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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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Complex velocity of line vortex, simplification

I am considering the potential flow where I have a uniform flow past wing and two line vortices, one at the origin and one at a position $(x_1,y_1)$ relative to a wing of chord $D$. I'm using the ...
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If $f_n:\Omega\to\Omega$ are homeomorphisms of a planar domain $\Omega$ such that $f_n\to f$, $f_n^{-1}\to g$ in $L^1$, is $f=g^{-1}$?

In Marchioro and Pulvirenti's book Mathematical Theory of Incompressible Nonviscous Fluids, the proof of global well-posedness of the 2D Euler equation in a bounded domain $\Omega\subset\mathbb R^2$ ...
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1answer
26 views

Vorticity constant along a streamline

In the case of a 2D incompressible flow $\omega = - \nabla^2 \psi$ where $\psi$ is the stream function. From the vorticity equation of motion I am able to find the following: $$ \frac{\mathrm{d}\...
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19 views

Imitation Dynamics: Rest Points and Jacobian

Questions: Show that for a given matrix $A$ the imitation dynamics in the following equations: $$\dot{x}_i=x_i\sum_j x_j\psi((A\mathbf{x})_i-(A\mathbf{x})_j) $$ $$\dot{x}=x_i((A\mathbf{x})_i-\mathbf{x}...
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Angular Momentum Zero Produce of Irrotational Fluid Proof

A cute problem that I spent a bit time on. Please help. Consider flow domain smooth manifold in $n$ dimension $\Omega$ with boundary and some boundary data. Constant density is assumed. Assume ...
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11 views

When Linearised Model of Finite Depth Waves is Not a Sufficient Model

I have been investigating the linearised model of water wave motion in a finite depth fluid. In my particular case the flow is Inviscid, Irrotational and Incompressible and surface tension effects are ...
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20 views

Absence of Pressure gradient in Boundary Layer?

Could someone with knowledge in fluid mechanics please help me in understanding the argument for why dp/dy, the pressure gradient in the boundary layer, equals zero? I have watched his video for a few ...
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19 views

Struggling with Proof of Prandtl's Boundary Layer Equations

Would someone with knowledge in fluid mechanics please help me in understanding this man's argument for why dp/dy, the pressure gradient in the boundary layer, must equal zero? I would greatly ...
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22 views

About the derivative of the Jacobian in fluid dynamics

I was studying a book on the mathematics of fluid dynamics in which there was a lemma on how to find the derivative of the Jacobian. The explanation is as follows (Sorry if it's too long): There is a ...
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25 views

What's the mathematics behind the volume of fluid method (VOF)?

I've come across the volume-of-fluid method (when reading Sethians "Level Set Methods and Fast Marching Methods" where the evolution of fluids is described with fractions in cells. However, the ...
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1answer
69 views

Solving a Navier-Stokes equation

I was reading a paper in solving a Navier-Stokes equation applied for 1D fluid flow in hydraulic fracturing. The below part I couldn't understand. Anyone please explain for me how do they get (4) from ...
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23 views

Show that integral of a jacobian $J(A^2/2,B) = 0$ assuming periodicity

I'm trying to prove the enstrophy conservation of a inviscid barotropic turbulence. This boils down showing that the following integral of a Jacobian vanishes $$\int_0^\pi \int_0^{2\pi} \bigg({\...
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1answer
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Conservation law $A_t + (A^{3/2})_x = 0$ for flood water wave

The flood wave in a river follows the conservation law $$ A_t + (A^{3/2})_x = 0 $$ where $A(x,t)$ is the cross sectional area of the water at the location $x$ and time $t$. A sudden heavy rain ...
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25 views

Deriving a mathematical model for a fluid system.Help.

ive been trying to derive a mathematical model for the system shown in the image The case is a pump, pumps water into a Column. The drain of the column is blocked. Thus there is only flow in. i know ...
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1answer
70 views

Complex potential describing an inviscid flow

I'm working through a homework sheet for a Fluid Mechanics module. The question is given: Consider the flow described by the complex potential $$w=4z+\frac{8}{z}.$$ Determine $\psi$, $\phi$,...
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1answer
50 views

Use Milne Thomson circle theorem to show complex potential for this flow

I was wondering if anyone could help me with the following problem, as I'm unsure on how to begin. Any suggestions would be appreciated, thanks for reading this. I just don't understand how to apply ...
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1answer
34 views

