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A short class with Sarah and her students. The other ways are under More.
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Welcome class! Today, we're building on what we've learned about fluid properties. Can anyone remind me what shear strain rates and extensional strain rates are?
Shear strain rates are related to the change in shape of a fluid element without a change in volume, like how honey flows.
And extensional strain rates measure how much the fluid is stretching or compressing!
Exactly! Now we express these rates in terms of partial derivatives. Would someone like to write down how we denote epsilon for shear and extensional?
Epsilon xx for extensional, and epsilon xy for shear!
Great memory! Remember that these strain rates are pivotal for our next concepts. Let’s summarize: extensional strains involve changes in volume, while shear strains involve shape changes. Now, let’s explore the three laws of conservation!
Moving to conservation laws, can anyone name the three fundamental equations we rely on in hydraulics?
Conservation of mass, momentum, and energy!
That's right! The continuity equation is derived from the conservation of mass. Can someone explain why this is vital in fluid dynamics?
It ensures that the mass entering any volume is equal to the mass exiting, maintaining a consistent flow!
Excellent! There’s also conservation of momentum, which builds on Newton's second law. Why do you think it's essential?
It helps us understand how forces affect flowing fluids!
Exactly! Energy conservation is less covered in our course, but it's fundamental, especially in thermodynamics. Let’s review everything before we derive the continuity equation.
Next, let’s dig into thermodynamic variables. Who can name the main variables we discussed?
Density (ρ), enthalpy (H), viscosity (μ), and thermal conductivity (K)!
Correct! And what makes these variables interesting in our studies?
They depend on pressure and temperature under thermodynamic equilibrium!
Well done! These temperatures and pressures affect our calculations. What can we assume about density and viscosity in our course?
We can assume they stay constant for certain analyses!
Exactly! This simplifies our calculations, allowing us to focus on essential aspects of fluid flow. Let’s discuss this further with examples in next session.
Let’s move on to deriving our continuity equation! Who remembers what we start from?
We use the conservation of mass!
Right! Now, if mass equals density (ρ) times volume (V), what can we say about the changes that occur?
The rate of change of mass over time can be set to zero for a steady flow.
Correct! We apply the chain rule to relate volume changes to fluid velocity. Now, what happens when we sum up the strain rates?
We get a relation that leads us to the divergence of velocity equaling zero when assuming incompressible flow!
Excellent insight! That's the continuity equation. It forms the basis of many fluid dynamics applications. Let’s recap this derivation at the end of our session.
Now we’ll introduce stream functions. What purpose do stream functions serve in our calculations?
They help simplify computations for 2D flows!
Exactly! They allow us to convert our continuity equation into a more manageable form. Can anyone express this in terms of the variables we discussed?
If we define, rho u as delta psi by delta y and rho v as negative delta psi by delta x, it satisfies the continuity!
Well put! Continuity can be satisfied using these expressions. Finally, let’s summarize today’s key points before we conclude.
The principle that mass cannot change within a closed system.
Mathematical expression ensuring mass conservation in fluid flows.
Measures of deformation in fluids under stress.
A function that simplifies flow calculations in two-dimensional scenarios.
In flows so fast, mass stays in line, the continuity keeps it all just fine!
Imagine a river flowing steadily, where every drop of water entering at one point exits at another, just like how every story has a beginning and end, ensuring no water is lost in between.
Remember 'MOM' for Momentum, Observed Mass, and Energy in hydraulics!
SOME – Stream function, Observed mass, Momentum, Equation of continuity.
Term
What is the continuity equation?
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Term
What are shear strain rates?
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Define stream function.
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Rate of deformation due to shear stress in a fluid.
Rate of deformation due to tensile or compressive stress.
Principle stating that mass cannot be created or destroyed.
Mathematical expression representing conservation of mass in fluid flow.
A mathematical function that allows the description of flow in two dimensions.
This section covers the foundational concepts of fluid properties and the conservation laws critical for hydraulic engineering, including mass, momentum, and energy.
In this section, the basic fluid properties are revisited, focusing on the derivation of shear strain rates and the classification of fundamental conservation equations. Key concepts such as the conservation of mass, momentum, and energy, as well as the thermodynamic variables relevant in hydraulic engineering, are also discussed.
This section introduces crucial principles of hydraulic engineering, starting with fluid properties, particularly viscous flow. Key concepts, including the rate of shear strain, are explored to establish a foundation for further analysis.
We summarize the classification of fundamental equations governing fluid behavior:
Different thermodynamic variables such as density (ρ), energy (h), viscosity (μ), and thermal conductivity (K) are discussed with respect to their dependence on pressure (P) and temperature (T) in thermodynamic equilibrium. The section culminates in the derivation of the continuity equation and introduces the stream function to describe fluid flow in two dimensions.