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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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Existence of Smooth Periodic Solutions for Navier-Stokes Equations in Three Dimensions [closed]

This paper establishes the existence of smooth periodic solutions for the Navier-Stokes equations in three-dimensional space. The Navier-Stokes equations describe the motion of incompressible viscous ...
Biruk A.'s user avatar
2 votes
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How do I find the absolute maximum and minimum values of the Lamb-Oseen Vortex?

I am researching alternative solutions to Stokes equations and I came across a problem with the Lamb-Oseen vortex I cannot solve that I hope will allow easier derivations of vortex functions with ...
Tayler Montgomery's user avatar
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Exercise 7.6 of Robinson, Rodrigo, Sadowski: Smoothness of Navier-Stokes on Bounded Domains

My question is about Exercise 7.6 of the excellent book 'The Three-Dimensional Navier-Stokes Equations' by Robinson, Rodrigo and Sadowski. More generally, it is about higher regularity in space of ...
RiaDoog's user avatar
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Space of functions, Banach spaces, reference books to find basic properties of Bochner integral, Laplace and Fourier transforms.

I'm looking for references where I can find definitions and basic properties of Bochner Integral in Banach Spaces and its basic properties, such as: Every continuous function is integrable, ...
Silvinha's user avatar
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2 answers
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2D Laplace equation analytical solution

I am trying to solve a simple Poiseulle Flow in 2D in Cartesian coordinates numerically and analytically. For the analytic part, I am stuck at the following: Suppose we have a 2D Laplace equation $$ \...
kirkos73's user avatar
1 vote
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Can this property that I formulated be used to find solutions to Navier Stokes in cylindrical coordinates?

Let's say I want to derive a vortex flow function $$\psi (r,t)$$ using a scalar field bell surface of the form below, where $W(t)$ controls the width (but is not actually the width) as a function of ...
Tayler Montgomery's user avatar
0 votes
1 answer
28 views

Derivation of Continuity Equation for an Incompressible flow

Good day guys, I was playing around with the following form of the continuity equation: $$ \frac{\partial \rho}{\partial t} - \nabla \cdot (\rho \vec{v}) = 0 $$ For an incompressible fluid: $\frac{D\...
RMS's user avatar
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Complete orthonormal basis of divergence free vector fields

I'm working on a problem in fluid dynamics. I need to find a complete basis of orthonormal 3D vector fields. My "inner product" between vectors $\mathbf{v}_1$, $\mathbf{v}_2$ is a dot ...
davenpi's user avatar
2 votes
1 answer
26 views

Evans's scalar conservation law order of operation? $\left( u_t + F(u)_x \right)$

This is a small question (taken from Evans's PDE book, page 8). I generally avoid writing parenthesis for a function operating on some character to its right. In absence of parenthesis, I read my ...
Nate's user avatar
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Finding the trajectories of fluid particles

I have completed the question up to when it asks me to show that up to leading order, the trajectories of fluid particles for these waves are ellipses. I am slightly confused what I am looking for; ...
idk31909310's user avatar
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1 answer
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Solving for a linear equation from a nondimensionalisation

Whilst doing my homework I came across the following question and got particularly stumped at question c). I do not know how I could possibly derive a precise linear relationship. Any help? When a ...
Geralt's user avatar
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Expanding the barotropic nondivergent PV equation: Which vector calculus property/identity to apply for dot product and del operator? [closed]

I am trying to expand the barotropic nondivergent potential vorticity (PV) equation [link] $$\frac{\partial \zeta}{\partial t} = -\vec{V} \cdot \nabla(\zeta + f)$$ where $\zeta$ is the relative ...
Brian Añano's user avatar
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Free-Surface kinematic BC - Azimuthal suspended flow - cylindrical coordinates - How to define the free surface and derive its impermeability?

I am working on a linear stability analysis of an azimuthal free surface flow. However, I am stuck in the derivation of the kinematic BC to apply for the impermeability of the azimuthal free surface (...
Virgil 11's user avatar
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1 answer
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Green's Formula for vector fields in the Navier Stokes Weak Formulation

I am currently studying the weak formulation of the Navier-Stokes equations and came across the following equation: \begin{equation} \int_{\Omega} \mathbf{v} \cdot \Delta \mathbf{u} \, dx = -\int_{\...
Luigi's user avatar
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Calculating the numerical flux in local Riemann problem

I am reading the following material: https://www.i2m.univ-amu.fr/perso/kai.schneider/PDF-FILES/gdsmd_apnum_2015final.pdf I am trying to figure out how to obtain the equation 7 from equations 5 and 6. ...
Marcos's user avatar
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In what sense is $\int (u \cdot \nabla) u \cdot u dx$ an energy flux?

