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A short class with Sarah and her students. The other ways are under More.
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Welcome back, students! Today we will derive the Navier-Stokes equations for viscous fluid flow. Can anyone remind me what viscous flow is?
Isn't it the flow of fluids that have a measurable viscosity, showing resistance to flow?
And I think it involves shear stress and deformation, right?
Exactly! Viscous flow is characterized by the internal friction due to viscosity. This internal friction can be described using equations we will derive today.
Now, let's discuss the difference between thermodynamic pressure and mechanical pressure. Who can explain how they differ?
I believe thermodynamic pressure relates to the state of the fluid, while mechanical pressure depends on stresses in the fluid?
Right! The mechanical pressure is the average of normal stresses in the fluid. Can someone tell me the formula for mechanical pressure?
It's p bar = -(1/3)(τxx + τyy + τzz) where τxx, τyy, and τzz are the normal stresses!
Great job! As we've discussed, these pressures only align under specific conditions, including the Stokes Hypothesis.
Next, we will derive the Navier-Stokes equations. Who can remind us why we assume incompressible flow?
Because in many hydraulic applications, like water flow, the density remains constant!
And it makes our calculations simpler, especially with the divergence of velocity being zero.
Exactly! Incompressible flow allows us to simplify the equations. Let’s see how that works in our equations.
Finally, we explore inviscid flow. What happens to our equations if we neglect the viscous terms?
We get the Euler equation, which applies to frictionless flows!
Correct! It's a first-order equation that simplifies our computations even further. It leads us to Bernoulli's equation.
Can you remind us what Bernoulli’s equation represents?
Of course! Bernoulli’s equation relates pressure, velocity, and height in a flowing fluid. Excellent participation today, everyone!
Foundational equations of fluid dynamics that incorporate viscosity.
Pressure that stems from the state of the fluid in thermodynamic equilibrium.
Pressure derived from the fluid's internal stresses.
Flow where density remains constant, allowing simplification of the Navier-Stokes equations.
Flow described by Euler's equations, where viscous effects are negligible.
Flow so smooth, no stress in view, Euler equations tell us what fluids do.
Imagine a river flowing down a mountain. The water travels swiftly (like inviscid flow) and adjusts with minimal friction, gathering speed before flowing into calm waters below (Bernoulli's principle in effect).
P.V.V - Pressure, Velocity, Viscosity - Remember the key components when dealing with fluid behavior!
NAVIER - Newtonian, Average pressure, Viscous terms, Incompressible relation, Equations of motion, Relevant to fluid flow.
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What are Navier-Stokes Equations?
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What is the Stokes Hypothesis?
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What is incompressible flow?
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What does the Euler equation describe?
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A fluid that exhibits viscosity, leading to resistance to flow.
Equations that describe the motion of viscous fluid substances.
Pressure associated with the thermal state of a fluid.
Pressure determined by the stresses within the fluid.
A hypothesis stating that mechanical pressure equals thermodynamic pressure under specific conditions.
An equation governing the motion of inviscid (non-viscous) fluids.
Flow in which the density of the fluid remains constant.
This section discusses the derivation of the Navier-Stokes equations, the difference between thermodynamic and mechanical pressure, and conditions under which they are equivalent.
In this lecture, the Navier-Stokes equations for viscous flow are derived, highlighting the distinction between thermodynamic pressure and mechanical pressure. The Stokes Hypothesis is introduced to explain conditions under which these pressures can be considered the same, especially for incompressible flow.
In this lecture, Prof. Mohammad Saud Afzal elaborates on the derivation of the Navier-Stokes equations that govern the behavior of viscous fluids. The instructor begins by revisiting the deformation law for Newtonian viscous fluids. It emphasizes the distinction between thermodynamic pressure and mechanical pressure, defining the mechanical pressure derived from the average of normal stresses. The concepts of the Stokes Hypothesis are introduced, highlighting its rare satisfaction in real fluids.
The importance of incompressible flow is then discussed, revealing that the Navier-Stokes equations can be simplified when certain assumptions (such as constant viscosity and density) are made. Finally, the lecture covers the concept of inviscid flow and derives the Euler equation leading to Bernoulli's equation, establishing the foundation for computational fluid dynamics to be discussed in subsequent modules.