fundamental in fluid dynamics and are used to analyze the behavior of fluids under various forces and conditions. The forces considered in the Navier-Stokes equation govern the dynamics of fluid flow and are integral to understanding fluid behavior in engineering and natural systems.

1. Gravity Forces: Gravity is a body force acting on the fluid due to the weight of the fluid particles. It is represented as a gravitational acceleration term multiplied by the fluid density. Gravity plays a significant role in fluid flow, especially in natural systems like rivers, oceans, and the atmosphere. In the Navier-Stokes equation, the gravity force is accounted for by the body force term, which contributes to the overall momentum balance.

2. Pressure Forces: Pressure forces arise due to the variation in pressure within the fluid. These forces act perpendicular to the surface of a fluid element and are responsible for driving fluid motion in many situations, such as in pipelines, pumps, and atmospheric flows. The pressure gradient term in the Navier-Stokes equation captures the effect of pressure forces on fluid motion.

3. Viscous Forces: Viscous forces are internal frictional forces within a fluid that resist relative motion between adjacent fluid layers. These forces are a result of the fluid's viscosity and play a crucial role in determining the flow characteristics, such as laminar or turbulent flow. The viscous forces are represented by the viscous stress tensor in the Navier-Stokes equation, which accounts for the shear stresses due to velocity gradients within the fluid.

Navier-Stokes Equation:

The Navier-Stokes equation can be expressed in its general form as:

ρ (∂v/∂t + v · ∇v) = -∇p + μ∇²v + ρg

Where:

• ρ = Fluid density
• v = Velocity vector
• t = Time
• p = Pressure
• μ = Dynamic viscosity
• g = Gravitational acceleration
• ∇ = Gradient operator

Concept:

To determine the velocity at a specific point in a two-dimensional flow field, we need to analyze the given velocity field equation and evaluate it at the specified point and time.

Given:

• Point coordinates: (1, 2)
• Time: 2 seconds

Step 1: Identify the velocity field equation

The problem mentions a velocity field equation, but it is not explicitly provided in the given content. Typically, a two-dimensional velocity field is expressed as:

\(\vec{V} = u(x,y,t)\hat{i} + v(x,y,t)\hat{j}\)

where u and v are the velocity components in the x and y directions, respectively.

Step 2: Evaluate the velocity components at (1, 2) and t = 2s

Without the specific velocity field equation, we cannot compute the exact velocity components. However, the problem provides multiple-choice options, suggesting that the calculation leads to one of these values.

Step 3: Calculate the velocity magnitude

The magnitude of the velocity vector is given by:

\(|\vec{V}| = \sqrt{u^2 + v^2}\)

Explanation:

Uniform but Unsteady Flow in Fluid Dynamics

• In fluid dynamics, a uniform flow refers to a flow in which the velocity of the fluid at any given point in the space does not vary with position. However, in an unsteady flow, the velocity can vary with time. Therefore, a uniform but unsteady flow is one where the fluid velocity is the same at every point in the space at any given instant but can change over time.
• In a uniform but unsteady flow, the primary characteristic is that the flow parameters (e.g., velocity, pressure) are the same at every point in a given cross-section at any instant.
• However, these parameters can change as time progresses.
• This type of flow is often seen in situations where external conditions influencing the flow change over time, such as varying wind speeds or fluctuating pressure conditions.

Smoke rising uniformly but varying in velocity from a chimney over time.

• This option correctly illustrates a uniform but unsteady flow. The smoke rises uniformly, meaning its velocity is consistent across any horizontal cross-section at a given height at any instant. However, the velocity of the smoke changes over time, indicating that the flow is unsteady. This can happen due to changes in the temperature of the chimney, variations in the wind speed, or other external factors affecting the velocity of the smoke.

Explanation:

Streamlines in Fluid Flow

Streamlines are an important concept in fluid dynamics, representing the paths that fluid particles follow in a steady flow. Understanding streamlines is crucial for analyzing fluid behavior.

Analyzing the Given Options

1. Option 1: "Lines that are parallel to the equipotential line." (Incorrect)

   • Equipotential lines are perpendicular to streamlines in potential flow, not parallel.

2. Option 2: "Lines along which the stream function is constant." (Correct)

   • In fluid dynamics, a streamline is defined as a line that is tangent to the velocity vector of the flow at every point, which corresponds to a line where the stream function is constant.

3. Option 3: "Equipotential lines along which the velocity potential is constant." (Incorrect)

   • Equipotential lines are lines along which the velocity potential is constant, but these are not streamlines. Equipotential lines are orthogonal to streamlines in irrotational flow.

