# 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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### Choosing a timestepping condition for a thermal simulation using output of a SPH simulation

I am a math noob so please maybe this question might be completely stupid. I am trying to take in data from a SPH (smooth particle hydrodynamics) fluid stimulation and utilize it for further work. The ...
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### Fluid in a rotating cylinder

I am so stuck over a partial differential equation. I have the following problem A liquid in a spinning cylinder has $u_r=u_h=0$ and $u_\phi = u$, a time $t=0$ the cylinder stops and this his the ...
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### Understanding the Definition of the Koopman Operator

Consider a continuous time dynamical system $$\dot x(t) = F(x(t)),$$ where $x(t)$ is a coordinate vector of state and the right side of the equation $F$ is a non-linear smooth function. Let the state ...
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### Show that the drainage is defined by this ode

A water tank with a rectangular cross-section and one trapezoidal side as shown in the figure contains a volume of water $𝑉(ℎ)$ [$\text{m}^3$] when the depth of the water in the tank is $ℎ ~\text{m}$ ...
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### Where i can to find material about Diffeomorphism Groups as Metric Spaces

My professor give me one document where it speak about Diffeomorphism Groups as Metric Spaces but it does no has calculations and i need to explain something about it.(The reference book is The ...
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### How can I interpret the 2D advection equation?

I want to model the advection of debris rocks with a thickness h_d on top of a glacier through ice flow with velocity components u and v. Can anybody explain the difference between these 2 equations ...
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### How to check if a function is a stream function?

I am completely stuck on the following question: Given the function $\psi=\frac{1}{2}({x^2-y^2})$, check whether it is a stream function for some $F=\langle y, x\rangle$. Also show that the above $F$ ...
1 vote
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### Continuity equation and the square of density, velocity product

I have two questions related to the continuity equation. (1) In fluid mechanics, we have the continuity equation $$\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho v) = 0$$ I am interested in ...
1 vote
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### What is the derivation for the fluid dynamics continuity equation

In my Fluid Dynamics module I have begun learning about mass flux. In my most recent lecture I have been presented with the following text and equation: For a general fluid in some volume $V$, ...
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### Modeling of blood flow using PDEs?

I'm in an intro to PDE class, and so far we've only covered the Diffusion and wave equations. I'm working on a final project, and I'm very interested in how PDEs are used to model blood flow. I've ...
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### Sobolev inner product equality

I've been reading the book "Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models" by Boyer, and on the proof of existence for the stokes problem (...
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### Invariant form of material derivative

The material derivative in vector form is, $$\frac{DQ}{Dt}=\frac{\partial Q}{\partial t}+(V\cdot\nabla) Q$$ Where $Q$ is the fluid property. I didn't understand the following thing from here The form ...
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### Finding the center manifold for a 2D dynamical system

I solved many cases for the following dynamical system $\dot{x} = x (1-x-ay)$ and $\dot{y} = c y (1- b x -y)$. However, I reached the case where $c>0$ and $a>1$, $b=1$ and I ended up with the ...
1 vote
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### How to get equations of fluid flow velocity with vorticies

I am looking for the equation of the velocity field $\textbf{u} = u(x,y)\hat{i} + v(x,y)\hat{j}$ of a two-dimensional steady flow, which streamlines look similarly to this image: example image of the ...
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### Navier Stokes equation incompressible to compressible

This is a basic question. I am reading through my university printed notes about Navier Stokes equation. I am learning it myself so there is alot of confusion please help me clear it. Check whether ...
1 vote
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### When does global solution for Burger's equation exists?

Consider Burger's equation $$\begin{cases} u_t+uu_x=0 \\ u(x,0) = u_0(x) \end{cases}$$ Can someone explain to me what is meant by "global solution" to this Burger equation? How does that ...
1 vote
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### Examples of Pitchfork Bifurcation in nature?

I was just wondering if anyone had some nice examples of pitchfork bifurcation in nature! For Hopf Bifurcations, for me the classical example is cylinder flow. What is the same for a pitchfork ...
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### Does this equation arising in parametric oscillation have a name?

I'd like to investigate the system of equations of the form $\left[\frac{\partial}{\partial t} + u(t) \cdot \nabla_x \right]^2 n(x,t) = a\ n(x,t) + b\ m(x,t)$, where $u(t)$ is given and does not ...
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### Geometry of orbits in regular grazing point.

Consider a piecewise smooth impact dynamic system of a field $F \in C ^ r$, with discontinuity boundary $\Sigma = H ^ {-1} (0)$, where $0$ regular value of H. Let $x ^ *$ be a regular grazing ...
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### How to differentiate a multivariable function $S = S(\xi, \delta, \epsilon)$ [closed]

Consider the equation $-cS' + SS' + \delta S''' - \epsilon S'' = 0$. The function $S = S(\xi, \delta, \epsilon)$, where $\xi = x - ct$ satisfies this ODE. In this case, ' denotes differentiation with ...