Questions tagged [fluid-dynamics]
For questions about fluid dynamics which studies the flows of fluids and involves analysis and solution of partial differential equations like Euler equations, Navier-Stokes equations, etc. Tag with [tag:mathematical-physics] if necessary.
1,055
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How to find a solution of the form $u=\mathfrak{R}(f(y)e^{i\omega t})$ to the PDE $\frac{\partial u}{\partial t}=\nu\frac{\partial^2u}{\partial y^2}$?
I'm working on an unassessed course problem,
Let $$\frac{\partial u}{\partial t} = \nu\frac{\partial^2u}{\partial y^2}.$$ By seeking a solution of the form $$u=\mathfrak{R}(f(y)e^{i\omega t}),$$ ...
- 1,891
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17
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Do we have 3 Dimensional solution for navier stokes equations? IF notI have the solution I published four years ago with small attention. [closed]
Navier Stokes 3 dimmensional version of solution
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2
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38
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How to produce this graph in COMSOL?
I am trying to model generalised Couette flow in COMSOL. My goal is to reproduce this textbook graph in COMSOL.
To cue in:
Two parallel plates are in $h$ distance apart. The steady laminar flow of ...
- 131
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0
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13
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Effect of point source perturbation on lift for 2D subsonic flows past airfoils
I am trying to determine the effect of inserting a stationary point source perturbation on the lift exerted on an airfoil inmersed in an inviscid, compressible, subcritical flow (no shock waves, hence ...
- 1
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0
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14
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Finite Volumes: How to discretize an equation that has only a pressure differential and source term?
I have what is probably a very easy question. It is the first part in a longer, harder question about the SIMPLE algorithm. I feel that I understand the rest of the parts of the question (not shown). ...
- 2,297
1
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2
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51
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Deducing a $\frac{Dv}{Dt}$=$\frac{∂v}{∂t}$+$\frac{1}{2}$∇ (|v|$^2$) for an irrotational flow.
A flow has been defined in my textbook as irrotational if skew(∇v(x, t)) ≡ 0, i.e. if the velocity gradient is a symmetric tensor.
An exercise left at the end of the chapter tasks one to deduce $\frac{...
- 159
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53
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How can we write the Navier-Stokes equations on rotating sphere intrinsically?
The Navier-Stokes equations in $\mathbb{R}^n$ read $$u_t+u\cdot \nabla u -\mu \Delta u +\nabla p =f$$
$$\nabla \cdot u =0.$$
I had a look at a paper by Maryam Samavaki and Jukka Tuomela titled "...
- 31
3
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0
answers
92
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What is the vorticity of a velocity field?
If $u:\mathbb{R}^3\to \mathbb{R}^3$ is a velocity field one defines the vorticity as the curl
\begin{align}
\omega= \text{curl}(u).
\end{align}
I just read that vorticity measures the rotation of the ...
- 63
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49
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Stokes' Theorem, Circulation, Vorticity and Cylindrical Polar Coordinates
So, the aim of this question is to show that Stokes' Theorem holds in this context, but I'd like some help understanding how this relates to cylindrical polar coordinates (something I struggle with ...
- 306
2
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1
answer
53
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(When) does $\text{body force}=\text{pressure gradient}=0\neq\text{fluid velocity}\implies0<\text{drag}$?
I'm working on an unassessed course problem,
A cylinder with radius $a_1$ moves parallel to its axis with constant positive velocity $U$ inside a stationary coaxial cylinder with radius $a_2(> a_1)...
- 1,891
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0
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26
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Converting vector into polar coordinates
In the following paper, the surface velocity for a moving, spherical particle is given as (eq 1):
$$\textbf{v}_s(\hat{\textbf{r}}) = \sum {2\over{{n(n+1)}}} B_n (\hat{\textbf{e}} \cdot \hat{\textbf{r}}...
- 87
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Why does $\frac{\text{d}}{\text{d}t}\int_V\rho u_i\text{ d}V=\int_V\left(\frac{Du_i}{Dt}\right)\rho\text{ d}V$?