4. Option 4: "Lines along which vorticity is zero." (Incorrect)

   • Vorticity being zero indicates irrotational flow, but it does not define streamlines. Streamlines can exist in both rotational and irrotational flows.

Concept:

For flow along a stream line acceleration is given as

If V = f(s, t)

Then, \(dV = \frac{{\partial V}}{{\partial s}}ds + \frac{{\partial V}}{{\partial t}}dt\)

\(a = \frac{{dV}}{{dt}} = \;\frac{{\partial V}}{{\partial s}} \times \frac{{ds}}{{dt}} + \frac{{\partial V}}{{\partial t}}\) 

For steady flow \(\frac{{\partial V}}{{\partial t}} = 0\)

Then \(a = \frac{{\partial V}}{{\partial s}} \times \frac{{ds}}{{dt}}\) 

Since V = f(s) only for steady flow therefore \(\frac{{\partial v}}{{\partial s}} = \frac{{dv}}{{ds}}\)

Therefore \(a = V \times \frac{{dV}}{{ds}}\)

Calculation:

Given, V_A = 2 m/s, V_B = 5 m/s, and distance s = 1 m

\(\frac{{dV}}{{ds}} = \frac{{\left( {5 - 2} \right)}}{1} = 3\)

So acceleration of fluid at B is

\({a_B} = {V_B} \times \frac{{dV}}{{ds}} = 5 \times 3 = 15\)

Concept:

Vortex flow:

The motion of a fluid in a curved path is known as vortex flow.

When a cylindrical vessel containing some liquid is rotated about its vertical axis, the vortex flow will be followed by liquid.

Vortex motion is of two types:

1. Forced vortex:

• In the forced vortex, fluid moves on the curve under the influence of external torque.
• Due to the external torque, a forced vortex is a rotational flow.
• As there is the continuous expenditure of energy, Bernoulli's equation is not valid for forced vortex.
• For forced vortex, v = rω is applicable.
• Examples: 
  • The flow of water through a runner of the turbine.
  • Rotation of water in the washing machine.

2. Free vortex:

• When no external torque is required to rotate the fluid mass, that type of flow is called a free vortex.
• As there is no torque in the free vortex, so free vortex is an irrotational flow.
• For free vortex, a moment of momentum is constant i.e. vr = constant.
• Examples:
  • The flow of liquid through a hole provided at the bottom of a container.
  • Draining the bathtub.

∴vortex flow is both rotational and irrotational flow depending on the torque applied.

Explanation:

Incompressible flow: It is that type of flow in which the density is constant for the fluid flow. Liquids are generally incompressible while gases are compressible.

Mathematically, ρ = Constant.

These can be laminar or turbulent, external or internal.

Laminar and Turbulent flow is considered to be incompressible if the density is constant or the fluid expands with little energy in compressing the flow. Hence a flow with varying density (Incompressible) flow could be Laminar & Turbulent.

Additional Information

Compressible flow: The flow in which the density of the fluid changes from point to point or the density is not constant for the fluid

Mathematically, for compressible flow ρ ≠ Constant

Explanation:

Velocity Potential function

• This function is defined as a function of space and time in a flow such that the negative derivation of this function with respect to any direction gives the velocity of the fluid in that direction.

Properties of Velocity Potential function:

• If velocity potential (ϕ) exists, there will be a flow.
• Velocity potential function exists for flow then the flow must be irrotational.
• If velocity potential (ϕ) satisfies the Laplace equation, it represents the possible steady incompressible irrotational flow.

Additional Information

Stream Function:

• It is the scalar function of space and time.
• The partial derivative of stream function with respect to any direction gives the velocity component perpendicular to that direction. Hence it remains constant for a streamline
• Stream function defines only for the two-dimensional flow which is steady and incompressible..

Properties of stream function:

1. If ψ exists, it follows continuity equation and the flow may be rotational or irrotational.
2. If ψ satisfies the Laplace equation, then the flow is irrotational.

Explanation:

Total acceleration of a flow is given by:

\(\frac{D\vec V}{Dt}=\frac{\partial\vec V}{\partial t}+u\frac{\partial\vec V}{\partial x}+v\frac{\partial\vec V}{\partial y}+w\frac{\partial\vec V}{\partial z}\)

The total derivative,

\(\frac{D}{Dt}=\frac{\partial}{\partial t}+u\frac{\partial}{\partial x}+v\frac{\partial}{\partial y}+w\frac{\partial}{\partial z}\)

The total differential D/Dt is known as the material or substantial derivative with respect to time.

The first term \(\frac{\partial}{\partial t}\) in the right hand side is known as temporal or local derivative which expresses the rate of change with time, at a fixed position.

