All questions of Fluid Kinematics for Mechanical Engineering Exam

Given stream function, ψ = 2x² y

Φ = x2 − y2 + C

Understanding the Stream Function in Cylindrical Coordinates
The stream function, denoted as ψ, is a useful concept in fluid dynamics, particularly in cylindrical polar coordinates (r, θ, z). It helps relate the fluid's velocity components to a scalar function, simplifying the analysis of flow fields.
Velocity Components in Cylindrical Coordinates
In cylindrical polar coordinates, the velocity components can be expressed as:
- Vr: Radial velocity component
- Vθ: Angular (or tangential) velocity component
Relationship Between Stream Function and Velocity Components
The correct relationship between the stream function ψ and the velocity components in cylindrical coordinates is given by option 'C':
- Vr = 1/r * ∂ψ / ∂θ
- Vθ = - ∂ψ / ∂r
Explanation of the Relationships
- Radial Velocity (Vr):
- The expression Vr = 1/r * ∂ψ / ∂θ indicates that the radial component of velocity is proportional to the rate of change of the stream function with respect to the angular coordinate θ. This means that as the flow rotates, the radial component is influenced by how the stream function varies in the angular direction.
- Tangential Velocity (Vθ):
- The expression Vθ = - ∂ψ / ∂r shows that the angular component of velocity is inversely related to the rate of change of the stream function with respect to the radial coordinate r. This negative sign indicates that an increase in the radial distance leads to a decrease in the tangential velocity, reflecting the conservation of angular momentum in the flow.
Conclusion
Understanding these relationships is crucial for analyzing fluid flow in cylindrical systems, providing insight into the behavior of the flow field using the stream function.

A

Introduction:
Free vortex flow refers to a type of fluid flow in which the fluid particles rotate about a central axis without any radial flow. In this flow pattern, the fluid particles move in concentric circles, and the tangential velocity decreases as the radius increases. The following options are given, and we need to identify the one that is not an example of free vortex flow.

Explanation:
Let's analyze each option and determine if it represents free vortex flow or not.

a) Flow of water through the runner of a turbine:
This option represents the flow of water through the runner of a turbine. In this case, the water enters the turbine with a certain velocity and rotational motion. As it passes through the runner, the water particles follow a curved path and rotate about the central axis. Therefore, this option represents free vortex flow.

b) Flow of liquid through a hole provided at the bottom:
This option describes the flow of liquid through a hole provided at the bottom. When a hole is opened at the bottom of a container filled with liquid, the liquid flows out due to the force of gravity. The liquid particles near the hole move with higher velocity, and the velocity decreases as we move away from the hole. However, the rotational motion of the particles is not significant in this case. Therefore, this option does not represent free vortex flow.

c) A whirlpool in a river:
A whirlpool in a river is a classic example of free vortex flow. It occurs when water flows in a circular path, creating a swirling motion. The water particles rotate about a central axis, and the tangential velocity decreases as the radius increases. Therefore, this option represents free vortex flow.

d) Flow of the liquid around a circular bend in a pipe:
When a liquid flows around a circular bend in a pipe, it experiences a combination of radial and tangential flow. The fluid particles near the inner side of the bend experience a higher velocity and tend to move away from the bend. The fluid particles on the outer side of the bend move slower and tend to move towards the bend. This combination of radial and tangential flow does not represent a pure rotational motion about a central axis, which is characteristic of free vortex flow. Therefore, this option does not represent free vortex flow.

Conclusion:
After analyzing each option, we can conclude that option 'A' (Flow of water through the runner of a turbine) is not an example of free vortex flow. The other options, including the flow of liquid through a hole, a whirlpool in a river, and the flow of liquid around a circular bend in a pipe, all represent free vortex flow in different scenarios.

Introduction:
Streamlines and equipotential lines are commonly used in fluid mechanics to visualize the flow pattern and potential distribution in a fluid field. The pattern of these lines varies depending on the type of flow, whether it is a source flow or a sink flow. In this context, the statement claims that the pattern for streamlines and equipotential lines is different for source and sink flow.

