All Questions
Tagged with vector-fieldsfluid-dynamics
85 questions
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38views
How can groundwater spread be found using gradient descent?
Suppose that a droplet of water in an aquifer flow has the position, $\gamma_t \in \mathbb{R}^3$ with a time-independent hydraulic head $h(x,y)$ (a watertable elevation map), where, \begin{align*} ...
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1answer
81views
When is the Lamb vector the gradient of a function?
There is something I am not seeing in this derivation of advanced Bernoulli's principle: https://open.oregonstate.education/intermediate-fluid-mechanics/chapter/bernoulli-equation/ The Lamb vector is ...
- 6,589
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Determine the maximum and minimum acceleration from the velocity field [closed]
If the speed distribution, in m/s, of a flow is given by $v = 2x^3 + 2y^2 - 3z$, then the acceleration of the fluid at the point with coordinates $[2, 1, 5]$, in meters, will be greater than $20, \...
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Momentum density of electromagnetic field
I'm reading a textbook deriving the expression for the momentum densisty of the EM field and it starts with the following concept. Let's define the vector field $\vec{p}_\text{me}(\vec{r},t)$ to be ...
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1answer
50views
Topology and parity in divergence free vector fields
If you consider a general divergence free vector field (not constant), in 2 and 3 dimensions it is easy to see that it is an axial vector and hence is not invariant under parity transformation (flip ...
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1answer
65views
How do I find the absolute maximum and minimum values of the Lamb-Oseen Vortex?
For an angular velocity function derived by Navier-Stokes, $$ \omega \left(r,t\right)=\frac{\omega _0R_0^2}{R\left(t\right)^2}exp\left(-\frac{r^2}{R\left(t\right)^2}\right)$$ from which the azimuthal ...
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45views
Expanding the barotropic nondivergent potential vorticity equation: Which vector calculus property/identity to apply for dot product and del operator?
I am trying to expand the barotropic nondivergent potential vorticity (PV) equation [link] $$\frac{\partial \zeta}{\partial t} = -\vec{V} \cdot \nabla(\zeta + f)$$ where $\zeta$ is the relative ...
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1answer
69views
Shapes of waves in the surface of a pond when the breeze blows
When I throw a stone into a pond while a breeze is blowing across its surface and the water is at rest, what would be the shapes of the waves? Before the wind, the shape of the waves is circular. Then ...
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How to make a parametric that matches a vector field?
So I have a vector field defined as $(X(x,y),Y(x,y))$ and I’m trying to make a parametric $(t,t)$ who’s derivative at a point is equal to the vector field at that point. for example the vector field $(...
2votes
1answer
269views
Stokes stream function derivation
I want to know a concrete derivation of 3D Stokes stream function. The statement is, for example in 3D spherical coordinates (with symmetry in rotation about the $z$-axis), if $$\nabla \cdot u=0\tag{...
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Physical significance of $\vec{w}$ $\times$ $($curl $\vec{v})$
I think if curl of a vector field $\vec{v}$ corresponds to an applied rotation, it's cross product with a velocity vector field $\vec{w}$ (say) should give something analogous to the resulting torque. ...
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103views
What does the notation $(k \cdot \nabla ) v$ mean? [duplicate]
I am reading a paper and it uses a notation I am not too familiar about. Although I saw it used elsewhere, I don't remember the meaning of it and I don't want to misinterpret it and realize after ...
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90views
How can the vortex be an elemental potential flow if there is a point of curl?
Aero is not my speciality at all so apologies if missed anything. But when looking at potential flows, i thought the whole point is for there to be no rotation at any point and its that reason the ...
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82views
No-slip condition tangential and normal component decomposition
No-slip condition on a corrugated surface (modelled by a sinusoidal function $b(x)$)) $\vec{ u} (x,b(x)) =u \vec{i}+ w \vec{k} = 0 \vec{i} + 0 \vec{k}$ expressing in terms of the stream function : $$\...
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2answers
166views
If fluid flow is in units $m/s$, what units are $div$ in?
Imagine a fluid flow given by the formula $\mathbf{v} = x\mathbf{i} + y\mathbf{j}$, this is a simple radial flow outward and the dimensions would be length/time: At a given point, what is the velocity ...
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