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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24 views

Amplitude of waves in water after a disturbance [closed]

Suppose I have a perfectly circular pool which is five meters in radius, three meters in depth, and filled with water. Say I drop a steel ball with a radius of five centimeters into the middle of the ...
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39 views

Simplifying the Navier-Stokes equation via clever choice of the pressure field

Is there any way to simplify the Navier-Stokes equation in $3$-dimensional space by choosing a clever pressure field? I don't know much about PDES but I thought of this question.
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Jacobian for signal speeds in local Lax-Friedrichs FVM for shallow water problem

This question is working on the same problem as mentioned here:. The original PDE is: $$\frac{\partial}{\partial t}\begin{bmatrix}h \\ hu \\ hv \end{bmatrix}+ \frac{\partial}{\partial x}\begin{bmatrix}...
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1answer
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How to prove this equation about calculation of matrix determinant?

How to prove the equation about the determinant of Matrix $M$, i.e., $|M|=\frac{(M \cdot a) \times (M \cdot b) \cdot (M \cdot c)}{a \times b \cdot c}$ where $a$, $b$ and $c$ are arbitrary vectors. ...
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1answer
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Using Runge Kutta in local lax friedrichs fvm for shallow water problem

This question is working on the same problem but using a new concept as mentioned here:. Now I am trying to expand this concept using the Runge kutta (RK4):. Now the original PDE is as follows: $$\...
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Unstable behavior of Cl (lift coefficent) in fluent

Using fluent for simulate the flow around a squared cilinder with rounded corners i get this wrong behavior (exponentially) for what concern the lift coefficient. The boundary conditions are uniform. ...
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20 views

Show that operator is admitted by the incompressible Navier Stokes equation

The NSE is $$(\mathbf{u} \cdot \nabla) \mathbf{u} + \frac{\partial \mathbf{u}}{\partial t} -\nu \Delta \mathbf{u} + \nabla p=0$$ $$\nabla \cdot \mathbf{u}=0,$$ that is $$\sum_i\left( u_i \frac{\...
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1answer
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Calculating fluxes in local Lax-Friedrichs FVM for shallow water problem

so I have a more programming and some applied maths background, but absolutely none in FVM or Fluid dynamics. I have to implement a solver using the local Lax-Friedrichs method, and so far kinda-so-...
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1answer
46 views

Fluid Mechanics - Sources/Sinks/Streamlines

My doubt is regarding the last part. How does one find the differential equation and then prove that they lie on the given surface? I am aware of the method, by writing dx/u = dy/v (or its equivalent ...
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Coordinate Transformation and Derivatives

I'm currently working on an assignment where I end up with the following differential equation in cylindrical coordinates: $0=\frac{\partial}{\partial r}\left(r \frac{\partial w_\varphi}{\partial r}\...
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Solving differential equation - mass conservation - Fluid dynamics

I'm trying to solve this differential equation - mass conservation law from Euler equations in 1D: $\frac{\partial n}{\partial t} + \nabla \cdot (n \mathbf{u}) = 0 $ (The one dimension I'm using is x):...
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If $u\in H_0^1(\Lambda,\mathbb R^2)\cap H^2(\Lambda,\mathbb R^2)$, is $\nabla^\perp\cdot\Delta u\in L^2(\Lambda)$?

Let $\Lambda\subseteq\mathbb R^2$ be open (and sufficiently regular for the subsequent consideration), $u\in H_0^1(\Lambda,\mathbb R^2)\cap H^2(\Lambda,\mathbb R^2)$ and $$w:=\nabla^\perp\cdot\Delta u,...
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Is $H_0^1\ni u\mapsto(u\cdot\nabla)u$ Fréchet differentiable?

Let $\Lambda\subseteq\mathbb R^2$ be bounded and open, $$V:=\left\{u\in H_0^1(\Lambda,\mathbb R^d):\nabla\cdot u=0\right\}$$ and $H$ denote the completion of $V$ with respect $\left\|\;\cdot\;\right\|...
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If $\nabla\cdot u=0$ and $w=\operatorname{curl}u$, then $\int w=0$

Let $\Lambda\subseteq\mathbb R^2$ be open, $u\in C^1(\Lambda,\mathbb R^2)$ with $\nabla\cdot u=0$ and $$w:=\frac{\partial u_2}{\partial x_1}-\frac{\partial u_1}{\partial x_2}.$$ How can we show that $...
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If $\nabla\cdot v=0$ and $w=\nabla^\perp\cdot v$, then $v=\nabla^\perp g\ast w$, where $g$ is the fundamental solution of the Poission equation

Let $\Omega\subseteq\mathbb R^2$ be open, $v:\Omega\to\mathbb R^2$ with $\nabla\cdot v=0$ (in a sense to be specified later), $$\nabla^\perp:=\left(-\frac\partial{\partial x_2},\frac\partial{\partial ...
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55 views