Explanation:
To understand why the statement is true, let's first define what source flow and sink flow are:

- Source Flow: In source flow, fluid is flowing outward from a central point, creating a radial flow pattern. It can be visualized as a fluid source that emits fluid particles in all directions.
- Sink Flow: In sink flow, fluid is flowing inward towards a central point, creating a converging flow pattern. It can be visualized as a fluid sink that absorbs fluid particles from all directions.

Now, let's analyze the pattern of streamlines and equipotential lines for each type of flow:

1. Source Flow:
- Streamlines: In source flow, the streamlines are radial lines that originate from the central point and spread outwards. The streamlines are equidistant from each other, indicating a uniform flow rate at every point along the streamlines.
- Equipotential Lines: The equipotential lines in source flow are concentric circles centered at the source point. These lines represent the potential distribution in the flow field. The equipotential lines are perpendicular to the streamlines, indicating that the potential gradient is perpendicular to the flow direction.

2. Sink Flow:
- Streamlines: In sink flow, the streamlines are converging lines that approach the central point. The streamlines become closer to each other as they approach the sink, indicating an increasing flow rate towards the center.
- Equipotential Lines: The equipotential lines in sink flow are also concentric circles centered at the sink point. Similar to source flow, the equipotential lines are perpendicular to the streamlines, indicating the potential gradient direction.

Conclusion:
In conclusion, the pattern for streamlines and equipotential lines is different for source and sink flow. Source flow exhibits radial streamlines and concentric equipotential lines, while sink flow exhibits converging streamlines and concentric equipotential lines. Understanding these patterns is crucial for visualizing and analyzing fluid flow in different scenarios.

The streamlines are defined by

Given flow field: u = 3xy^2 + 2x - y^2 and v = x^2 - 2y - y^3

To determine whether the flow field is physically possible, we need to consider certain properties of the flow field such as continuity equation, irrotational flow, and compressibility.

1. Continuity Equation:
The continuity equation for an incompressible flow is given by ∇ · V = 0, where V is the velocity vector field. Let's calculate ∇ · V for the given flow field:

∇ · V = ∂u/∂x + ∂v/∂y
= (3y^2 + 2) + (2x - 2y - 3y^2)
= 2x - 2y

From the continuity equation, it can be observed that the flow field is incompressible only if ∇ · V = 0. In this case, the flow field is incompressible as the equation evaluates to 2x - 2y.

2. Irrotational Flow:
An irrotational flow field has zero vorticity, which means the curl of the velocity vector field (∇ × V) should be zero. Let's calculate ∇ × V for the given flow field:

∇ × V = (∂v/∂x - ∂u/∂y) i + (∂u/∂x + ∂v/∂y) j
= (-2 + 2x - 6y^2) i + (3y^2 + 2 - 2x) j

From the expression for ∇ × V, it can be observed that the flow field is rotational as it has non-zero vorticity.

3. Compressibility:
To determine compressibility, we need to check if the flow field satisfies the compressible flow equations, which are not given in the question. As the equations are not provided, we cannot determine the compressibility of the flow field.

Conclusion:
Based on the analysis above, we can conclude that the given flow field is possible for steady incompressible rotational flow.

U = 2x2 + y2, v = −4xy

Flow rate for gases is commonly measured in kgf/s (kilogram-force per second).
Here's an explanation of why the correct answer is option 'D':

Unit Analysis:
To understand why the unit for flow rate of gases is kgf/s, let's break down the units:

- 'kgf' stands for kilogram-force, which is a unit of force.
- 's' stands for seconds, which is a unit of time.

Flow Rate Definition:
Flow rate refers to the amount of gas that passes through a given point in a system per unit time. It is commonly expressed in terms of mass or volume per unit time.