Material Derivative in Cylindrical Coordinates

I'm testing myself on my knowledge from this book by taking the material derivative of velocity in cylindrical coordinates: $$\frac{D\mathbf u}{Dt}=\mathbf u\cdot \nabla \mathbf u$$ Which, in tensor ...
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24 views

Relation between vector gradient (double-dotted) and divergence

I've been working on a fluid mechanics problem and have come across the following conundrum. I am asked to show two constants must satisfy $a>0$ and $(b+\frac{2}{3}a)>0$ given that $$2a\mathbf{...
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1answer
42 views

Linear Hydrodynamic Stability - Periodicity of Perturbations

I am currently taking a course in Hydrodynamic Stability, based around Drazin's 'Introduction to Hydrodynamic Stability'. Up until this point, we have considered linear stability, where we assume a ...
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1answer
115 views

Including rotational motion into a reaction-diffusion model

The reference below describes a system of hypothetical sub-particle units or etherons, diffusing from a region of high to low concentration using Fick’s law of diffusion. How would one introduce ...
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1answer
67 views

Coordinate Transformation and Differential Operators

In this paper the following relation can be found: \begin{align} \nabla\phi=\mathbf{g}^j\frac{\partial \phi}{\partial \xi^j}= J\frac{\partial}{\partial \xi^j}\left(\frac{\mathbf g^j}{J}\phi\right) \...
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Reference request on an equation while studying Stokes steam-functions

I came across this equation while studying Stokes steam-functions and I'm not sure how its derived. The equation is $$(r^2-\frac{a^3}{r})\sin^2(\theta)=b^2$$ I believe it is the equation of a ...
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38 views

Is there any easy numerical integration method for advection in a 2-d non-linear vector field?

I am trying to simulate a fluid running through a 2-d rough landscape. The landscape is represented by a regular grid, and heights are just given as values to each cell (0,1,2 m). For the sake of ...
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54 views

Shallow water equation entropy concept

I am a beginner in shallow water equation. I am interested in the equation $$h_t+(hu)_x=0$$ $$(hu)_t+(hu^2+\frac{1}{2}gh^2)_x=0$$ I have the following doubts 1)Weak solutions are not unique in ...
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73 views

Uniqueness of the potential flow past a cylinder

I have a question regarding the uniqueness of the potential flow past a cylinder. Consider a two dimensional uniform potential flow in $x_1$-direction past the cylinder $B_R = \{ x = (x_1, x_2) \in \...
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1answer
33 views

Clarification on the Lid-Driven Cavity Problem in CFD

I need some clarifications on The Lid-Driven Cavity Problem. What does it actually mean? I know cavities are bubbles created when a fluid moves through liquid in low pressure zones, but what does the ...
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2answers
106 views

Laplacian of $1/r$ in a tensor

As we know the $$\nabla^2(1/r) =- 4 \pi \delta^3(r).$$ However, I recently was readling an hydrodynamic book (An introduction to dynamics of colloids By J.K.G Dhont). The Oseen tensor is defined as: ...
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71 views

Derivation of Burgers' Equation

I'm aware that it is possible to reduce Navier-Stokes to Burgers by neglecting pressure, and that one can derive the inviscid form by considering an ideal gas and concluding that the convective ...
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24 views

Thin film flow boundary conditions

How to derive the conditions for shear stress balance and normal stress balance at interface (at z = h) for thin film flow over a rotating disk.?
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Derivation of 2D Korteweg-de-Vries equation

Coming from engineering rather than mathematics, I am recently dealing with non-linear partial differential equations e.g. like the well known Korteweg-de-Vries equation: $$u_{t} + uu_x + u_{xxx} = 0$$...
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1answer
24 views

Pulse vertices of a quad from a center point?

I'm working in Java, using LWJGL3. I'm creating a 3D world, and would like a quad (that represents water) to have its vertices "fluctuate" away from the center of the terrain, in all directions. I was ...
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1answer
41 views

Taking moments of a fluid equation

I understand that this question is maybe better placed in the physics stack exchange but thought maybe I would find some help here as well. Given the following term that is taken from the Vlasov ...
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22 views

Change in unit normal of moving surface

Suppose I have a smooth open region $W\subseteq \mathbb{R}^3$ that is being continuously acted upon by a family of diffeomorphisms $\{\phi_t:\mathbb{R}^3\to\mathbb{R}^3\ : \ t\in\mathbb{R}\}$, where $\...
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0answers
19 views

deformation of a fluid

Given a 2D flow $u=[Sx, Sy, 0]^T$, show that a fluid element which was initially circular will be deform into an ellipse with major axes aligned $\frac{\pi}{4}$ with the $x-$ axis. I have started by ...
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1answer
51 views