Can Navier-Stokes equation be derived from Cauchy momentum equation?

$$\frac{{\partial (\rho {v}_x})}{\partial t}+\nabla\cdot{(\rho {v}_x\mathbf{v}) = -(\nabla P)_x + \rho g_x+\nabla\cdot\mathbf{\tau_x}}$$ This is the $x$ component of the Cauchy momentum equation, ...
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1answer
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Notation issue with the Cauchy momentum equation

The conservation form of the Cauchy momentum equationis$$\frac{\partial}{\partial t}(\rho\vec{u})+\nabla\cdot(\rho\vec{u}\vec{u}^T)=-\nabla p+\nabla\cdot\vec{\tau}+\rho\vec{g}$$The second term on the ...
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Steady state solution to continuity equation

We have the continuity equation $$ \partial_t \, q(\mathbf{x},t) = -\nabla \cdot[ \, q(\mathbf{x},t) \, \mathbf{w}(\mathbf{x},t) \, ] $$ where the conserved scalar $q$ is advected by the field $\...
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Show that for all $w$, there is a unique $v$ with $\nabla\cdot v=0$ and $w=\nabla\wedge v$

Let $\Omega\subseteq\mathbb R^2$ be open and $w:\Omega\to\mathbb R$. I've read that there is a unique $v:\Omega\to\mathbb R^2$ with \begin{align}\nabla\cdot v&=0,\tag1\\\int v(x)\:{\rm d}x&=0,\...
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Analogue of $(f\cdot\nabla)f=\frac12\nabla|f|^2-f\times\operatorname{curl}f$ in the two-dimensional case

If $\Omega\subseteq\mathbb R^3$ is open and $f:\Omega\to\mathbb R^3$ is differentiable, there is the identity $$(f\cdot\nabla)f=\frac12\nabla|f|^2-f\times\operatorname{curl}f\tag1,$$ where $$\...
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17 views

Finte Element on Cahn-Hilliard equation

I was trying to discretize the Cahn-Hilliard equation using the finite element. Assume its one dimension setting. $ \begin{align*} &\int_{\Omega} \frac{c_{t+1}-c_{t}}{\Delta t} \phi_j \...
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44 views

Calculating differentials between cell velocities

If I have a 2D cell-based fluid simulator, which uses velocities and pressure, how can I find the change in pressure between the neighboring cells for a cell? I might be missing something big here, ...
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1answer
48 views

Trouble with fluid mechanics question (Bernoulli equation for streamlines)

Here is the problem statement: A liquid is in the annular space between two vertical cylinders of radii $\kappa R$, $R$, and the liquid is open to the atmosphere at the top. Show that when the inner ...
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21 views

Is there a basic example of method of matched asymptotic expansions?

I am studying a paper where the method of matched asymptotic expansions is used. Unfortunately I have never seen this method before and am not sure how it works. I don't really need the rigorous ...
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27 views

Angular Momentum Balance in Fluid Mechanics

When researching a derivation of the angular momentum balance equations, I came across these excellent online notes from a course on continuum mechanics. While this is helpful, I would appreciate a ...
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22 views

Integral representation of the pressure of the Stokes flow

I'm currently reading these three books: S. Kim, S. J. Karrila - Microhydrodynamics: Principles and Selected Applications O. A. Ladyzhenskaia -The Mathematical Theory of Viscous Incompressible Flow ...
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Can we calculate a vector field from the pressure field with Navier Stokes?

I've been looking for weeks and can't seem to wrap my head around CFD or the computation of fluid and its movement in 2D with Navier Stokes. Without an initial velocity field, and given ONLY a ...
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35 views

Cylindrical coordinates, time derivatives, and the Navier-Stokes equations.

I am trying to derive the Navier-Stokes equations in cylindrical coordinates, but am having trouble with the material derivative. Some background. The unit vectors in Cartesian coordinates are given ...
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What is the relation between between the eigenvalues of $M\equiv (I-Q)A$ and the eigenvalues of $MBM$ where $B\equiv QA$?

I have the following spatial operators A, and Q where $$ A \equiv L + C$$ with $L$ being the discrete Laplacian operator and $C$ being the discrete advection operator. The operator $Q$ is given as ...
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215 views

Attempting to prove a conserved quantity for incompressible Navier-Stokes PDE

I am looking for feedback on this attempt to prove a conserved quantity for incompressible Navier-Stokes $$(\mathbf{u} \cdot \nabla) \mathbf{u} + \frac{\partial \mathbf{u}}{\partial t} = \nu \Delta \...
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16 views

Finding stream function from boundary shape

How might one go about finding the stream function of a piece-wise function which is: $x=0$ for $\left \{ 3< y \right \}$ $(x-3)^2+(y-3)^2=9$ for $\left \{ 0<y< 3 \right \}\left \{ 0<x&...
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Navier-Stokes: formal definition/intuition of “mean flow”?