- Mass Flow Rate: This is the amount of gas passing through a point in a system per unit time, measured in kilograms per second (kg/s). It represents the mass of gas flowing per unit time.
- Volume Flow Rate: This is the amount of gas passing through a point in a system per unit time, measured in cubic meters per second (m³/s). It represents the volume of gas flowing per unit time.

Conversion between Mass Flow Rate and Volume Flow Rate:
To convert between mass flow rate and volume flow rate for gases, one needs to consider the gas density. The density of a gas is defined as its mass per unit volume. By multiplying the volume flow rate by the gas density, one can obtain the mass flow rate.

- Mass Flow Rate (kg/s) = Volume Flow Rate (m³/s) × Gas Density (kg/m³)

Explanation of the Correct Answer:
The correct answer, option 'D' (kgf/s), represents the flow rate of gases in terms of force per unit time. The use of kilogram-force (kgf) in this context is due to the historical convention of expressing force in terms of the gravitational force exerted on a mass of one kilogram.

- 1 kgf is defined as the force exerted by gravity on a mass of 1 kilogram.
- The unit kgf/s represents the force per unit time, which is equivalent to the mass flow rate (kg/s) when considering the acceleration due to gravity.

Conclusion:
In summary, the unit for flow rate of gases is kgf/s. This unit represents the force per unit time and is equivalent to the mass flow rate when considering the acceleration due to gravity.

To determine the condition for the flow field to be continuous, we need to consider the continuity equation, which states that the rate of mass flow into a control volume must be equal to the rate of mass flow out of the control volume. In two-dimensional flow, this equation can be expressed as:

∂(ρu)/∂x + ∂(ρv)/∂y = 0

where ρ is the density of the fluid.

Now let's substitute the given expressions for u and v into the continuity equation and see if we can simplify it.

1. Substitute u = ax^2 + bxy and v = bxy + ay^2 into the continuity equation:

∂(ρ(ax^2 + bxy))/∂x + ∂(ρ(bxy + ay^2))/∂y = 0

2. Expand the derivatives:

2ax(ρa + ρbx) + 2by(ρb + ρay) = 0

3. Factor out the common terms:

2(ρa + ρbx)(ax + by) = 0

4. Equate each factor to zero:

ρa + ρbx = 0 (1)
ax + by = 0 (2)

Now let's analyze each factor separately.

Condition for (ρa + ρbx) = 0:

The factor ρa + ρbx represents a linear combination of constants (a and b) and variables (x and y). For this factor to be equal to zero, it means that the coefficients of the constants and variables must sum up to zero. Therefore, this condition is independent of the constants (a and b) but dependent on the variables (x and y).

Condition for (ax + by) = 0:

The factor ax + by represents a linear combination of variables (x and y) only. For this factor to be equal to zero, it means that the coefficients of x and y must sum up to zero. Therefore, this condition is independent of the constants (a and b) but dependent on the variables (x and y).

Conclusion:

Since both factors in the continuity equation have conditions that are independent of the constants (a and b) but dependent on the variables (x and y), the condition for the flow field to be continuous is independent of the constants (a and b) but dependent on the variables (x and y). Therefore, the correct answer is option 'A'.

The Laplace equation is a partial differential equation that describes the behavior of many physical phenomena, such as heat flow, electrostatics, and fluid dynamics. In the context of fluid dynamics, the Laplace equation is used to describe the behavior of a fluid flow that is both steady and irrotational.

Velocity Potential Function:

A velocity potential function is a scalar field that describes the velocity of a fluid flow as a function of position. It satisfies the Laplace equation, which means that it describes a fluid flow that is both steady and irrotational.

Stream Function:

A stream function is a scalar field that describes the flow of a fluid in terms of its paths or streamlines. It also satisfies the Laplace equation, which means that it describes a fluid flow that is both steady and irrotational.

Answer:

Option 'B' Only Velocity Potential function satisfies the Laplace Equation. The reason is that the velocity potential function is a scalar field that describes the velocity of a fluid flow as a function of position. It satisfies the Laplace equation, which means that it describes a fluid flow that is both steady and irrotational. The stream function also satisfies the Laplace equation, but it describes the flow of a fluid in terms of its paths or streamlines, not its velocity. Therefore, the correct answer is option 'B'.

