Test: Euler's Equation - Civil Engineering (CE) MCQ

Fluid Kinematics:

• Study of the motion of the fluid without any reference of force and moment is known as fluid Kinematics. There are two descriptions to define the fluid motion.
• Lagrangian Description.
• Eulerian Description.

​Lagrangian Description:

• The Lagrangian Description is one in which individual fluid particles are tracked, much like the tracking of billiard balls in a highschool physics experiment.
• In the Lagrangian description of fluid flow, individual fluid particles are "marked," and their positions, velocities, etc. are described as a function of time.
• The physical laws, such as Newton's laws and conservation of mass and energy, apply directly to each particle.
• If there were only a few particles to consider, as in a high school physics experiment with billiard balls, the Lagrangian description would be desirable.
• However, fluid flow is a continuum phenomenon, at least down to the molecular level. It is not possible to track each "particle" in a complex flow field.
• Thus, the Lagrangian description is rarely used in fluid mechanics.

​Eulerian Description:

• The Eulerian Description is one in which a control volume is defined, within which fluid flow properties of interest are expressed as field
• In the Eulerian description of fluid flow, individual fluid particles are not identified. Instead, a control volume is defined.
• Pressure, velocity, acceleration, and all other flow properties are described as fields within the control volume.
• In other words, each property is expressed as a function of space and time, as shown for the velocity field in the diagram.
• In the Eulerian description of fluid flow, one is not concerned about the location or velocity of any particular particle, but rather about the velocity, acceleration, etc. of whatever particle happens to be at a particular location of interest at a particular time.
• Since fluid flow is a continuum phenomenon, at least down to the molecular level, the Eulerian description is usually preferred in fluid mechanics.
• Note, however, that the physical laws such as Newton's laws and the laws of conservation of mass and energy apply directly to particles in a Lagrangian description. Hence, some translation or reformulation of these laws is required for use with an Eulerian description.
• Either description method is valid in fluid mechanics, but the Eulerian description is usually preferred because there are simply too many particles to keep track of in a Lagrangian description.

• Euler’s equation of motion of an ideal fluid, for a steady flow along a streamline, is basically a relation between velocity, pressure, and density of a moving fluid.
• Euler’s equation of motion is based on the basic concept of Newton’s second law of motion.
• The Euler’s equation of motion is a moment of momentum equation.

Euler's equation of motion:

The Euler's equation for a steady flow of an ideal fluid along a streamline is a relation between the velocity, pressure and density of a moving fluid. It is based on the Newton's Second Law of Motion which states that if the external force is zero, linear momentum is conserved. The integration of the equation gives Bernoulli's equation in the form of energy per unit weight of the following fluid.

It is based on the following assumptions:

The fluid is non-viscous (i,e., the frictional losses are zero)
The fluid is homogeneous and incompressible (i.e., the mass density of the fluid is constant)
The flow is continuous, steady and along the streamline.
The velocity of the flow is uniform over the section.
No energy or force (except gravity and pressure forces) is involved in the flow.
As there is no external force applied (Non-viscous flow), therefore linear momentum will be conserved.

Euler's equation of motion:

• Euler's equation for a steady flow of an ideal fluid along a streamline is a relation between the velocity, pressure, and density of a moving fluid.
• It is based on Newton's Second Law of Motion which states that if the external force is zero, linear momentum is conserved.
• The integration of the equation gives Bernoulli's equation in the form of energy per unit weight of the following fluid.

It is based on the following assumptions:

1. The fluid is non-viscous (i,e., the frictional losses are zero)
2. The fluid is homogeneous and incompressible (i.e., the mass density of the fluid is constant)
3. The flow is continuous, steady and along the streamline.
4. The velocity of the flow is uniform over the section.
5. No energy or force (except gravity and pressure forces) is involved in the flow.
6. As there is no external force applied (Non-viscous flow), therefore linear momentum will be conserved.

