# 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.

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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 $$u=\mathfrak{R}(f(y)e^{i\omega t}),$$ ...
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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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### 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 ...
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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 ...
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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). ...
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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: ...
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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 \...
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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 ...
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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 ...
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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 ...
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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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### 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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### 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 $\mathbf{H}$ ...
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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 ...
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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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### 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 ...
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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{...
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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.
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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 ...