Flow through a Doublet
When a uniform flow is flowing through a doublet, the resultant flow obtained is a combination of the effects of both the uniform flow and the doublet.

Resultant Flow
The resultant flow obtained is a combination of both the flow past a Rankine oval of equal axes and the flow past a circular cylinder.

Flow past a Rankine Oval
A Rankine oval is a theoretical shape used to represent the flow around a body in potential flow. When a uniform flow passes a doublet, it creates a flow pattern similar to that past a Rankine oval of equal axes.

Flow past a Circular Cylinder
In addition to the flow around a Rankine oval, the flow past a doublet also resembles the flow around a circular cylinder. The combination of these two flow patterns gives rise to the resultant flow obtained when a uniform flow passes through a doublet.

Conclusion
Therefore, the correct answer is option C - All of the mentioned, as the resultant flow obtained when a uniform flow is flowing through a doublet exhibits characteristics of both flow past a Rankine oval and flow past a circular cylinder.

Introduction:
Ideal fluid flow refers to the flow of a fluid that is assumed to have no viscosity or internal friction. In ideal fluid flow, the fluid particles move without any resistance, and the flow is governed by the principles of conservation of mass and energy. Several types of ideal fluid flow exist, but one of them is not a case of ideal fluid flow.

Forced Vortex Flow:
Forced vortex flow is a type of fluid flow in which the fluid particles move in circles or concentric paths around a central axis. This type of flow occurs when a fluid is forced to rotate within a confined space, such as in a vortex chamber. In forced vortex flow, the tangential velocity of the fluid particles increases as the distance from the central axis increases, while the radial velocity remains constant. This type of flow is a case of ideal fluid flow.

Uniform Flow:
Uniform flow refers to the type of fluid flow in which the velocity of the fluid remains constant at every point in the flow field. In uniform flow, the fluid particles move parallel to each other and have the same velocity magnitude. This type of flow is a case of ideal fluid flow.

Sink Flow:
Sink flow, also known as source flow, refers to the type of fluid flow in which the fluid particles converge towards a central point or sink. In sink flow, the fluid particles move radially inward towards the sink, and the velocity magnitude decreases as the distance from the sink decreases. This type of flow is a case of ideal fluid flow.

Superimposed Flow:
Superimposed flow refers to the type of fluid flow that occurs when two or more flows are combined or superimposed on each other. This can happen, for example, when two fluid streams with different velocities and directions are combined. Superimposed flow can result in complex flow patterns, but it is still a case of ideal fluid flow.

Conclusion:
Out of the given options, the case that is not a case of ideal fluid flow is the forced vortex flow (option A). Forced vortex flow involves a rotating fluid, which introduces rotational motion and non-uniform velocity profiles. In ideal fluid flow, the fluid particles do not experience any resistance or internal friction, and the flow is characterized by uniform velocity and no rotational motion.

Streamlines of free vortex flow

Definition:
Free vortex flow is a type of fluid flow where the fluid particles rotate about a central axis in a circular path without any radial motion. The streamlines in free vortex flow are the paths followed by fluid particles in the flow field.

Nature of streamlines:
The nature of streamlines in free vortex flow is concentric. This means that the streamlines are circular and concentric around a central axis. The circular streamlines indicate that the fluid particles are rotating about the central axis, forming a vortex.

Explanation:
In free vortex flow, the fluid motion is induced by the rotation of a solid body or due to an external force. The streamlines in this type of flow are determined by the balance between the centrifugal force and the pressure gradient.

When a fluid particle is in free vortex flow, it experiences a centrifugal force due to its rotation about the central axis. This centrifugal force causes the fluid particle to move away from the central axis. However, the pressure gradient in the flow field is directed towards the center, acting as a restoring force. This pressure gradient balances the centrifugal force, resulting in a circular path followed by the fluid particle.

