why do you define a dyadic rho u u if you don’t have to, why not just use rho u times n dot u
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 use of the dyadic tensor \rho \vec{u} \vec{u} in fluid mechanics, particularly in the context of momentum flux, provides a more comprehensive and flexible framework for analyzing fluid flow. Here’s why it is advantageous to use the dyadic tensor rather than just \rho \vec{u} (\hat{n} \cdot \vec{u}):
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Directional Aspects: The dyadic tensor \rho \vec{u} \vec{u} inherently captures both the direction of the momentum being carried and the direction in which it is being transported. This dual-directional nature is crucial for understanding how momentum is distributed across different planes in a flow field.
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Matrix Representation: The dyadic tensor can be represented as a matrix, which allows for straightforward mathematical manipulation and integration into the Navier-Stokes equations. This representation is particularly useful in computational fluid dynamics, where matrix operations are common.
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Generalization: Using the dyadic tensor provides a more general form that can be applied to various boundary conditions and flow configurations. It allows for the analysis of momentum flux across any arbitrary surface, not just those normal to a specific direction.
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Symmetry and Commutativity: For symmetric dyadics like \rho \vec{u} \vec{u}, the dot product with a vector is commutative, meaning \hat{n} \cdot \rho \vec{u} \vec{u} = \rho \vec{u} \vec{u} \cdot \hat{n}. This property simplifies calculations and ensures consistency in the analysis of momentum flux.
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Physical Interpretation: The dyadic tensor provides a clear physical interpretation of momentum flux as a volumetric quantity, which is essential for understanding the dynamics of fluid flow and the forces acting on fluid elements.
In summary, while \rho \vec{u} (\hat{n} \cdot \vec{u}) might suffice for specific cases, the dyadic tensor \rho \vec{u} \vec{u} offers a more robust and versatile approach for analyzing momentum transport in fluid mechanics.
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:
- Appendix D Fundamentals: Why … (HIGHLY RELEVANT)
- Appendix B.3: Dyadic Tensor Algebra Part 3 (HIGHLY RELEVANT)
- Appendix F.3: Volumetric Viscous Force Part 1 (HIGHLY RELEVANT)
- Chapter 7.3: Integral Conservation of Momentum Part 1 (HIGHLY RELEVANT)
- Appendix B.1: Vector Algebra Part 6 (HIGHLY RELEVANT)
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PK Bot’s 1 and 5 are meaningful.
2-4 are a lot of wasted words. both forms can be written as matrices. both forms can be applied to a variety of boundary condidtions. both forms commute. PK Bot apparently thinks it is paid by the word.
what you need depends on what question is being asked.
if the question is --what is the flux of momentum across a surface, \hat{n}\cdot\rho\vec{u}\vec{u} is correct but unnecessary; in that case \rho\vec{u}(\hat{n}\cdot\vec{u}) is sufficient.
if the question is – what is the state of momentum flux at a point, the answer is \rho\vec{u}\vec{u} and there is no vector representation that can represent that quantity.
fluid mechanics involves a vector result, so dyadics are typically a means to an end. the dyadics themselves have meaning (state of stress at a point, state of momentum flux at a point, rotation at a point, deformation at a point) but we can often find a way to get an answer without them. however, I prefer the depth of understanding that comes from using the dyadics.