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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Is $\nabla_j\nabla^iu^j=(\operatorname{grad}\operatorname{div}u)^i$?

When dealing with the Navier-Stokes equations, one typically models fluids in which the viscous stress depends linearly on the symmetric velocity gradient, $$\tau^{ij}=2\mu(\nabla u)^{(ij)}=\mu\left(\...
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Why isn't the minus sign cancelled in this change of variables?

This integral comes from fluid dynamics, specifically it's the fluid velocity when said fluid is subject to small perturbations. Regardless, my question is mathematical, and probably trivial, but I'm ...
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Dot product with nabla

It has come up in my studies that nabla dotted with a vector field is the divergence. However, when studying vorticity I have recently seen w . nabla, where the nabla second. Is this still divergence ...
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Compressible Navier-Stokes vs Incompressible, which is 'Easier' or usually has shorter computation times for finding numerical solutions?

This is a broad question which might not have a clear answer. I am aware that the incompressible NS equations can be used to approximate the compressible when the Mach number is low, and that this ...
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How do I show a function is the separable solution to a partial differential equation (The Laplacian)?

In the context of fluid dynamics I am given a partial differential equation (in polar coordinates) $$\frac{1}{r} \frac{\partial}{\partial r}\left( r \frac{\partial \phi}{\partial r}\right) + \frac{1}{...
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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 $F$ ...
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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 ...
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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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Derivation of the balance of kinetic energy from the momentum balance using the continuity equation

Derive the balance of kinetic energy from the momentum balance using the continuity equation. The momentum equation in 3D reads as follows: $$ \frac{\partial\rho \mathbf{v}}{\partial t} + \nabla\cdot(...
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How to sketch and determine direction of streamline functions?

In my fluid dynamics course, I am required to know how to sketch streamlines and showing their direction like two examples below. The only method I know is by plotting through using a table with ...
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Any follow ups to the Friedlander, Vishik preprint on Lax pair for Euler equations?

There is a 1990 IMA preprint, #691 by Friedlander and Vishik, Lax pair formulation for the Euler Equation, that I became aware of recently. This seems interesting but a (short) search has not turned ...
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A Clarification on Computing the Riemann Invariant for the Shallow Water Equation

The shallow water wave equation is given by: $$ \begin{bmatrix} h \\ u \end{bmatrix} _t + \begin{bmatrix} u & h \\ g & u \end{bmatrix} \begin{bmatrix} h \\ u \end{bmatrix} _x $$ The ...
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Difference in Flux for Shallow Water

I am looking for some help understanding if there is a difference in these two forms of the shallow water equations: $\partial_t(\eta u)+\partial_x((\eta u)^2+\frac{1}{2}g\eta^2)=0$ (Momentum) $\...
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Why can $v_x,v_y$ in ${\bf{v}}=(z,z,z)$ contain the $z$ variable?

I am rather confused why the vector field: $${\bf{v}}=(z,z,z)$$ can contain the $z$ variable in the $x$ and $y$ components any help with explaining this would be great.
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Shock Capturing Methods for Shallow Water Equations

I am looking for some help finding a numerical solution to the shallow water equations: $\partial_tu+\partial_x(u^2/2+g\eta)=0$ $\partial_t \eta+\partial_x(u\eta)=0$. where $u$ is the depth averaged ...
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How do you integrate this tensor?

The original math can be found here Original Derivation. My question is how does one derive expression (9)? In other words the equation $$v({\textbf{r}})={f \over{8\pi \eta r}}[\textbf{e}+(\hat{r} \...
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Why Fourier transform to solve this PDE

I'm reading a paper and has the linearized NS equation and follows it by getting the solution through a Fourier transform. What is the thought behind this? Meaning, why use a Fourier transform? $ \rho\...
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3 votes
1 answer
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Couette flow with infinite depth

Consider a fluid below $x$ axis or $xy$ plane. Its top layer starts to move with velocity $v$ at and after time $t_0$. If the flow is not fully developed and evolving from standing water, I receive $...
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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 ...
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Determinant of the strain tensor?