As a result, the streamlines in free vortex flow are concentric circles around the central axis. Each streamline represents a different circular path followed by the fluid particles in the flow. The radius of the streamlines increases as we move away from the central axis, indicating an increase in the velocity of the fluid particles.

Conclusion:
In summary, the nature of streamlines in free vortex flow is concentric. The streamlines are circular and concentric around a central axis, representing the paths followed by the fluid particles in the flow field.

Flow is laminar inside the boundary layer and turbulent outside

The statement that flow is laminar inside the boundary layer and turbulent outside is true. This is a characteristic of boundary layer flow, which is the thin layer of fluid that forms along a solid surface when it is exposed to a moving fluid. The boundary layer is divided into two regions: the laminar sublayer and the turbulent layer.

Explanation:

1. Boundary Layer Flow:
- When a fluid flows over a solid surface, such as air flowing over an aircraft wing or water flowing over a ship's hull, a boundary layer is formed.
- This boundary layer is a thin layer of fluid that adheres to the surface due to viscous effects.
- The boundary layer flow can be divided into two regions: the laminar sublayer and the turbulent layer.

2. Laminar Sublayer:
- The laminar sublayer is the region of flow closest to the solid surface.
- In this region, the flow is smooth and ordered, with the fluid particles moving in parallel layers.
- The fluid viscosity dominates in this region, and the flow is characterized by low velocity gradients and smooth streamlines.
- The flow in the laminar sublayer is called laminar flow.

3. Turbulent Layer:
- Outside the laminar sublayer, there is a region called the turbulent layer.
- In this region, the flow is characterized by chaotic motion, with the fluid particles moving in irregular patterns.
- The velocity gradients are higher, and the streamlines become more complex and unpredictable.
- The flow in the turbulent layer is called turbulent flow.

4. Transition from Laminar to Turbulent Flow:
- As the fluid flows over the surface, the laminar flow in the boundary layer may transition to turbulent flow.
- This transition occurs due to factors such as flow velocity, surface roughness, and pressure gradient.
- Once the flow becomes turbulent, it remains turbulent as it moves away from the surface.

5. Flow Characteristics:
- Inside the boundary layer, where the fluid is close to the surface, the flow is influenced by the smooth and ordered laminar sublayer.
- Therefore, the flow inside the boundary layer is laminar.
- Outside the boundary layer, where the flow is further away from the surface, the flow is influenced by the chaotic and irregular turbulent layer.
- Therefore, the flow outside the boundary layer is turbulent.

In conclusion, the statement that flow is laminar inside the boundary layer and turbulent outside is true. This is a characteristic of boundary layer flow, where the flow is smooth and ordered inside the laminar sublayer and chaotic and irregular outside the boundary layer.

Acceleration in radial direction is given by,

To determine the velocity at a given point (2, 3) in a 2D flow, we can use the relationship between the stream function (ψ) and the velocity components (u, v) given by:

u = ∂ψ/∂y
v = -∂ψ/∂x

Given that the stream function ψ = 3xy, we can calculate the velocity components at the point (2, 3) as follows:

1. Calculating ∂ψ/∂y:
∂ψ/∂y = 3x

Substituting x = 2:
∂ψ/∂y = 3(2) = 6

2. Calculating ∂ψ/∂x:
∂ψ/∂x = -3y

Substituting y = 3:
∂ψ/∂x = -3(3) = -9

3. Calculating the velocity components:
u = ∂ψ/∂y = 6
v = -∂ψ/∂x = -(-9) = 9

Therefore, the velocity components at the point (2, 3) are u = 6 and v = 9.

To find the magnitude of the velocity, we can use the following formula:

Velocity magnitude = √(u^2 + v^2)

Substituting the values of u and v:
Velocity magnitude = √(6^2 + 9^2)
Velocity magnitude = √(36 + 81)
Velocity magnitude = √117
Velocity magnitude ≈ 10.82 units

So, the correct answer is option C) 10.82 units.

Ψ = x2 − y 2