Due to the nature of this question I have cross-listed it on physicsSE. Let $u$ be either a solution to either the Euler equations or Navier-Stokes equations over a domain $\Omega$. In fluid dynamics ...
CBBAM's user avatar
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How to recognise a Kármán-Pohlhausen problem

I'm working on a problem, A fluid moves in a steady two-dimensional flow in the region defined by $x\geq0$, $y\geq0$. The boundary with equation $y=0$ is occupied by a stationary flat plate. Given ...
mjc's user avatar
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2 votes
1 answer
180 views

How to actually find the stream function for a simple Laplace problem?

Let's assume we're solving a $2D$ Laplace problem, in a domain (if necessary simply connected) $\Omega \subset \mathbb{R}^2$, with a Dirichlet boundary $\Gamma_D$ and a Neumann boundary $\Gamma_N$: $$\...
Collapse's user avatar
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Interpretation of the timescale non-dimensionalizing $\partial_t c(x,t) = - \frac{\dot{s}x}{s}\partial_x c + D \partial_x^2c$

Background: A thin filament of fluid mixes into a flow field in accord with the so-called "compression-diffusion equation" (CDE), which is essentially an advection-diffusion equation with a ...
kevinkayaks's user avatar
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Divergence of a Point

I am currently reading a book titled Notes on Computational Fluid Dynamics: General Principles, which is provided by OpenFOAM at https://doc.cfd.direct/notes/cfd-general-principles/contents. In ...
Anil Celik Maral's user avatar
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0 answers
5 views

Calculation of density distribution function for infinite (in x and y) static isothermal slab supported by gas pressure and it's own gravity

Let us cosider an infinite (in x and y) static isothermal slab, symmetric about z = 0, supported by gas pressure and under it's own gravity. No other forces are acting. To calculate the density ...
Mon's user avatar
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Navier Stokes Eqns: Boundary Conditions and Pressure Coupling

I have made it a personal long-term goal of mine to numerically solve the 1D transient Navier-Stokes equations for simple incompressible and compressible flow scenarios. I want to do it for a few ...
UserHandel's user avatar
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0 answers
22 views

Discretising and solving a partial differential equation temporally

Essentially, I've come across a PDE for reaction-diffusion which describes the concentration of a substance at some given point in space: $$\frac{\partial c_a}{\partial t} = \mu_a\nabla^2 c_a + R_a(...
shahendra's user avatar
0 votes
1 answer
73 views

History of the Leray Projection [closed]

I am hoping to cite the first use of the Leray projection, $\mathbb{P}$, onto the divergence-free vector-field. I tried scanning Leray's famous 1934 paper, 'Sur le mouvement...,' but couldn't find it, ...
gbnhgbnhg's user avatar
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1 vote
1 answer
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Relating Navier Stokes diffusion term with summations of the rate of strain tensor for incompressible flows

So the diffusion term in NS in Einstein notation: $$\upsilon \frac{\partial^2 u_i}{\partial x_j \partial x_j}$$ I saw in some textbooks that the term can also be written as a summation with the rate ...
Carlo Filippo Capano calippo's user avatar
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0 answers
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How to determine the exact value of the constant in Poincaré's inequality?

I am reading a paper about Navier-Stokes Equations, in which the author gives the inequality bellow $$\|u\|_{L^2(O_R)}\le CR^{\frac{1}{2}}\|\nabla u\|_{L^2(O_R)}$$ where $u$ is a $3D$ vector field, $...
Yamato's user avatar
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1 answer
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Is $\frac{d(A^{-2})}{dy} = \frac{1}{A^3} \frac{dA}{dy}$?

Equation 9-4 in "Open-channel hydraulics" by Chow (1959) (a true classic, THE most important textbook for open-channel hydraulics) states (excluding an ever-present correction coefficient): $...
OlaHH's user avatar
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1 vote
1 answer
61 views

Fluid Dynamics - Streamline Separation and Flow Velocity

Consider the following quote: "... The velocity distribution outside the boundary layer can be determined from the spacing of the streamlines. Since there can be no flow across streamlines, we ...
Kevin's user avatar
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0 votes
1 answer
43 views

Flow Rate based on Pipe Combination without considering Hydraulic Effects

I would really appreciate help on the following: Let us assume that we have n pumps pumping water through n pipes. The pumps are running at a constant power. It doesn't change. The flow rates in the ...
user22644's user avatar
2 votes
1 answer
68 views

Fundamental stress and pressure tensors of the Stokes system in $\mathbb{R}^3$.

