3.2 Conservation of Momentum and Cauchy's Equation
4 min read•august 16, 2024
in fluids is a crucial concept that links fluid motion to external forces. It's the foundation for understanding how fluids behave under different conditions, from simple pipe flows to complex atmospheric phenomena.
takes this idea further, providing a mathematical framework to analyze fluid behavior in detail. It considers all forces acting on a fluid element, making it a powerful tool for solving real-world fluid dynamics problems across various fields.
Conservation of Momentum in Fluids
Fundamental Principles and Mathematical Representation
Top images from around the web for Fundamental Principles and Mathematical Representation
Index notation with Navier-Stokes equations - Physics Stack Exchange View original
Is this image relevant?
Transformation of the Navier-Stokes Equation to the Cauchy Momentum Equation Using a Novel ... View original
Is this image relevant?
Derivation of the Navier–Stokes equations - Wikipedia, the free encyclopedia View original
Is this image relevant?
Index notation with Navier-Stokes equations - Physics Stack Exchange View original
Is this image relevant?
Transformation of the Navier-Stokes Equation to the Cauchy Momentum Equation Using a Novel ... View original
Is this image relevant?
1 of 3
Top images from around the web for Fundamental Principles and Mathematical Representation
Index notation with Navier-Stokes equations - Physics Stack Exchange View original
Is this image relevant?
Transformation of the Navier-Stokes Equation to the Cauchy Momentum Equation Using a Novel ... View original
Is this image relevant?
Derivation of the Navier–Stokes equations - Wikipedia, the free encyclopedia View original
Is this image relevant?
Index notation with Navier-Stokes equations - Physics Stack Exchange View original
Is this image relevant?
Transformation of the Navier-Stokes Equation to the Cauchy Momentum Equation Using a Novel ... View original
Is this image relevant?
1 of 3
Conservation of momentum in fluid dynamics equates the rate of change of momentum within a control volume to the sum of external forces acting on the fluid
Applies to both compressible and incompressible fluids underpinning understanding of fluid motion and interactions
Expressed mathematically using the material derivative accounting for local and convective acceleration of fluid particles
Leads to the derivation of the governing fluid motion
Closely related to Newton's Second Law of Motion applied to a continuous fluid medium
Represented in tensor notation as a set of partial differential equations highlighting the multi-dimensional nature of fluid dynamics
Material derivative in conservation of momentum expressed as DtDv=∂t∂v+(v⋅∇)v
External Forces and Their Categorization
External forces acting on a fluid categorized into and surface forces
Body forces act throughout the fluid volume (gravity, electromagnetic forces)
Surface forces act on the fluid boundaries (pressure, viscous stresses)
Gravity as a body force expressed as Fg=ρg where ρ is fluid and g is gravitational acceleration
Pressure as a surface force represented by Fp=−∇p where p is pressure
Viscous stresses in Newtonian fluids described by τij=μ(∂xj∂vi+∂xi∂vj) where μ is dynamic
Cauchy's Equation for Fluids
Derivation Process and Components
Derived by applying conservation of momentum to an infinitesimal fluid element
Considers all forces acting on the fluid element including body forces and surface forces
Surface forces represented using the stress tensor encompassing normal and shear stresses
Rate of change of momentum for the fluid element expressed using material derivative of velocity
Divergence theorem applied to transform surface integrals into volume integrals during derivation
Resulting Cauchy's equation relates material acceleration of fluid to divergence of stress tensor and body forces
General form of Cauchy's equation: ρDtDv=∇⋅σ+f where σ is the stress tensor and f represents body forces
Mathematical Representation and Notation
Final form typically expressed in indicial notation or vector form emphasizing applicability to three-dimensional fluid motion
Indicial notation: ρDtDvi=∂xj∂σij+fi
Vector form: ρDtDv=∇⋅σ+f
Stress tensor components: σij=−pδij+τij where p is pressure, δij is the Kronecker delta, and τij represents viscous stresses
For Newtonian fluids, viscous stress tensor: τij=μ(∂xj∂vi+∂xi∂vj−32δij∇⋅v)
Analyzing Forces on Fluid Elements
Application and Analysis Techniques
Cauchy's equation determines net force on fluid element by considering stress distributions and body forces
Enables analysis of normal and shear stresses in fluids providing insights into fluid deformation and flow characteristics