Preamble (updated)
My course notes have
Now consider a closed volume $V$ in the fluid which is bounded by a surface $S$. Let $V$ move with the fluid, so no fluid moves in or out of $V$. [...] Now $\...
- 1,891
3
votes
1
answer
41
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Why does incompressibility imply zero material derivative?
My course notes define
The material or substantive derivative, $$\frac{D}{Dt}=\frac{\partial}{\partial t}+\textbf{u}\cdot\nabla,$$ where u is the fluid velocity
then add,
In very many circumstances ...
- 1,891
2
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0
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34
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What's the divergence of Oseen tensor for the Zimm model?
While studying the Zimm model on the Doi & Edwards book "The Theory of Polymer Dynamics", I faced equation 4.41 which states that:
$$
\frac{\partial}{\partial\boldsymbol{R}_j}\cdot\...
1
vote
1
answer
22
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Help with understanding indices for Navier Stokes Equation
I am using Kundu and Cohen's textbook on Fluid Mechanics. I am not a mechanical engineering major, but I am trying to understand the indices for a project that I am doing. This is what I have so far:
...
- 63
1
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0
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27
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Why stream-line formulation is not good to solve a 2D navier-stokes problem?
The Navier-Stokes equation of an incompressible fluid is
$$\begin{align} \nabla \cdot \mathbf{u} & = 0 \tag{1}\label{1}\\ \dfrac{\partial \mathbf{u}}{\partial t} + \left(\mathbf{u} \cdot \nabla\...
- 1,136
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20
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Non Conformal Points of Joukowsky Transform
I am studying the use of the Joukowsky transform on the shape of airfoils. The transform is defined as
$$
f(\zeta) = \zeta + \frac{b^2}{\zeta}.
$$
I have found that $f$ is not conformal at $\zeta = \...
- 305
1
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1
answer
33
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How to transform the boundary conditions between formulations of PDE? 2D fluid flow
Question: How do I transform the boundary conditions of the speed formulation $(u, \ v)$ into the vortex-stream formulation $(\psi, \ w)$?
Cavity flow problem:
A incompressible Newtonian fluid is in ...
- 1,136
2
votes
2
answers
56
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acceleration of a particle moving along a streamline using tensor calculus
In a steady flow, the streamline coincides with the particle trajectory. In a book on MHD, I saw that if I pick a streamline $C$ and $s$ is a curvilinear coordinate measured along $C$, $V(s)$ is the ...
- 449
1
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0
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32
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Is Divergence without a dot product symbol a gradient on vector?
Divergence operating on a vector field ($\mathbf{u}=[u, v, w]\in\mathbb{R}^3$) outputs a scalar field:
$\nabla\cdot \mathbf{u} = \frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+\frac{\...
0
votes
1
answer
109
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Navier Stokes equation in cylindrical coordinates
A book on fluid mechanics gives the NS:
$$\dfrac{\partial\boldsymbol{u}}{\partial t}+(\boldsymbol{u}\cdot\boldsymbol{\nabla})\boldsymbol{u}=-\frac{1}{\rho}\nabla p+\nu\boldsymbol{\nabla}^2\boldsymbol{...
- 17
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0
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22
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Derivation of singular surface tension force term in Cahn-Hilliard-Navier-Stokes model
I am currently dealing with the Cahn-Hilliard-Navier-Stokes model
\begin{align}
\partial_t \phi + \nabla\cdot (u\phi) & = \nabla \cdot (M\nabla \mu) \\
\mu &= f'(\phi) - \lambda \nabla^2 \phi \...
- 161
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1
answer
59
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Calculating volume of water flowing over an edge
I'm working on a Unity project to simulate water flow over a grid with varying elevation. I'm trying to accurately reflect how water would spill over from one cell to the next.
For example, cell A in ...
0
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0
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16
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books/paper recommendation on computational thermal-turbulence by using FEM
I have just learned basic FEM for 2D N-S euqation, now my teacher let me to do the following problem, the document of this problem is in large fluid problem, the system of equations is listed in that ...
