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In this report, we will investigate the behavior of fluid dynamics, focusing on fluids in motion. Specifically, we will initially consider ideal fluids, which are defined as fluids with zero viscosity, making them inviscid. Inviscid fluids experience no resistance to movement, whether past solid objects or adjacent portions of the fluid moving at different velocities.
Our analysis begins with the examination of a differential volume of fluid, which remains a coherent unit of mass as it moves through the system.
We refer to this unit as a "particle" of fluid. Different fluid particles in a system may move in various directions and at different velocities at any given instant. Additionally, a single fluid particle can change its direction and speed over time.
To visualize the behavior of fluid particles, imagine capturing a snapshot at a specific instant, denoted as t1, displaying the velocity vectors of all fluid particles in the system. If we were to draw an arrow in the direction of the velocity vector for a particle as it enters the system and then move a differential distance in the direction of the velocity vector, we could continue this process until reaching the outlet of the system, connecting all the differential-length arrows.
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This would create a continuous line representing the velocity vectors from the inlet to the outlet, known as a streamline. Each point at the system's inlet originates a different streamline. While most streamlines continue to the system's outlet, some may dead-end within the system, intersecting a solid surface or an immobile fluid layer perpendicular to the fluid particle's velocity vector.
These dead-ends are called stagnation points.
Mathematically, streamlines can be described by the following equation:
[
frac{dx}{dt}i + frac{dy}{dt}j + frac{dz}{dt}k = 0
]
Streamlines provide insight into the velocity distribution within the system at a given instant. However, fluid particles may not strictly follow a particular streamline throughout their journey within the system. The path followed by a fluid particle over time is referred to as a pathline.
If streamlines remain unchanged in shape and location over time, the velocity at any given point in the system remains constant between different time intervals. In such cases, all fluid particles passing through a particular location will have the same velocity, and particles entering the system at a specific location will follow the same pathline, regardless of when they enter. This phenomenon characterizes steady flow, where streamlines and pathlines coincide.
Now, let's consider the movement of a particle along a pathline in an ideal fluid. We define the distance along the pathline using a coordinate 's'. A fluid particle at any point on the pathline (s1) at one instant will be at a different point on the same pathline (s2) at some later time. During this movement from s1 to s2, the particle may experience changes in three parameters that affect its energy:
These changes influence the energy associated with the particle, including its kinetic energy (KE), gravitational potential energy (PEgrav), and mechanical potential energy (PEmech). The mathematical representations of these forms of energy are as follows:
KE = 1/2mv2
PEgrav = mgz
PEmech = pV
Where 'm' represents the mass of the particle, 'v' is its velocity, 'z' is the elevation, 'g' is the acceleration due to gravity, 'p' is the pressure, and 'V' is the volume of the particle. To facilitate comparisons, we normalize these energy values by dividing them by the particle mass:
KE/m = 1/2v2
PEgrav/m = gz
PEmech/m = pV/m
As we have established that the fluid under consideration is ideal and possesses zero viscosity, no energy is lost as heat due to friction as the particle travels from point s1 to s2. In the absence of any other processes that add or remove energy from the particle during its journey, the principle of conservation of energy dictates that the total energy of the particle remains constant between these two locations. This implies that any increase in one form of energy carried by the particle must be counterbalanced by decreases in the other forms. The sum of the three identified forms of energy, denoted as Etot, allows us to express the energy conservation principle in the following ways:
Etot/m + PV/m + 1/2v2 + gz = Constant
P1V1/m + 1/2V12 + gz1 = P2V2/m + 1/2V22 + gz2
Here, 'P' represents pressure, 'V' is volume, 'v' is velocity, 'g' is the acceleration due to gravity, 'z' is elevation, and 'm' is the mass of the fluid particle.
When dealing with incompressible fluids, the velocity at points 1 and 2 along the pathline is equal, denoted as 'V.' In this case, we can simplify the equations by multiplying through by the fluid's density, resulting in Equation 9. By dividing the resulting equation by the specific weight, we obtain Equation 10:
(Etot/V + PV + frac{1}{2}v^2 + gz = text{Constant})
(P_1V_1 + frac{1}{2}V_1^2 + gz_1 = P_2V_2 + frac{1}{2}V_2^2 + gz_2)
Here, 'P' represents pressure, 'V' is volume, 'v' is velocity, 'g' is the acceleration due to gravity, 'z' is elevation, and 'm' is the mass of the fluid particle.
In Equations 9 and 10, 'p' represents the static pressure, 'γz' is the hydrostatic pressure, and (frac{1}{2}ρv^2) is the dynamic pressure. The static pressure corresponds to the pressure measured when the measuring device moves with the fluid particle or is placed in the flow path without converting any kinetic energy to mechanical energy during measurement. The dynamic pressure, on the other hand, represents the pressure measured when all kinetic energy is converted to mechanical energy at the measurement point. It is the highest pressure that can be generated in the fluid at the given elevation.
The terms in Equation 10 are also assigned special names and are collectively known as various forms of head:
- '(p/γ)' is called the pressure head
- '(frac{v^2}{2g})' is the velocity head
- 'z' is the elevation head
Furthermore, the sum of the elevation head and the pressure head is referred to as the piezometric head, and the sum of all three forms of head is known as the total head. Conceptually, the head signifies the elevation gain that could be achieved in the fluid if all the associated energy were converted into gravitational potential energy. For example, a velocity head of 6 meters implies that, if all kinetic energy were converted to potential energy, the fluid would rise by a height of ∆z = 6 meters.