Allows study of fluid behavior in various flow regimes (laminar, turbulent)
Simplifies for specific cases (inviscid flows, incompressible fluids) to analyze particular fluid dynamics problems
Forms basis for deriving other important fluid dynamics equations (Navier-Stokes equations, Euler equations)
Utilized in numerical methods (finite element analysis, computational fluid dynamics) as fundamental governing equation
Applied to analyze fluid-structure interactions where forces exerted by fluids on solid boundaries are of interest
Practical Examples and Simplifications
Simplified Cauchy's equation for inviscid flow: ρDtDv=−∇p+ρg
Incompressible Navier-Stokes equations derived from Cauchy's equation: ρ(∂t∂v+(v⋅∇)v)=−∇p+μ∇2v+ρg
Analysis of pressure distribution in a rotating fluid (centrifugal pump): ∂r∂p=ρω2r where ω is angular velocity and r is radial distance
Examination of shear stress in pipe flow: τw=8fρV2 where f is friction factor and V is average velocity
Significance of Cauchy's Equation
Comprehensive Description and Versatility
Provides comprehensive description of fluid motion by relating fluid acceleration to internal and external forces
Serves as bridge between microscopic behavior of fluid particles and macroscopic properties of fluid flow
Applicable to both Newtonian and non-Newtonian fluids making it versatile tool in fluid dynamics analysis
Forms foundation for studying complex fluid phenomena (turbulence, boundary layer flows, multiphase flows)
Crucial for developing and interpreting advanced fluid dynamics models and simulations
Formulation in terms of stress tensor allows incorporation of various constitutive relations enabling study of different fluid behaviors
Analysis of turbulent flow using Reynolds-averaged Navier-Stokes (RANS) equations derived from Cauchy's equation
Study of non-Newtonian fluid behavior in polymer processing using modified stress tensor in Cauchy's equation
Investigation of multiphase flows in oil and gas pipelines using extended forms of Cauchy's equation for each phase
Examination of magnetohydrodynamic flows by incorporating electromagnetic forces into Cauchy's equation
Analysis of blood flow in arteries using Cauchy's equation with non-Newtonian fluid models
Study of atmospheric dynamics and weather prediction using Cauchy's equation in geophysical fluid dynamics models
Investigation of ocean currents and tidal flows using Cauchy's equation in oceanographic models
Key Terms to Review (17)
Aerodynamics: Aerodynamics is the study of the behavior of air as it interacts with solid objects, like aircraft, vehicles, and buildings. It focuses on understanding forces like lift, drag, and thrust that are crucial in designing structures that move through or are influenced by air. A deep understanding of these forces is essential for developing efficient and effective designs in various applications.
Body forces: Body forces are forces that act throughout the volume of a body, as opposed to surface forces, which act on the surface. These forces can include gravitational, electromagnetic, and inertial forces, and they play a critical role in understanding how fluid elements move and interact with one another. In fluid dynamics, body forces are essential in formulating the equations of motion, influencing how momentum is conserved within a fluid system.
Cauchy's Equation: Cauchy's Equation is a fundamental expression in fluid dynamics that relates the stress tensor to the rate of strain in a fluid. This equation is essential in understanding how fluids respond to external forces, and it provides a mathematical framework for analyzing momentum conservation in fluid flow. By linking stress to strain, Cauchy's Equation highlights the relationship between force and motion, which is crucial for analyzing both Newtonian and non-Newtonian fluids.
Conservation laws: Conservation laws are fundamental principles in physics that state certain physical quantities remain constant within a closed system over time. These laws are crucial for understanding the behavior of fluids and other physical systems, as they help describe how quantities like mass, momentum, and energy are preserved in interactions and transformations.
Conservation of Momentum: Conservation of momentum is a fundamental principle stating that the total momentum of a closed system remains constant over time, as long as no external forces are acting on it. This principle is essential for understanding how fluids behave in various scenarios, such as shock waves and vortex dynamics, and connects to the underlying equations that govern fluid motion and stress relationships.
