Glossary

7.1 Navier-Stokes Equations

3 min read•july 19, 2024

The are the backbone of fluid mechanics. They describe how fluids move and interact, using principles of mass and momentum conservation. These equations are powerful tools for understanding everything from blood flow to weather patterns.

However, the Navier-Stokes equations have limitations. They assume fluids are continuous and Newtonian, which isn't always true. For complex flows like turbulence, we often need simplified models to make calculations manageable.

Derivation and Components of the Navier-Stokes Equations

Derivation of Navier-Stokes equations

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  • Conservation of mass principle expressed by the ∂ρ∂t+∇⋅(ρu⃗)=0\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \vec{u}) = 0 relates the change in fluid density ρ\rho over time tt to the divergence of the mass flux ρu⃗\rho \vec{u}
  • Conservation of momentum principle based on Newton's second law F⃗=ma⃗\vec{F} = m\vec{a} considers surface forces (pressure and viscous stresses) and body forces (gravity, electromagnetic forces) acting on a fluid element
  • Navier-Stokes equations derived by applying conservation of mass and momentum to a fluid element result in the general form ρ(∂u⃗∂t+u⃗⋅∇u⃗)=−∇p+μ∇2u⃗+ρg⃗\rho \left(\frac{\partial \vec{u}}{\partial t} + \vec{u} \cdot \nabla \vec{u}\right) = -\nabla p + \mu \nabla^2 \vec{u} + \rho \vec{g} where pp is pressure, μ\mu is dynamic , and g⃗\vec{g} represents body forces per unit mass (gravity)

Components of Navier-Stokes equations

  • Local acceleration term ρ∂u⃗∂t\rho \frac{\partial \vec{u}}{\partial t} represents the change in velocity over time at a fixed point in the fluid
  • Convective acceleration term ρ(u⃗⋅∇u⃗)\rho (\vec{u} \cdot \nabla \vec{u}) accounts for the change in velocity due to the motion of the fluid itself
  • Pressure gradient term −∇p-\nabla p represents the force acting on the fluid due to spatial variations in pressure
  • Viscous diffusion term μ∇2u⃗\mu \nabla^2 \vec{u} represents the force acting on the fluid due to viscous stresses arising from velocity gradients
  • Body force term ρg⃗\rho \vec{g} represents the force acting on the fluid due to external body forces such as gravity or electromagnetic fields

Application and Limitations of the Navier-Stokes Equations

Application to fluid flow problems

  • Simplify the Navier-Stokes equations based on problem assumptions and constraints
    1. Steady-state flow ∂u⃗∂t=0\frac{\partial \vec{u}}{\partial t} = 0 eliminates the local acceleration term
    2. Incompressible flow assumes constant density ρ\rho simplifying the continuity equation
    3. Newtonian fluid assumption relates shear stress τ\tau linearly to strain rate ∂u∂y\frac{\partial u}{\partial y} via dynamic viscosity μ\mu
  • Apply appropriate boundary conditions such as no-slip (zero fluid velocity at solid boundaries) and free surface (continuous normal and shear stresses across the interface) conditions
  • Obtain analytical solutions for simple geometries and flow conditions (Poiseuille flow through pipes, Couette flow between parallel plates)

Limitations of Navier-Stokes equations

  • Continuum assumption treats fluid as a continuous medium which may not be valid for rarefied gases or flows with high Knudsen numbers (ratio of mean free path to characteristic length)
  • Newtonian fluid assumption may not hold for non-Newtonian fluids with complex rheology (blood, polymers)
  • Turbulence modeling is required as direct numerical simulation (DNS) of turbulent flows using Navier-Stokes equations is computationally expensive, necessitating approximate models (RANS, LES)
  • Compressibility effects require additional terms and equations (energy equation, equation of state) to extend Navier-Stokes equations to compressible flows

Key Terms to Review (16)