The last three terms \(u\frac{\partial}{\partial x}+v\frac{\partial}{\partial y}+w\frac{\partial}{\partial z}\) in the right hand side of  are together known as convective derivative which represents the time rate of change due to change in position in the field.

Explanation:

Free vortex

When the fluid mass is rotating about an axis without any external torque is known as a free vortex and free vortex motion is irrotational flow.

Forced vortex

When an external force is required to rotate the fluid mass at a constant angular velocity about an axis is known as a forced vortex.

Concept:

Continuity equation: It is the conservation of mass flow rate.

• ρ1A1V1 =  ρ1A1V1

For incompressible fluid density will be constant thus continuity equation will be:

• A1V1 = A2V2  

where, A1, A2 = area of section 1 & 2 respectively, V1, V2 = velocity of section 1 & 2 respectively

The flow rate of liquid is equal to Q = AV.

Calculation:

Given:

Area: A1 = 0.25 m2, A2 = 0.1 m2.

Flow rate: Q = 0.1 m3/s.

Q = A1V1 = A2V2 

\(V_1 = \frac{Q}{A_1} = \frac{0.1}{0.25} = 0.4\ m/s\)

\(V_2 = \frac{Q}{A_2} = \frac{0.1}{0.1} = 1\ m/s\)

∴ The velocity of flow in the reduced pipe is 1 m/s

Concept:

Vortex flow:

The motion of a fluid in a curved path is known as vortex flow.

When a cylindrical vessel containing some liquid is rotated about its vertical axis, the vortex flow will be followed by liquid.

Vortex motion is of two types:

1. Forced vortex:

• In the forced vortex, fluid moves on the curve under the influence of external torque.
• Due to the external torque, a forced vortex is a rotational flow.
• As there is the continuous expenditure of energy, Bernoulli's equation is not valid for forced vortex.
• For forced vortex, v = rω is applicable.
• Examples: 
  • The flow of water through a runner of the turbine.
  • Rotation of water in the washing machine.

2. Free vortex:

• When no external torque is required to rotate the fluid mass, that type of flow is called a free vortex.
• As there is no torque in the free vortex, so free vortex is an irrotational flow.
• For free vortex, a moment of momentum is constant i.e. vr = constant.
• Examples:
  • The flow of liquid through a hole provided at the bottom of a container.
  • Draining the bathtub.

∴ Vortex flow is both rotational and irrotational flow depending on the torque applied.

Explanation:

General Continuity equation:

\(\begin{array}{l} \frac{{\partial \left( {\rho u} \right)}}{{\partial x}} + \frac{{\partial \left( {\rho v} \right)}}{{\partial y}} + \frac{{\partial \left( {\rho w} \right)}}{{\partial z}} + \frac{{\partial \rho }}{{\partial t}} = 0\\ \frac{{\partial \rho }}{{\partial t}} + \vec \nabla.\left( {\rho \vec V} \right) = 0 \end{array}\)

For incompressible and steady flow:

\(\frac{{\partial u}}{{\partial x}} + \frac{{\partial v}}{{\partial y}} + \frac{{\partial w}}{{\partial z}} = 0\)

\(\vec \nabla \cdot {\rm{\vec V}} = 0\)

∴ The flow needs to be steady and incompressible.

Concept:

Any field which satisfied the continuity equation given below is considered as flow field

\(\frac{{\partial u}}{{\partial x}} + \frac{{\partial v}}{{\partial y}} + \frac{{\partial w}}{{\partial z}} = 0\)

Where u, v, ω are x, y and z components of velocity field of flow.

Further ; The flow is said to be rotational is curl of velocity vector (i.e. \(\nabla \times \vec U\) is not equal to zero, otherwise flow is irrotational.

i.e. If \(\nabla \times \vec U = 0\) then flow is irrotational.

Calculation:

Given:

\(\vec U = \left( {2{x^3}} \right)i + \left( {6{x^2}y} \right)\hat I\)

Since given  has component in x and y direction and no component in ‘z’ direction, so it is a 2D velocity field.

Further \(\vec u = 2{x^3}\;\& \;U = 6{x^2}y\) 

\(\frac{{\partial u}}{{\partial x}} = 6{x^2}\;;\frac{{\partial u}}{{\partial y}} = 6\;x^2\)

Since \(\frac{{\partial u}}{{\partial x}} + \frac{{\partial v}}{{\partial y}} = 6{x^2} + 6\;x^2 \ne 0 \Rightarrow \) not a flow.

Since, the given velocity vector did not satisfied the continuity equation, so it did not represent the flow or flow is not possible.

Also, is there is no flow, hence is no sense of saying whether the flow is rotational or irrotational flow.