15.3.1. System Level Equations

As I derived in the last class, one is the system level equations and the other is the control volume level equations.

The Reynolds Transport Theorem (RTT) is a fundamental principle in fluid mechanics that relates the rates of change of a property (like mass, momentum, or energy) in a system to that of a control volume. It separates system-level equations (which focus on a specific quantity in a defined system) from control volume equations (which account for fluxes across a defined volume).

Consider tracking water flow in a tank. The system level would focus on how much water is in the tank at any moment, while the control volume would look at how water flows in and out of the tank through its pipes.

B: extensive property of fluid (m, ṁ, E) b: corresponding intensive property (1, v, e)

In fluid mechanics, properties are categorized into extensive and intensive. Extensive properties, like mass (m), momentum (ṁ), and energy (E), depend on the system size. On the other hand, intensive properties, such as specific volume (v = 1/rho) and specific energy (e), are independent of the system size and provide a per unit measure.

Think of extensive properties like the total amount of ingredients in a pot (mass), while intensive properties resemble how concentrated a solution is (specific volume).

The change of the extensive properties within the control volume with respect to time is related to the net outflow through the control surface.

In a control volume, the net change of extensive properties (like mass) can be calculated by evaluating the net outflow through the control surface. This means that as mass flows in and out, the internal storage of mass and the net outflow must balance to maintain conservation.

Imagine a bank account: the money added (inflow) and withdrawn (outflow) balance each month to keep a steady account balance (storage) unless new conditions change.

Let us have two simplifications or assumptions which are valid for fluid flow problems most of the time: steady and incompressible flow.

When we consider steady flow, we assume that the properties of the fluid (like velocity and density) do not change with time, allowing us to simplify calculations. Incompressible flow assumes that the fluid density remains constant, further simplifying equations involving flow rates across surfaces.

Think of water flowing through a pipe at a constant speed. If the speed and temperature remain unchanged over time (steady flow), and the density doesn't vary (incompressible flow), calculations for water pressure become much easier.

If I have the velocity vectors, consider the control surface in such a way that your normal vectors and your velocity vectors either will have the same or opposite direction.

When evaluating the interaction of fluid velocity with a control surface, it's crucial to align the control surface's normal vectors with the velocity vectors of the fluid. This alignment simplifies the mathematical computations required to describe the flux of fluid through the surface.

Picture sprinklers in a garden. The direction of the water flow (velocity vectors) should ideally match the direction of the plants for effective watering (control surface alignment).

Now considering moving control volume like a ship which is moving with a velocity V.

In scenarios where the control volume moves (e.g., a ship), the relative velocity of fluid relative to the vessel needs to be considered. This relative velocity is crucial in calculating the mass flow across the moving control surface because it affects how you evaluate inflow and outflow rates.

Think about a car driving through rain. The speed of the car and the falling rain's speed combine to determine how wet the windshields get—this interplay of velocities impacts the overall experience.

Another case involves control volumes moving with a variable velocity, meaning, the control volume has a velocity that changes with time.

For control volumes that are not only moving but also vary in speed, the calculations become more complex as the velocity must be treated as a function of both position and time. This introduces new parameters into the equations, requiring more advanced mathematical tools.

Consider an escalator system: as you move from the top to the bottom, the escalator's speed may change, requiring adjustments in timing for how quickly you need to step on or off.

We need to have three equations: conservation of mass, conservation of linear momentum, and conservation of energy.

In fluid dynamics problems, the most fundamental equations involve conservation laws—mass, momentum, and energy conservation principles guide the design and analysis of fluid systems. Understanding these helps in predicting system behavior and future states during fluid flow.

Think of a three-legged race. Each participant (mass, momentum, energy) must keep pace with one another to successfully move forward—if one stumbles, the entire team falters.

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