show me what is different about integral vs differential forms of conseratiion of mass and ocnservation of momentum
This text is generated by AI trained by Professor Brian J. Kirby on his lectures, videos, notes, message board responses, and textbook
PK Bot’s answer:
The integral and differential forms of conservation of mass and momentum serve different purposes and are used in different contexts within fluid mechanics. Here’s a breakdown of the differences based on the source material:
Conservation of Mass
Integral Form:
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Purpose: The integral form of conservation of mass is used to analyze a control volume, which is a fixed region in space through which fluid flows. It is particularly useful for problems where we know the fluid properties at the boundaries of the control volume.
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Equation: The integral conservation of mass equation is given by:
\frac{\partial}{\partial t} \int_\mathcal{V} \rho \, \mathrm{dV} + \int_\mathcal{S} \boldsymbol{\hat{n}} \cdot \rho \boldsymbol{\vec{u}} \, \mathrm{dA} = 0 -
Application: This form is used to relate the inflow and outflow of mass across the boundaries of a control volume. It is particularly useful for systems like pipe networks or tanks where the flow is well-defined at the boundaries.
Differential Form:
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Purpose: The differential form of conservation of mass applies to an infinitesimal fluid element and is used to describe the flow field at every point in space. It is essential for deriving field equations that describe fluid motion.
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Equation: For incompressible flow, the differential conservation of mass equation is:
\nabla \cdot \boldsymbol{\vec{u}} = 0 -
Application: This form is used to impose a kinematic constraint on the velocity field, ensuring that there is no net volumetric expansion or contraction in the flow. It is crucial for analyzing complex flow fields and is often used in computational fluid dynamics.
Conservation of Momentum
Integral Form:
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Purpose: The integral form of conservation of momentum is used to analyze the forces and momentum changes within a control volume. It is derived from Newton’s second law and is useful for problems involving forces on objects due to fluid flow.
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Equation: The integral conservation of momentum equation is:
\frac{\partial}{\partial t} \int_\mathcal{V} \rho \boldsymbol{\vec{u}} \, \mathrm{dV} + \int_\mathcal{S} \left(\rho \boldsymbol{\vec{u}}\right) \left(\boldsymbol{\hat{n}} \cdot \boldsymbol{\vec{u}}\right) \, \mathrm{dA} = \sum \boldsymbol{\vec{F}} -
Application: This form is used to calculate forces on objects, such as the force on a bend in a pipe or the lift on an airfoil, by considering the momentum fluxes across the control volume boundaries.
Differential Form:
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Purpose: The differential form of conservation of momentum, often referred to as the Navier-Stokes equations, describes the momentum balance at every point in the fluid. It is essential for detailed analysis of fluid flow.
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Equation: The Navier-Stokes equations for incompressible flow are:
\rho \frac{\partial \boldsymbol{\vec{u}}}{\partial t} + \rho \boldsymbol{\vec{u}} \cdot \nabla \boldsymbol{\vec{u}} = -\nabla p + \eta \nabla^2 \boldsymbol{\vec{u}} + \rho \boldsymbol{\vec{g}} -
Application: These equations are used to predict the velocity and pressure fields in fluid flow problems. They are fundamental to computational fluid dynamics and are used to solve complex flow problems analytically or numerically.
In summary, the integral forms are used for control volume analysis, focusing on boundary conditions and overall system behavior, whereas the differential forms provide detailed pointwise descriptions of the flow field, essential for understanding the internal dynamics of fluid systems.
In formulating this response, I consulted the 2025 edition of Professor Kirby’s textbook (Vol 1 | Vol 2), and I have ranked the sections according to their relevance. If they are marked as relevant, I encourage you to explore them:
Textbook sections:
- Chapter 7.5: When to use Integral Conservation (HIGHLY RELEVANT)
- Chapter 8.2: Conservation of Mass (HIGHLY RELEVANT)
- Chapter 7.2: Integral Conservation of Mass (HIGHLY RELEVANT)
- Chapter 7.3: Integral Conservation of Momentum Part 1 (HIGHLY RELEVANT)
- Appendix D (HIGHLY RELEVANT)
Disclaimer:
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worth noting that the momentum flux integral can be written in a way that shows a more clearly parallel structure with mass:
\frac{\partial}{\partial t} \int_\mathcal{V} \rho \boldsymbol{\vec{u}} \, dV + \int_\mathcal{S} \boldsymbol{\hat{n}}\cdot\rho\boldsymbol{\vec{u}\vec{u}} \, dA = \sum \boldsymbol{\vec{F}}
here \rho\boldsymbol{\vec{u}\vec{u}} is the momentum flux dyadic, which compares to \rho\boldsymbol{\vec{u}}, which in the mass conservation equation is the mass flux vector.