In (computational) fluid dynamics there is a notion of mean flow that I don't quite understand: E.g. in the Wikipedia entry for turbulence modeling it says: Averaging the equations gives the ...
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51 views

Euler equation in a Riemannian manifold

I am trying to prove that the Euler equation for a Riemannian 3-manifold $M$ $$ \frac{\partial u}{\partial t} + \nabla_{u} u = - \operatorname{grad} p \qquad \text (1) $$ is equivalent to $$ \frac{\...
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1answer
54 views

What are the Rankine Hugoniot Jump Conditions for quasilinear equations?

When we consider a conservation law $$ q_t + f(q)_x = 0, $$ the Rankine-Hugonoit conditions are given by $$ s(q^+ - q^-) = f(q^+) - f(q^-) . $$ However, how does this change if we are given some ...
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1answer
166 views

Help solving shallow water equations initial value problem?

The question is as followed: "Consider the initial value problem (IVP) for the linearised shallow water equations (1) $\frac{\partial h}{\partial t} + H_0 \frac{\partial u}{\partial x} = 0$ ; (2) ...
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1answer
24 views

Prandtl mixing length theory represented by a second order, non-linear ODE with boundary conditions

$$-K=\nu \frac{1}{r}\frac{\partial u}{\partial r}-a\varepsilon^{2}\frac{\partial u}{\partial r}\left ( 2\frac{\partial^2 u}{\partial r^2}+\frac{1}{r}\frac{\partial u }{\partial r} \right )$$ $$u(r=R)=...
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36 views

Question regarding derivation from Hydraulics

Context: Fundamental properties (behavior) of infinitesimal disturbances to flow. The coordinate system is placed on the bottom of a channel. Basic linearized equations per unit width are denoted as: ...
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While solving the motion in plane problems (dynamics) how to figure out whether the radial accelration is 0 or mgsin(theta)?

There is a problem : A straight smooth tube revolves with constant angular velocity W in a horizontal plane about one extremity which is fixed. If at zero time the tube be horizonal and a particle ...
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16 views

Vector identity: dot product of a vector with biharmonic firction

I would like some guidance / clarification on my workings for this. Here is my attempt at expanding the following, $$\underline u \cdot [A\nabla^4 \underline u] = \underline u \cdot [A (\nabla \cdot ...
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1answer
27 views

Controllable system

Let (A, B) is controllable and that rank B = n , we want to show that the geometric multiplicity of each eigenvalue of A is at most n. Any help
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21 views

Asymptotic Model for 2D Navier-Stokes

I am reading some texts on fluid dynamics problems and am unsure what the exact benefit is to using what is called an 'asymptotic model'. For example, when considering a two-dimensional flow between ...
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38 views

Image vortex for a point vortex inside a circle

Need help with the specific question related to the circle theorem. Let $z$ be a complex number. Show that the image of a line vortex of strength $2 \pi \kappa$ located at $z = b$ inside the circle $|...
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Proving that Stokeslet form of solution actually solves Stoke's flow

Stokes problem is \begin{equation} \nabla^{2}\mathbf{u} + \nabla P = \mathbf{f} \end{equation} has solution \begin{eqnarray} \mathbf{u}(\mathbf{r}) & = & \int_{V} \mathbf{f}(\mathbf{r'})\cdot\...
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67 views

Inclined Rankine Oval

I'm trying to plot Inclined Rankine Oval in Matlab. I use quite simple code: ...
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65 views

For a fluid flow in converging duct find velocity component shear strain rate rotional or irrtional

Consider a viscous fluid flow by pressure drop in a converging duct. The duct’s bottom and top walls are taken to be fixed and flat. The rectangular coordinates system for duct is shown in below ...
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1answer
69 views

What is wrong with my bilinear interpolation in Navier-Stokes model?

I'm modeling density of a fluid with a color map using numerical methods to compute the Navier-Stokes equations. This is the code which computes the advected velocity and density based on the ...
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1answer
46 views

Numerical simulation of water layer over terrain

I'm looking for a grid-based numerical method that allows simulating water over a grid-based terrain, presumably something like shallow water equations. I have a square grid of terrain elevation ...
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11 views

Control Volume for Momentum Integral in flow of fast jet

When trying to solve the usual problem of finding the force on a vertical wall given the surface area $A$ and speed of the horizontal flow $U$, I use the momentum integral: $$\frac{d}{dt}\int_V\rho \...
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21 views

Deriving the Poisson equation for pressure

I am trying to derive the Poisson equation for pressure, specifically that $$\bigtriangledown^2(p + \frac{1}{2}|\vec{u}|^2) = \bigtriangledown \cdot(\vec{u}\times \vec{w}) + \varepsilon \kappa^2\psi\...
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2answers
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Flow and flow rate… Halp!

I'm so confused... I think I got the meaning of flux, it's a scalar that indicates the "quantity of a vector field (of field lines)" that crosses a surface of a given area. So no time relation ...

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