All Questions
7 questions
1
vote
1
answer
85
views
Deriving smoothing kernels
I'm watching a video on smoothed particle hydrodynamics it just blindly claims that these smoothing kernels are pretty good.
$$W(r-r_b,h)\equiv\dfrac{315}{64\pi h^9}\left(h^2-|r-r_b|^2\right)^3$$
$$\...
- 114
0
votes
0
answers
70
views
What do equations involving infinitesimals say?
I am reading this note on the Bernoulli equations with the following derivations:
I am struggling to find a calculus based meaning for the above equations involving the infinitesimal $\delta V$: I ...
- 101
1
vote
1
answer
122
views
Fluid mechanics - Particular derivative
I've got a problem in MHD where I need to develop the following derivative (where $\vec{B}$ is the magnetic field and $\rho$ the density, both are functions of $(\vec{r},t))$: $$ \frac{\mathrm{d}}{\...
- 11
1
vote
2
answers
125
views
Partial derivative in hydrostatic equilibrium in star
In a simple model, a gaseous, non-rotating star consists of many thin, concentric spherical shells with radius $r$ and mass $\text{d}m$. The total mass of the shells within radius $r$ is $m$. The ...
- 1,169
0
votes
1
answer
78
views
Particle paths - the distance moved by a particle in a velocity field
This question is is context to particle paths.
Particle paths are trajectories of a given particle in the velocity field:
$$\boldsymbol{u}(\boldsymbol{x},t)$$
A particle location at position $\...
- 187
0
votes
1
answer
835
views
Relationship Between Differentiation in Two Frames
In section 10.3 of Principles of Ideal-Fluid Aerodynamics by Karamcheti, he writes the following:
$\qquad$ Denote by $K_1$ a reference frame fixed with respect to the moving body. We shall denote ...
- 149
6
votes
2
answers
5k
views
What does $(\mathbf{u}\cdot\nabla)\mathbf{u}$ mean in the Navier-Stokes equation?
I am studying the Navier-Stokes equations and I have the equation in the form:
$$\rho \dfrac{\partial{\mathbf{u}}}{\partial{t}} + \rho (\mathbf{u}\cdot\nabla)\mathbf{u} - \mu\nabla^2\mathbf{u} + \...