Continuity equation: The continuity equation is a mathematical expression that represents the principle of conservation of mass in fluid dynamics. It states that for an incompressible fluid, the mass flow rate must remain constant from one cross-section of a flow to another, which leads to the conclusion that the product of the cross-sectional area and fluid velocity is constant. This fundamental principle connects various phenomena in fluid behavior, emphasizing how mass is conserved in both steady and unsteady flow conditions.
Control Volume Analysis: Control volume analysis is a method used in fluid dynamics to study the behavior of fluids within a defined space, known as the control volume. This approach focuses on the mass, momentum, and energy exchanges across the boundaries of the control volume, allowing for a systematic examination of fluid motion and forces acting on it. By applying conservation laws to this volume, we can derive key equations, such as the continuity equation and the momentum equation, which are foundational in understanding fluid behavior.
Density: Density is defined as the mass of a substance per unit volume, typically expressed in kilograms per cubic meter (kg/m³). This property plays a crucial role in understanding how fluids behave under various conditions, affecting buoyancy, flow rates, and pressure distributions.
Free surface condition: The free surface condition refers to the boundary condition applied at the interface between a fluid and a surrounding medium, typically air or another fluid, where the stress is zero. This condition implies that the pressure exerted on the fluid's surface is equal to atmospheric pressure, and it plays a crucial role in the behavior of fluids in motion, especially when analyzing fluid dynamics involving interfaces. Understanding this condition helps in deriving equations that govern fluid flow and analyzing how fluids interact with their environment.
Hydrodynamics: Hydrodynamics is the branch of fluid dynamics that deals with the study of fluids in motion, particularly focusing on the forces acting on and the motion of fluids. It connects fluid behavior to physical principles like pressure, velocity, and flow rates, and plays a crucial role in understanding various flow patterns and phenomena in both natural and engineered systems.
Navier-Stokes Equations: The Navier-Stokes equations are a set of nonlinear partial differential equations that describe the motion of viscous fluid substances. They express the fundamental principles of conservation of mass, momentum, and energy in fluid dynamics, providing a mathematical framework to analyze various flow phenomena.
No-Slip Condition: The no-slip condition is a fundamental principle in fluid dynamics stating that a fluid in contact with a solid boundary will have zero velocity relative to that boundary. This means that the fluid 'sticks' to the surface, causing the velocity of the fluid layer at the boundary to equal the velocity of the boundary itself, typically resulting in a velocity gradient in the fluid adjacent to the surface.
Pressure Gradient: A pressure gradient is the rate at which pressure changes in a specific direction within a fluid, driving fluid motion from areas of high pressure to low pressure. This concept is fundamental to understanding fluid dynamics, as it influences flow behavior and is essential in various equations and solutions that describe how fluids behave under different conditions.
Steady flow: Steady flow refers to a condition in fluid dynamics where the velocity of the fluid at any given point does not change over time. In such a state, all properties of the fluid, including velocity, pressure, and density, remain constant as the fluid moves through a system. This concept is crucial as it simplifies the analysis of various flow situations and enables the application of fundamental conservation laws.
Unsteady flow: Unsteady flow refers to fluid motion where the velocity of the fluid at a given point changes with time. This contrasts with steady flow, where conditions remain constant over time. Unsteady flow is crucial in understanding how forces and energy are distributed in a fluid, influencing momentum conservation, the behavior of velocity fields, and the representation of streamlines and paths taken by particles in the fluid.
Viscosity: Viscosity is a measure of a fluid's resistance to deformation and flow, essentially describing how 'thick' or 'sticky' a fluid is. It plays a crucial role in determining how fluids behave under different conditions, affecting flow rates and the interaction between layers of fluid.
Viscous forces: Viscous forces are internal forces that arise within a fluid due to its viscosity, which is the measure of a fluid's resistance to deformation and flow. These forces play a crucial role in the motion of fluids, as they affect the velocity profiles and momentum transfer in both single-phase and multiphase flow scenarios. Understanding viscous forces is essential for analyzing how fluids behave under various conditions and is integral to equations governing fluid dynamics.