- 149
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1
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62
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How to solve this biharmonic equation? (Viscous fluid flow)
I am investigating lid-driven cavity flow, demonstrated in the below diagram:
We have a square (two dimensional) domain, with fully Dirichlet conditions for the velocity and fully Neumann conditions ...
- 10.6k
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0
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15
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how to formulate the stability problem in the incompressible inviscid limit and find the dispersion relation
Folks,
I'm trying to formulate the stability problem in the incompressible inviscid limit and find the dispersion relation in the Couette flow regime. As shown, the 2 infinite plates move one against ...
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0
answers
36
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Converting Integral to Matrix Operator Form
I'm going through this paper here: https://brian-f-farrell.fas.harvard.edu/publications/three-dimensional-optimal-perturbations-viscous-shear-flow%C2%A0.
Specifically, my question is related to the ...
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3
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1
answer
62
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Delta functions within integrals
If I have a velocity integral defined by the following:
$$\mathbf{v}(\mathbf{r}) = \int \mathbf{H}(\mathbf{r} - \mathbf{r}') \cdot \mathbf{f}(\mathbf{r}')~\mathrm d^3 \mathbf r' $$
where $\mathbf{H}$ ...
- 87
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25
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How to implement Newton iteration in Navier stokes equation
Suppose the nonlinear Navier stokes equation
$$
\left\{\begin{array}{l}
(\mathbf{u} \cdot \nabla) \mathbf{u}-\nabla \cdot \mathbb{T}(\mathbf{u}, p)=\mathbf{f} \quad \text { in } \Omega \\
\nabla \cdot ...
- 149
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1
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95
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Understanding the notation $\nabla$
I came across this problem when going over some material related to shear stress vector.
As far as I know the symbol $\nabla$ has a couple of different meanings.
Let $\vec{i},\vec{j},\vec{k}$ be the ...
- 1,466
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1
answer
40
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Setting up Very Simple PDE Resembling Ice Flow Full Stokes for a Toy FEM Problem
My main goal is to come up with a very very simple model that resembles a simple version of the Full Stokes equations used in ice flow modeling. I would like to take the viscosity to be constant in my ...
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23
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How to specify the value of pressure in a point in solving steady stokes equation numerically?
I try to solve the steady-Stokes equation numerically on , that is
\begin{aligned}
-\nabla \cdot \mathbb{T}(\mathbf{u}, p) & =\mathbf{f} \quad \text { on } \Omega=[0,1]*[-0.25,0], \\
\nabla \cdot \...
- 149
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1
answer
57
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Taking the curl of advective part of navier-stokes equation to get vorticity in index notation
I need to take the curl of
$\frac{\partial u_i}{\partial t} = -\frac{1}{\rho}\frac{\partial p}{\partial x_i}-\frac{\partial}{\partial x_j}u_i u_j+\nu\frac{\partial^2 u_i}{\partial x_i^2}$
to get
$\...
- 11
1
vote
1
answer
54
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Why isn't this divergence theorem question all equal to zero?
I have the following conditions for flow velocity and stress tensor:
$$\nabla \cdot \textbf{u} = 0$$ $$\nabla \cdot \mathbf{\sigma} = 0$$
The flow velocity is a vector field, stress tensor a rank-2 ...
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3
votes
0
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92
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Kernel generalized divergence convergence
Let $\Omega \subset \mathbb{R}^n$ be bounded, open, Lipschitz. Let $A_n,A^{-1}_n \in L^{\infty}(\Omega,\mathbb{R}^n \times \mathbb{R}^n )$ such that:
$$A_n,A^{-1}_n \to I_d \quad \text{uniformly}$$
We ...
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1
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37
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Prove that the following is incompressible (ie divergence is zero)
I am trying to prove that the following fluid flow is incompressible (in other words, its divergence is zero):
$$\textbf{u}( \textbf{r, S}) = \alpha {(\textbf{r}\cdot\textbf{S}\cdot \textbf{r})\textbf{...
