in some cases we write \int (\rho u) \cdot (u \cdot n) and in other places we write \int n \cdot \rho u u , which is right? or are they the same?
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:
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PURPOSE OF THE EXERCISE:
Demonstrate that the two notations for the convective momentum flux—
\int \rho \,\boldsymbol{\vec{u}} (\boldsymbol{\vec{u}}\cdot\hat{n})\,dA and \int \hat{n}\cdot\left(\rho \,\boldsymbol{\vec{u}}\boldsymbol{\vec{u}}\right)\,dA are equivalent representations of the same physical quantity and must be interpreted correctly when performing control-volume analyses. -
ENGINEERING CONTEXT:
In the momentum conservation equation for a control volume, the net momentum flux is expressed as the surface integral of the convective flux. Whether you write this as \int \rho \,\boldsymbol{\vec{u}} (\boldsymbol{\vec{u}}\cdot\hat{n})\,dA or \int \hat{n}\cdot\left(\rho \,\boldsymbol{\vec{u}}\boldsymbol{\vec{u}}\right)\,dA, you are formulating the same term. This understanding is essential for correctly applying the momentum conservation laws in practical designs such as piping systems, nozzles, and airfoils. -
PHYSICAL CONTEXT:
Both notations represent the transport of momentum due to fluid motion across a boundary. They can be shown to be identical by commutation identities. The first notation separates momentum and speed normal to a control surface. The second notation gathers momentum and its transport into a momentum flux dyadic tensor. -
ASSUMPTIONS:
Assume that the fluid is a continuum with well-defined velocity and density fields and that the operations of dot product and dyadic (outer) product follow the standard algebraic rules; these assumptions are standard in classical fluid mechanics analyses. -
SOLUTION TECHNIQUE:
The flux of momentum across a surface is given by the product of the momentum with the component of the velocity normal to the surface:
\int \rho \,\boldsymbol{\vec{u}} (\boldsymbol{\vec{u}}\cdot\boldsymbol{\hat{n}})\,dA
dot products of vectors commute:
\int \rho \,\boldsymbol{\vec{u}} (\boldsymbol{\hat{n}}\cdot\boldsymbol{\vec{u}})\,dA
now \rho is a scalar and (\boldsymbol{\hat{n}}\cdot\boldsymbol{\vec{u}}) is a scalar also, so multiplicatoin by these scalars commute also and I can move them around:
\int \rho \,(\boldsymbol{\hat{n}}\cdot\boldsymbol{\vec{u}})\boldsymbol{\vec{u}} \,dA
now, dot products of scalars with symmetric dyadics commute, so we can change the order of operation:
\int \rho \,\boldsymbol{\hat{n}}\cdot(\boldsymbol{\vec{u}}\boldsymbol{\vec{u})} \,dA
and again we can move the \rho around:
\int \,\boldsymbol{\hat{n}}\cdot(\rho\boldsymbol{\vec{u}}\boldsymbol{\vec{u})} \,dA
so this form is equivalent to the \int \rho \,\boldsymbol{\vec{u}} (\boldsymbol{\vec{u}}\cdot\boldsymbol{\hat{n}})\,dA form. This form creates a dyadic \rho\boldsymbol{\vec{u}}\boldsymbol{\vec{u}} which is a dyadic tensor that describes the state of momentum flux at any point in space. It is used analogously to the stress tensor, which describes the state of stress at any point in space. We dot \boldsymbol{\hat{n}} into these dyadics to identify the momentum flux through a control surface or the surface stress (traction) at a control surface.
- REFLECTION:
The equivalence of these two expressions confirms that the momentum flux can be written in either form, provided the notation is interpreted carefully. This flexibility in representation is particularly useful when employing the divergence theorem to transform between surface and volume integrals in the derivation of the differential conservation equations used 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:
- Chapter 7 Fundamentals: Reynolds Transport Theorem (HIGHLY RELEVANT)
- Chapter 7.2: Integral Conservation of Mass (HIGHLY RELEVANT)
- Appendix F.3: Volumetric Viscous Force Part 1 (HIGHLY RELEVANT)
- Appendix D Fundamentals: Why … (HIGHLY RELEVANT)
- Appendix B.4: Vector Calculus Part 3 (HIGHLY RELEVANT)
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