Help setting up integral for circulation

I just need a bit of help with a problem. I'm being asked to evaluate the circulation of a velocity field. I've just finished calculating the curl of the vector field in question and I know from my ...
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2answers
35 views

Finding the Closest Point to the Centre of a Rotating Vector Field

I am working on the analysis of a data set that resulted from a simulation of a Tornado. I need to be able to transform the vectors for the velocities at each measured point in the space from ...
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1answer
78 views

Fluid Mechanics Questions

A steady two-dimensional flow (pure straining) is given by $u = k x$, $v = -k y$, for $k$ constant. Find the equation for a general streamline of the flow. At $t = 0$,the fluid on the curve $x^2 + y^...
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1answer
42 views

Vorticity for the Navier-Stokes equations

The definition that I know of is the vorticity $\omega$ is the curl of the velocity $u$. Now I'm reading a note saying $\omega$ is defined to be the $d\times d$ antisymmetric matrix: $$\omega = \frac{...
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1answer
52 views

Showing that stream function models a situation

I'm working on my assignment for fluid mechanics and I'm a bit lost as to how to respond to a question. I believe I can get part a tomorrow when I sit down to work it out, but I'm not sure exactly how ...
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26 views

Inviscid Taylor Couette Flow

I'm trying to solve a simple problem where I have an INVISCID fluid between two cylinders and they are rotating with some angular velocity $\Omega_1$ and $\Omega_2.$ The cylinders have radii $R_1$ and ...
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1answer
55 views

Solving Differential Equation With Bessel Functions

trying to solve $$\frac{d}{dr}\bigg(\frac{1}{r} \frac{d}{dr}(r\bar u_r)\bigg) - k^2 \bar u_r = \frac{k^2}{\omega^2}\phi(r) \bar u_r $$ subject to $$R_1 \leq r \leq R_2,$$ $$\bar u_r(R_1) = \...
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Modified equation for KdV using 2.Order Discretization scheme

The modified equation of linear PDEs can be found in a systematic manner (https://www.sciencedirect.com/science/article/pii/0021999174900114). However, it does not seem to be that easy for nonlinear ...
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1answer
31 views

Question on free surface elevation of water wave

A semi-infinite channel of finite depth is occupied by an ideal fluid layer initially at rest . the vertical finite end of the channel is fixed and only a part of the horizontal bottom , with finite ...
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2answers
74 views

Fluid Dynamics, free surface boundary condition derivation.

I'm stuck on part of the derivation for a boundary condition for the free surface of a fluid. The fluid concerns a rectangular trough of water whose surface is at a height $$ z=h(x,t) $$ (Problem in ...
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2answers
62 views

Does this velocity field have a potential?

Let $\Psi=x-\frac{x^3y}{2}$ be the stream function. Then, by definition: $$ v_x=-\frac{\partial \Psi}{\partial y}, \, v_y=-\frac{\partial \Psi}{\partial x} $$ To determine whether this field have a ...
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2answers
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Intuition of definition of divergence

Intution : The divergence of a three-dimensional vector field is the extent to which the vector field flow behaves like a source at a given point. But if my vector field is $F=\langle P,Q,R\rangle$ ...
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37 views

Kinetic energy of incompressible fluid as quadratic form on tangent space.

I have started working through "Topological Methods in Hydrodynamics" by V.I. Arnold and I am confused by the following located on page 2: "The kinetic energy of an [incompressible] fluid [with ...
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1answer
57 views

Viscous Fluids at a Slope (Navier-Stokes)

An in-compressible viscous fluid flows down a flat slope of angle θ to the horizontal under the force of gravity, with g the acceleration due to gravity. What are the boundary conditions for the fluid ...
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21 views

Quick question involving Computational Fluid Dynamics

So I'm a high school student with some free time on my hands and I've been trying to learn about the Navier-Stokes equation. Anyway, this set of lecture notes from UCLA is a bit hard to understand. ...
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Weak Formulation of Navier-Stokes

I was looking into the Navier-Stokes Weak formulation Let $f\in L^2(\Omega_T)$, $u_0 \in H(\Omega)=\lbrace u\in L^2(\Omega):\text{div }u=0\text{ in }\Omega;u\cdot n|_{\partial \Omega}=0\rbrace$. ...