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0
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17
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Turbulence modeling for mathematicians
I am currently doing my masters degree in Computational Fluid Dynamics (CFD). While I am learning a lot of skills in regards to coding and meshing, the course content is a bit unsatisfactory at least ...
- 10.6k
2
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0
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32
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Condition for being a ejecting vector field
Consider the two images in this link: ejecting vector field image.
In the upper picture: a fluid is smoothly flowing in a two dimensional pipe (i.e. a velocity vector field passing without any ...
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38
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How can I compute the Jacobian of the diffusive flux in Shallow Water System?
In the conservation form of shallow water system
$$ \dfrac{\partial h}{\partial t}+\dfrac{\partial q}{\partial x}=0$$
$$ \dfrac{\partial q}{\partial t}+\dfrac{\partial }{\partial x}(\dfrac{q^2}{h}+\...
- 1
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0
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23
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Second order ode with non-constant coefficients (non-homogeneous)
I have been trying to solve an equation from a research paper given as,
$$R^2P_0^{\prime\prime}+(R+3R^3/H)P_0^{\prime}-P_0 = -6R^3/H^3$$
($R$ is the radial coordinate and $H = 1+R^2/2$), given the ...
- 21
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1
answer
127
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Pressure values at the boundary in Navier Stokes equations
An interesting and important question: what boundary conditions should be used for the pressure field in Navier-Stokes system of equations?
I'm currently trying to find a way to derive the pressure ...
- 330
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1
answer
64
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Galilean invariance of Burgers Equation
Proposition 1.1, Section 1.2 of Majda and Bertozzi reads as follows:
Let $v,p$ be a solution to the Euler or Navier Stokes Euquations. Then the following tranformations also yield solutions:
For any $...
- 3,295
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1
answer
33
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Explain why the convective derivative of the position vector is the velocity.
Can someone please explain why the convective derivative of the position is the velocity. It's not very obvious to me from the definition.
Thanks
- 307
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1
answer
82
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General solution for vector differential equation
I encounter a vector differential equation in solving poroelastic flow as follows:
$$\nabla\times\nabla\times\boldsymbol{\Phi}=\mathbf{A}+\nabla(\frac{r^2}{2}\chi)$$
where $\boldsymbol{\Phi}$ is an ...
- 351
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0
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28
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Galilean Invariance of Navier Stokes Equation
I'm struggling to work out some simple formalism. I'm reading something that says if the quadruple $(v_i(x_i,t),p(x_i,t))$ solves the navier stokes equations so does the quadruple $(v_i^c(x,t),p^c(x,t)...
- 3,295
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votes
1
answer
26
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Total Time Derivative and partial time derivative
Hi guys i have some problems starting from $(1)$ deriving the $(2)$:
$$\frac{d}{dt}\int_V \rho e dV=\int_V \rho \pmb{f}\cdot \textbf{u}dV+\int_{V}\rho~s~dV+\int_{\partial V}(\pmb{\sigma} \cdot \pmb{u})...
- 3
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0
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45
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Euler–Tricomi equation in Canonical Forms $\;u_{xx}+xu_{yy}=0$
I need help reducing the Euler-Tricomi equation to it's first and second canonical forms, in the region where it is hyperbolic. It's below in both notations:
\begin{align}
Leibniz\;notation:&\quad ...
0
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0
answers
47
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no-flux boundary condition of advection-diffusion equation for both velocity field and scalar field
I am trying to find a no-flux boundary condition for the advection-diffusion equation in a bounded region $\Omega$, both for the velocity field and scalar filed $\theta$.
$$\partial_{t}\theta+\mathbf{...
- 1
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votes
0
answers
19
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Making three dimensional vector field divergence free.
In fluid dynamics, there is any general method/numerical method to make three dimensional vector field (momentum equation) divergence free.
- 569
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0
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54
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Calculate volume flux across the x = 1 plane
I believe I should be calculating $\int_P \textbf{u} \cdot \textbf{n}\,ds$, where $P$ is the plane $x = 1$. The velocity field is $\textbf{u} = (u(x,y), v(x,y), 0)$, as $\textbf{n} = (1,0,0)$ the ...
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