When a cylinder is rotated such that half of the liquid spills out. Then liquid left in a cylinder at height Z/2,

And liquid will rise at the wall of the cylinder by the same amount as it falls at the centre from its original level at rest.

Euler’s Equation of motion in the differential form given by the following equation:

This equation is based on the assumptions that the flow is ideal and viscous forces are zero.

The integration of the Euler’s equation of motion with respect to displacement along a streamline gives the Bernoulli equation.

Euler's equation of motion:

The Euler's equation for a steady flow of an ideal fluid along a streamline is a relation between the velocity, pressure and density of a moving fluid. It is based on the Newton's Second Law of Motion which states that if the external force is zero, linear momentum is conserved. The integration of the equation gives Bernoulli's equation in the form of energy per unit weight of the following fluid.

It is based on the following assumptions:

• The fluid is non-viscous (i,e.,inviscid fluid or the frictional losses are zero)
• The fluid is homogeneous and incompressible (i.e., the mass density of the fluid is constant)
• The flow is continuous, steady and along the streamline.
• The velocity of the flow is uniform over the section.
• No energy or force (except gravity and pressure forces) is involved in the flow.

As there is no external force applied (Non-viscous flow), therefore linear momentum will be conserved.

Concept

Euler's equation of Motion

• Euler's equation for a steady flow of an ideal fluid along a streamline is a relation between the velocity, pressure, and density of a moving fluid.
• It is based on Newton's Second Law of Motion.
• The integration of the equation gives Bernoulli's equation in the form of energy per unit weight of the following fluid.
• Euler equation of motion in which the forces due to gravity and pressure are taken into consideration

Assumptions:

1. The fluid is non-viscous (i,e., the frictional losses are zero)
2. The fluid is homogeneous and incompressible (i.e., the mass density of the fluid is constant)
3. The flow is continuous, steady and along the streamline
4. The velocity of the flow is uniform over the section
5. No energy or force (except gravity and pressure forces) is involved in the flow

Stokes Law: 

• According to Stokes law, the backward viscous force acting on a small spherical body of radius 'r' moving with uniform velocity V through the fluid of viscosity η is given by 

F = 6 π η r V

Where η = Coefficient of viscosity, r = radius of the sphere, and V= Velocity 

From above it is clear that,  the viscous drag force as F = – 6πηav called Stokes' Law. Therefore option 3 is correct.

• From the figure, it is clear that the drag force fd is in the opposite direction fg
• Stoke's law is applicable for spherical particles only. The fine clay particles are not spherical in shape.
• While applying Stoke's law, the concept of equivalent diameter is used. The equivalent diameter of a soil particle is defined as the diameter of an imaginary sphere that has the same specific gravity as the soil particle and settles with the same terminal velocity as that of the soil particle.

​Applications of Stoke's Law

1. To calculate the terminal velocity of a falling sphere and hence the viscosity of the fluid.
2. Desilting river flow
3. Separating the coolant from metal chips in machining operations
4. Sanitary engineering - treatment of raw water and sewerage etc.

Important Points

Reynolds equation is given by

Note: By neglecting viscous force and force due to turbulence,
Euler’s equation, max = (fg)x + (fp)x 

Euler's equation of motion:

The Euler's equation for a steady flow of an ideal fluid along a streamline is a relation between the velocity, pressure and density of a moving fluid. It is based on the Newton's Second Law of Motion which states that if the external force is zero, linear momentum is conserved. The integration of the equation gives Bernoulli's equation in the form of energy per unit weight of the following fluid.

It is based on the following assumptions:

• The fluid is non-viscous (i,e.,inviscid fluid or the frictional losses are zero)
• The fluid is homogeneous and incompressible (i.e., the mass density of the fluid is constant)
• The flow is continuous, steady and along the streamline.
• The velocity of the flow is uniform over the section.
• No energy or force (except gravity and pressure forces) is involved in the flow.

As there is no external force applied (Non-viscous flow), therefore linear momentum will be conserved.

The Navier-Stokes equations represent the conservation of momentum, while the continuity equation represents the conservation of mass.