(a) explain what each component of ρuu is (b) explain how to take the dot product n dot ρuu and what the result means (c) explain how to take the dot product ρuu dot n and what the result means
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PK Bot’s answer:
(a) Components of ρ\vec{u}\vec{u}
The term ρ\vec{u}\vec{u} represents the momentum flux dyadic tensor, where:
- ρ is the fluid density, a scalar quantity that represents mass per unit volume.
- \vec{u} is the velocity vector of the fluid, which has both magnitude and direction.
The dyadic tensor ρ\vec{u}\vec{u} has two directional aspects, the direction in which momentum is carried and the direction of the momentum itself. In a Cartesian coordinate system, the tensor can be represented by use of a matrix:
\left[\rho\vec{u}\vec{u}\right] = \begin{bmatrix} ρu u & ρu v & ρu w \\ ρv u & ρvv & ρvw\\ ρwu & ρwv & ρww \end{bmatrix}
where \left[\rho\vec{u}\vec{u}\right] is the matrix representation of \rho\vec{u}\vec{u}. Each component of \left[\rho\vec{u}\vec{u}\right] describes the flux of one component of momentum in one coordinate direction. For example the \hat{xy} component of the tensor describes how y-momentum is carried in the x-direction.
(b) Dot Product \hat{n} \cdot ρ\vec{u}\vec{u}
The dot product of \hat{n} with ρ\vec{u}\vec{u} extracts the part of the momentum flux that crosses a plane normal to \hat{n}.
To take the dot product of the unit normal vector \hat{n} with the dyadic tensor ρ\vec{u}\vec{u}, you can do it with matrix operations by multiplying the transpose of the dyadic tensor with the vector:
\left[\hat{n} \cdot (ρ\vec{u}\vec{u})\right] =
\begin{bmatrix} ρu u & ρvu & ρwu \\ ρuv & ρvv & ρwv\\ ρuw & ρvw & ρww \end{bmatrix} \begin{bmatrix}n_x\\ n_y\\ n_z \end{bmatrix}
The result gives the momentum flux across a surface normal to \hat{n}, indicating how much momentum is being transferred through that surface per unit area.
(c) Dot Product ρ\vec{u}\vec{u} \cdot \hat{n}
Taking the dot product of the dyadic tensor ρ\vec{u}\vec{u} with the unit normal vector \hat{n} is a fundamentally different operation that would lead to a different result if the dyadic tensor were not symmetric. However, because the dyadic tensor in this case is symmetric, the result (even though it has a different meaning) has the same value as the value for \hat{n}\cdot\rho\vec{u}\vec{u}.
- The operation ρ\vec{u}\vec{u} \cdot \hat{n} contracts \rho\vec{u}\vec{u} in a different way. Rather than finding the momentum flux across a plane normal to \hat{n}, ρ\vec{u}\vec{u} \cdot \hat{n} finds the fluxes in all directions of \hat{n}-momentum.
- it can be evaluated by taking the matrix multiplication of the dyad with that of the vector:
\left[ (ρ\vec{u}\vec{u})\cdot \hat{n}\right] =
\begin{bmatrix} ρu u & ρuv & ρuw \\ ρvu & ρvv & ρvw\\ ρwu & ρwv & ρww \end{bmatrix}\begin{bmatrix}n_x\\ n_y\\ n_z \end{bmatrix}
I wrote out the momentum flux so that you can see when I take the transpose, but of course the momentum flux is a symmetric dyad. \rho u v means something different from \rho v u, but of course those two expressions are equal to each other.
In summary, \rho \vec{u}\vec{u} is the momentum flux dyad. Dotting \hat{n} into the momentum flux gives the flux crossing an \hat{n}-plane, and dotting the momentum flux into \hat{n} gives the \hat{n}-momentum flux. These two things have the same value but are different conceptually; they are the same only because \rho\vec{u}\vec{u} is symmetric.
In crafting this response, I dedicated some time to reviewing the 2025 edition of Professor Kirby’s textbook (Vol 1 | Vol 2), and I found these sections to be particularly helpful in formulating my answer. I encourage you to explore the pertinent ones directly:
Textbook sections:
- Appendix B.1: Vector Algebra Part 1 (MARGINALLY RELEVANT)
- Appendix B.4: Vector Calculus Part 2 (MARGINALLY RELEVANT)
- Appendix B.3: Dyadic Tensor Algebra Part 1 (MARGINALLY RELEVANT)
- Appendix B Fundamentals: Unit Vectors and Dyads (MARGINALLY RELEVANT)
- Appendix B.3: Dyadic Tensor Algebra Part 2 (MARGINALLY RELEVANT)
- Appendix B.1: Vector Algebra Part 2 (MARGINALLY RELEVANT)
- Appendix B Fundamentals: Relation between tensors, tensor operators, and matrices (MARGINALLY RELEVANT)
- Appendix E Fundamentals: Relation between tensors, tensor operators, and matrices (MARGINALLY RELEVANT)
- Appendix D Fundamentals: Why … (MARGINALLY RELEVANT)
- Chapter 3.5: Hydrostatic Equation: Vector Differential Operator Form (MARGINALLY RELEVANT)
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