Aerodynamics: Aerodynamics is the branch of fluid mechanics that studies the behavior of air as it interacts with solid objects, such as aircraft, vehicles, and buildings. This field is crucial for understanding how lift, drag, and other forces affect the motion and stability of objects moving through the air, which connects deeply to historical advancements, types of fluid flow, rotation effects, governing equations, and boundary layer phenomena.
Continuity equation: The continuity equation is a fundamental principle in fluid mechanics that expresses the conservation of mass within a fluid flow. It states that the mass flow rate of a fluid must remain constant from one cross-section of a pipe or channel to another, provided there are no mass additions or losses. This concept connects with various aspects of fluid behavior and dynamics, playing a crucial role in understanding how fluids move and behave under different conditions.
Hydrodynamics: Hydrodynamics is the branch of fluid mechanics that deals with the behavior of fluids in motion. It focuses on understanding how fluids interact with solid boundaries and the forces acting on them, which is crucial for various applications ranging from engineering to environmental science. By analyzing fluid flow, hydrodynamics plays a vital role in predicting and optimizing performance in systems like aircraft, ships, and pipelines.
Incompressibility: Incompressibility refers to the assumption that the density of a fluid remains constant throughout its flow. This concept simplifies the analysis of fluid motion, particularly for liquids, since they exhibit minimal density changes under varying pressure. Understanding incompressibility is crucial when dealing with the Navier-Stokes equations and applying Euler's equation and Bernoulli's equation, as it allows for more straightforward calculations and predictions regarding fluid behavior.
Inlet boundary condition: An inlet boundary condition refers to the specific constraints or requirements placed on the flow characteristics at the entry point of a fluid domain in computational simulations. These conditions define how fluid enters the system, including parameters such as velocity, pressure, and temperature, which are critical for accurately modeling flow behavior. Properly defining inlet boundary conditions is essential for the stability and accuracy of numerical solutions in fluid dynamics.
Laminar Flow: Laminar flow is a fluid motion characterized by smooth, parallel layers of fluid that move in an orderly fashion, with minimal mixing between the layers. This type of flow typically occurs at low velocities and is influenced by the fluid's viscosity and density, which play a crucial role in determining the flow behavior.
Navier-Stokes Equations: The Navier-Stokes equations are a set of nonlinear partial differential equations that describe the motion of fluid substances. These equations are fundamental in fluid mechanics, capturing how velocity, pressure, temperature, and density of a fluid are related over time and space, making them essential for understanding various fluid behaviors and phenomena.
No-slip condition: The no-slip condition is a fundamental principle in fluid mechanics stating that a fluid in contact with a solid boundary will have the same velocity as that boundary. This means that at the interface between the fluid and the solid, the fluid's velocity becomes zero if the solid is stationary, ensuring a seamless interaction between different media. Understanding this concept is crucial for analyzing how fluids behave in various situations, particularly when applying conservation laws and governing equations.
Partial Differential Equations: Partial differential equations (PDEs) are mathematical equations that involve functions of multiple variables and their partial derivatives. They are crucial in describing various physical phenomena, particularly in fluid mechanics, where they model how quantities like velocity, pressure, and temperature change over space and time. PDEs help connect the underlying principles of fluid motion with observable behavior, making them essential for understanding dynamic systems.
Potential Flow: Potential flow refers to an idealized flow of an incompressible fluid where the flow velocity can be expressed as the gradient of a scalar potential function. In this context, potential flow simplifies the analysis of fluid motion by ignoring viscosity and allowing for irrotational flow, which means that the fluid has no vorticity. This concept is crucial for understanding various phenomena in fluid dynamics, especially when applying the Navier-Stokes equations under specific conditions where viscous effects are negligible.
Steady Flow: Steady flow refers to a condition where the fluid properties at a given point in a system do not change over time. This implies that parameters such as velocity, pressure, and density remain constant at specific points, which simplifies the analysis of fluid motion and allows for consistent behavior across different sections of the flow.
Stokes Flow: Stokes flow refers to the movement of a viscous fluid where inertial forces are negligible compared to viscous forces, leading to a linear relationship between stress and strain rate. This type of flow typically occurs at low Reynolds numbers, indicating that the fluid's motion is dominated by viscosity rather than inertia, which is particularly important in various applications like microfluidics and biological systems.
Turbulent flow: Turbulent flow is a type of fluid motion characterized by chaotic changes in pressure and velocity, leading to the formation of eddies and vortices. This flow regime significantly impacts various fluid mechanics principles, such as energy dissipation, momentum transfer, and the behavior of fluid particles within a system.
Vector Field: A vector field is a mathematical representation that assigns a vector to every point in a space, illustrating how quantities such as velocity or force change throughout that space. In fluid mechanics, vector fields are essential as they provide insights into the behavior of fluid flow, capturing both direction and magnitude at any given point, which is crucial for understanding the dynamics of fluids governed by equations like the Navier-Stokes equations.
Viscosity: Viscosity is a measure of a fluid's resistance to deformation and flow, essentially describing how thick or sticky a fluid is. This property plays a crucial role in understanding fluid behavior under different conditions and directly affects various phenomena, such as flow rates, pressure distribution, and energy loss in fluid systems.
Vorticity: Vorticity is a measure of the local rotation of fluid elements in a flow field, quantified as the curl of the velocity vector. It is crucial in understanding the behavior of fluid motion, especially in the presence of rotational flows, and helps in analyzing how vortices form and interact within a fluid. Vorticity plays a significant role in many fluid dynamics phenomena, including turbulence, circulation, and the dynamics of rotating bodies.
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Glossary
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7.2 Euler's Equation and Bernoulli's Equation