In fluid mechanics, we often decompose the tensor $\nabla\boldsymbol u$ into its symmetric and antisymmetric parts, called the strain rate tensor and the vorticity tensor respectively, $$\nabla\...
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Calculating the moment of water in a gutter - fluid dynamics

A gutter is in the form of half a cylinder and is full of water. $\textbf{a. }$Prove, by integrating surface forces, that the total force on the gutter (per unit length) is equal to the weight of ...
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Oseen's tensor derivation of Stokes' drag

I am going through some lecture notes and I have been absolutely stumped by the derivation of Stokes' drag (it is very short). The sphere is of radius $a$ and is instantaneously located at $\bf{x} = 0$...
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flow rate in a circular pipe

If the velocity distribution of a fluid flowing through a pipe is known, the flow rate Q can be calculated by $Q=\int vdA$, where v is the velocity and A is the area pipeline cross section. Consider a ...
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Solving the Euler equation $\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+\frac{\partial p}{\partial x}=0$

In solving the one dimensional Euler equation in fluid dynamics, we have two functions. What should I do If I want to solve it without considering that one of them is constant? $$\frac{\partial u}{\...
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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 ...
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Writing equations in the form of a conservation law

I'm presented with the following internal energy equation: $$ \frac{\partial (\rho e)}{\partial t} + \nabla \cdot (\rho e \vec{v}) = -p\nabla \cdot \vec{v} + \nabla \cdot (K\nabla T) + \varepsilon_V + ...
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2 votes
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Is an analytical solution required to solve the Clay institute 'Navier Stokes' problem?

I'm currently a PhD student in a field which heavily uses the Navier-Stokes equations. I have had for some time a question regarding the Clay Mathematics Institute 'Existence and Smoothness of the ...
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Decomposition of propagation function

In the book of Jean Prüss (Evolutionary Integral Equations and Applications, 2012, pg. 111) it says in the Proposition 4.10... Let $dc$ be completely positive, let $w(t;\tau)$ denote the associated ...
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Power Series of a compressible flow function

In compressible flow there is a formula for an angle: $$ \tan \left(\theta \right)=\frac{2\cot \left(\beta \right)\left(M_1\sin ^2\left(\beta \right)-1\right)}{M_1\left(k+\cos \left(2\beta \right)\...
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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 ...
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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 ...
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Why is the streamfunction constant at the boundary of a pipe?

Consider a long cylindrical pipe rotating about its long axis with an imcompressible fluid inside. Why is the streamfunction constant at the boundary inside the pipe?
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Prove that $n \cdot \frac{\partial u}{\partial n} = 0$ for divergence-free functions that vanishes on boundary

Prove that $n \cdot \frac{\partial u}{\partial n} = 0$ for divergence-free functions that vanishes on boundary. I just read that if $u$ is a divergence-free (vectorial) function in $\Omega$ with $u = ...
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2 votes
1 answer
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Derivation involving taking moment of Navier Stokes equation

The article under concern is: Germano, M. (1992). Turbulence: The filtering approach. Journal of Fluid Mechanics, 238, 325-336. doi:10.1017/S0022112092001733 On page 328, Germano considers the Navier-...
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2 votes
3 answers
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$\pi / 4 \not = \sum_{n, \text { odd }}^{\infty} \frac{1}{n} \sin \left(n \pi \frac{z}{h}\right)$ but the author argues otherwise

In the book of Theoretical microfluidics by Bruus (2008) at page 49 the equation 3.48, it is argued that $$-\frac{\Delta p}{\eta L}=-\frac{\Delta p}{\eta L} \frac{4}{\pi} \sum_{n, \text { odd }}^{\...
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What does $| A |$ denote?

Let us suppose we have a diagonalisable $5 \times 5$ matrix $A$. What is the meaning of $|A|$? I am a student of fluid dynamics. I am referring to the Blazek's Computational Fluid Dynamics: ...
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2 votes
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Conserved $L^p$ norm of vorticity of 2D Navier-Stokes equation in a regular, bounded domain?

I am now considering the vorticity formulation of the 2D incompressible Navier-Stokes equation in some regular, bounded domain $D$: \begin{align*} D_t\omega &= \nu\Delta \omega,\\ \omega(x,0) &...
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