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Euler’s Equation: The Equation of Motion of an Ideal Fluid
This section is not a mandatory requirement. One can skip this section (if he/she does not like to spend time on Euler's equation) and go directly to Steady Flow Energy Equation.Using the Newton's second law of motion the relationship between the velocity and pressure field for a flow of an inviscid fluid can be derived. The resulting equation, in its differential form, is known as Euler’s Equation. The equation is first derived by the scientist Euler.
Derivation:
Let us consider an elementary parallelopiped of fluid element as a control mass system in a frame of rectangular cartesian coordinate axes as shown in Fig. 12.3. The external forces acting on a fluid element are the body forces and the surface forces.
Fig 12.2 A Fluid Element appropriate to a Cartesian Coordinate System used for the derivation of Euler's Equation
Let Xx, Xy, Xz be the components of body forces acting per unit mass of the fluid element along the coordinate axes x, y and z respectively. The body forces arise due to external force fields like gravity, electromagnetic field, etc., and therefore, the detailed description of Xx, Xy and Xz are provided by the laws of physics describing the force fields. The surface forces for an inviscid fluid will be the pressure forces acting on different surfaces as shown in Fig. 12.3. Therefore, the net forces acting on the fluid element along x, y and z directions can be written as
Since each component of the force can be expressed as the rate of change of momentum in the respective directions, we have
the mass of a control mass system does not change with time, 
is constant with time and can be taken common. Therefore we can write Eqs (12.5a to 12.5c) as
Expanding the material accelerations in Eqs (12.6a) to (12.6c) in terms of their respective temporal and convective components, we get
The Eqs (12.7a, 12.7b, 12.7c) are valid for both incompressible and compressible flow. By putting u = v = w = 0, as a special case, one can obtain the equation of hydrostatics .
Equations (12.7a), (12.7b), (12.7c) can be put into a single vector form as
where 
the velocity vector and the body force vector per unit volume 
are defined as
Equation (12.7d) or (12.7e) is the well known Euler’s equation in vector form, while Eqs (12.7a) to (12.7c) describe the Euler’s equations in a rectangular Cartesian coordinate system.
The document Euler’s Equation: The Equation of Motion of an Ideal Fluid | Civil Engineering Optional Notes for UPSC is a part of the UPSC Course Civil Engineering Optional Notes for UPSC.
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FAQs on Euler’s Equation: The Equation of Motion of an Ideal Fluid - Civil Engineering Optional Notes for UPSC
| 1. What is Euler's Equation and how is it related to the motion of an ideal fluid? |  |
Ans. Euler's Equation is a fundamental equation in fluid dynamics that describes the motion of an ideal fluid. It relates the pressure distribution, velocity field, and density of the fluid to the acceleration of the fluid particles.
| 2. What are the key components of Euler's Equation? | |
Ans. The key components of Euler's Equation include the convective acceleration term, pressure gradient term, gravitational acceleration term, and any external force acting on the fluid.
| 3. How is Euler's Equation different from Navier-Stokes Equation? | |
Ans. Euler's Equation is a simplified form of the Navier-Stokes Equation, which includes viscosity and other dissipative effects. Euler's Equation is applicable to ideal fluids, while the Navier-Stokes Equation is used for real fluids.
| 4. How can Euler's Equation be used in practical applications? | |
Ans. Euler's Equation is used in various engineering applications, such as aircraft design, water flow in pipes, and weather prediction models. It helps in understanding the behavior of fluids in various scenarios.
| 5. What are the assumptions made in deriving Euler's Equation for ideal fluids? | |
Ans. The assumptions made in deriving Euler's Equation for ideal fluids include the fluid being inviscid (no viscosity), incompressible, and irrotational. These assumptions simplify the equations and make them easier to solve for ideal fluid flow scenarios.
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Nov 22, 2024 Last updated |
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