Fluid Mechanics/Analysis Methods

Pathlines and Streamlines
The path which a fluid element traces out in space is called a pathline. For steady non-fluctuating flows where a pathline is followed continuously by a number of fluid elements, the pathline is called streakline. A streamline is the imaginary line whose tangent gives the velocity of flow at all times if the flow is steady, however in an unsteady flow, the streamline is constantly changing and thus the tangent gives the velocity of an element at an instant of time. A common practice in analysis is taking some of the walls of a control volume to be along streamlines. Since there is no flow perpendicular to streamlines, only the flow across the other boundaries need be considered.

Hydrostatics
The pressure distribution in a fluid under gravity is given by the relation

$$ \frac{dp}{dz} = - \rho g $$

where dz is the change in the direction of the gravitational field (usually in the vertical direction). Note that it is quite straightforward to get the relations for arbitrary fields too, for instance, the pseudo field due to rotation.

The pressure in a fluid acts equally in all directions. When it comes in contact with a surface, the force due to pressure acts normal to the surface. The force on a small area dA is given by p dA where the force is in the direction normal to dA. The total force on the area A is given by the vector sum of all these infinitesimal forces.

Control Volume Analysis
A fluid dynamic system can be analysed using a control volume, which is an imaginary surface enclosing a volume of interest. The control volume can be fixed or moving, and it can be rigid or deformable. Thus, we will have to write the most general case of the laws of mechanics to deal with control volumes.

The first equation we can write is the conservation of mass over time. Consider a system where mass flow is given by dm/dt, where m is the mass of the system. We have,

$$ \dot m = \int_{CS} \rho \left(   \mathbf{V}\cdot\mathbf{n}  \right) dA $$

For steady flow, we have

$$ \int_{CS} \rho \left(   \mathbf{V}\cdot\mathbf{n}  \right) dA = 0 $$

And for incompressible flow, we have

$$ \int_{CS} \left(   \mathbf{V}\cdot\mathbf{n}  \right) dA = 0 $$

If we consider flow through a tube, we have, for steady flow,

$$ \rho_1A_1V_1 = \rho_2A_2V_2 $$

and for incompressible steady flow, A1V1 = A2V2.

Law of conservation of momentum as applied to a control volume states that

$$ \sum F = \frac{d}{dt} \left(   \int_{CV} \mathbf{V} \mathbf{\rho}  \right) + \int_{CS} \mathbf{V}\mathbf{\rho} \left(   \mathbf{V}\cdot\mathbf{n}  \right)dA $$

where V is the velocity vector and n is the unit vector normal to the control surface at that point.

Law of Conservation of Energy (First Law of Thermodynamics)

$$ \frac{d\mathbf{Q}}{dt} + \frac{d\mathbf{W}}{dt} = \frac{d}{dt} \left(   \int_{CV} e\mathbf{\rho}  \right) + \int_{CS} e\mathbf{\rho} \left(   \mathbf{V}\cdot\mathbf{n}  \right)dA $$

where e is the energy per unit mass.

Bernoulli's Equation
Bernoulli's equation considers frictionless flow along a streamline.

For steady, incompressible flow along a streamline, we have

$$ \frac{p}{\rho} + \frac{V^2}{2} + gz = constant $$

We see that Bernoulli's equation is just the law of conservation of energy without the heat transfer and work.

It may seem that Bernoulli's equation can only be applied in a very limited set of situations, as it requires ideal conditions. However, since the equation applies to streamlines, we can consider a streamline near the area of interest where it is satisfied, and it might still give good results, i.e., you don't need a control volume for the actual analysis (although one is used in the derivation of the equation).

Energy in terms of Head
Bernoulli's equation can be recast as

$$ \frac{p}{\rho g} + \frac{V^2}{2g} + z = constant $$

This constant can be called head of the water, and is a representation of the amount of work that can be extracted from it. For example, for water in a dam, at the inlet of the penstock, the pressure is high, but the velocity is low, while at the outlet, the pressure is low (atmospheric) while the velocity is high. The value of head calculated above remains constant (ignoring frictional losses).

Mechanical Energy Balance Equation
Yet another variation of the Bernoulli's equation is the mechanical energy balance equation. The mechanical energy balance equation is useful when needing to consider things such as work or losses due to friction, or if there are differences between the outlet and inlet (such as pressure, velocity, and height).

$$ \frac{\Delta p}{\rho} + \frac{\Delta (V^2)}{2} + g\Delta z + f = -w $$