While Equations 9 and 10 are valuable in conceptual terms, they can be cumbersome for practical use as they necessitate tracking the locations of individual fluid particles over time. This requirement can be eliminated by limiting the application of these equations to systems with steady flow. In such cases, all fluid particles passing through point s1 at any given time possess identical properties, and the same holds for particles passing through point s2. Therefore, analyzing the energy of a particle at s1 and s2 at a specific instant is equivalent to analyzing any particle that has traveled from s1 to s2. In other words, in systems with steady flow, Equations 9 and 10 apply to different particles at s1 and s2, as long as these points are on the same pathline. Steady flow implies that particles must be on the same streamline, and when applied to particles on a single streamline in steady flow, both Equations 9 and 10 collectively constitute the Bernoulli equation.
The constant value of Etot/V or Etot/mg in the Bernoulli equation is referred to as the Bernoulli constant.
It is essential to note that the Bernoulli equation can be applied without knowledge of the detailed path followed by the fluid particle from point 1 to point 2; the only requirement is that both points lie on the same streamline in a system with steady flow. Additionally, the equation remains applicable even if the distance between points 1 and 2 is infinitesimal, denoted as ds. In such cases, Equation 9 can be expressed as:
(d/ds(p + frac{1}{2}ρv^2 + γz) = 0)
Upon differentiation and utilizing the fact that 'v' can be represented as ds/dt, the equation simplifies to:
(0 = dp/ds + pfrac{ds}{dt} + γfrac{dz}{ds} = dp/ds + pfrac{ds}{dt} + γsinθ)
Here, 'ρas' represents the acceleration of the particle in the 's' direction, and 'Fs' is the force exerted on the particle in the same direction. The term 'dz/ds' can be expressed as 'sin θ,' where 'θ' is the angle of the streamline with the horizontal. Thus, the equation can be further simplified to:
(Fs/V = -dp/ds - γsinθ)
Equation 14 presents the Bernoulli equation as a force balance on the fluid particle, illustrating that the net force per unit volume in the 's' direction, responsible for any acceleration along the streamline, equals the sum of the gradients in the static and hydrostatic pressures in that direction.
It is essential at this point to review the assumptions made during the derivation that may constrain the application of the Bernoulli equation. Firstly, the equation is valid only along a pathline or a streamline within a system exhibiting steady flow. Secondly, the assumption is made that no energy is transferred into or out of the particle between points 1 and 2, and that all energy transformations within the particle between these points involve conversions among kinetic, mechanical (pressure-based), and gravitational energy.
Another critical aspect of the assumptions made during the derivation of the Bernoulli equation is the absence of energy transfers into or out of the fluid particle between points s1 and s2. This implies the absence of pumps or any other energy-delivering devices between these two locations. Additionally, there should be no energy losses, meaning that energy-removing devices such as turbines are not present in the flow path between s1 and s2. Moreover, the absence of energy loss due to friction is a fundamental assumption, indicating that the fluid under investigation is inviscid or ideal. Lastly, it was assumed that the fluid is incompressible, and this assumption played a role when equating 1 V to 2 V in the transition from Equation 8 to Equations 9 and 10. In cases where the fluid is compressible, a similar analysis can still be conducted, but the mathematics become more complex as mechanical energy would depend on both pressure and the variable particle volume.
As previously mentioned, the Bernoulli equation applies strictly along a streamline. In general, the total energy per unit volume (as represented by Equation 9) or per unit weight (as represented by Equation 10) can vary from one streamline to another. Consequently, it is inappropriate to apply the equation to describe the total flow across a collection of streamlines. Typically, our interest lies in macroscopic flow rather than the flow of individual fluid particles. Therefore, it is valuable to determine conditions under which the Bernoulli equation can be applied on a macroscopic scale.
Such conditions can be identified relatively easily: if the Bernoulli constant remains identical for all streamlines within the system, then all fluid particles throughout the system possess the same energy per unit volume or weight. Consequently, the Bernoulli equation can be applied to a collection of streamlines as effectively as it can be to a single streamline.
A physical system in which such a condition approximately applies is one in which the fluid is initially contained in a reservoir with negligible velocity. More precisely, if the velocity head is negligible compared to the piezometric head, this condition holds. In such cases, changes in pressure and elevation anywhere in the reservoir are equivalent to those for a static fluid. Consequently, the Bernoulli constant retains the same value (equal to ρ + PV^2/2 + zγ) for all the fluid in the system (Figure 1).
There are other physical arrangements that lead to the same outcome. In this course, we primarily consider systems in which all the fluid entering the system possesses equal energy per unit volume or weight. In instances where fluid enters the system at multiple points, the energy per unit volume or weight remains equal for all the fluid entering from each location.
The Bernoulli equation is a fundamental concept in fluid dynamics that describes the conservation of energy for fluid particles traveling along streamlines in systems with steady flow. It provides valuable insights into the behavior of ideal fluids, especially when applied to macroscopic flow conditions where the Bernoulli constant remains constant for all streamlines. While the equation has its limitations, including the assumption of incompressibility and the strict applicability along streamlines, it serves as a powerful tool for analyzing fluid motion and understanding the interplay between pressure, velocity, and elevation in various fluid systems.
Fluid Dynamics: Continuity Equation and Bernoulli's Equation. (2024, Jan 05). Retrieved from https://studymoose.com/document/fluid-dynamics-continuity-equation-and-bernoullis-equation
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