Let $\mathbf{\mathcal{G}}$ denote the Oseen-Burgers tensor and $\mathbf{\Pi}$ denote the fundamental pressure vector in $\mathbb{R}^3$, i.e. on components we have $$\mathbf{\mathcal{G}}_{jk}(\mathbf{x}...
math_is_hard's user avatar
0 votes
2 answers
63 views

Analytical Solution of Non-Linear Coupled ODE

I have the following coupled PDE, which describes the trajectory of a non-inertial particle in Taylor-Green Vortex Flow: $$ \frac{\partial}{\partial t} \begin{bmatrix} x_p \\ y_p \end{bmatrix} = e^{-2 ...
Jacob Ivanov's user avatar
0 votes
1 answer
48 views

Approximate solution to ODE potentially using perturbation theory

On the wikipedia article for stokes drift (https://en.wikipedia.org/wiki/Stokes_drift) they show that for: $\dot{\zeta} = u\sin(k\zeta - wt)$ has approximate solution (by perturbation theory): $\zeta(\...
Jamminermit's user avatar
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1 vote
0 answers
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Prove the following (vector) operator is self-adjoint

Define the operator $$F(\boldsymbol{\xi})=\nabla(\boldsymbol{\xi}\cdot\nabla p)$$ where $p$ is a given smooth function. Show that $F$ is self-adjoint, i.e., there exists a vector field $P$ such that $$...
Zeta's user avatar
  • 155
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0 answers
30 views

Indeterminate Form for Partial Derivative of Flow Variables

The following is a paraphrased version of the derivation within John D. Anderson's Fundamentals of Aerodynamics's section on the Method of Characteristics: The exact governing equation for two-...
Jacob Ivanov's user avatar
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0 answers
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Integrating by Parts $\int\limits^{2}_{0}pydy$ - A calculation related to flowrate.

Happy new year everyone! I have a question that I struggle related to integration by parts. Suppose I'm going to solve the following using integration by parts. $$\int\limits^{2}_{0}pydy$$ Where $p$ ...
Charith's user avatar
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0 answers
76 views

vector field PDE sandwich theorem

Consider the vector valued system of PDE's for $\textbf{f} = [f_1,f_2,f_3]$ and $\textbf{g} = [g_1,g_2,g_3]$ \begin{equation} \dfrac{\partial}{\partial t}\textbf{f}- F\left(\textbf{f}, \nabla f_i, \...
MrPie 's user avatar
  • 235
2 votes
1 answer
117 views

Integrating Factor for Vorticity Evolution

The Vorticity Evolution in 2D Cartesian Coordinates, assuming incompressibility, is as follows: $$ \frac{\partial \omega}{\partial t} = \nu \left( \frac{\partial^2 \omega}{\partial x^2} + \frac{\...
Jacob Ivanov's user avatar
1 vote
0 answers
21 views

Realizing a modified transport equation

Stated somewhat informally, the continuity equation or transport equation $\partial_t\rho_t = -\nabla\cdot(\rho_t v_t)$ describes the evolution of a density where each particle flows along a vector ...
Juno Kim's user avatar
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0 answers
62 views

Solving a second Order ODE with a complex coeffcient

I have been asked to derive a solution for F(y) in the form F(y) = A cos ky + B sin ky + C , where k = (1 + I)/δ , δ = sqrt(2ν/ω) and A,B and C are constants to be found. The ODE I have found is F''+(...
user1176200's user avatar
4 votes
0 answers
86 views

Transport equation and entropy conditon

Consider the transport equation with smooth coefficient $a \in C^1(\mathbb{R}\times \mathbb{R}^+)$ given by \begin{align} u_t+(a(x,t)u)_x=0. \end{align} A weak solusolution $u \in C(\mathbb{R}^+;L^{1}(...
Rosy's user avatar
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0 answers
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Energy Density Conservation for Compressible Euler Equations

I have the following system of equations (for an ideal gas, from Exercise 2.1 from Fundamentals of Computational Fluid Dynamics): $$ \rho_t + \gamma p u_x + up_x = 0 $$ $$ p = (\gamma - 1) \left( e - \...
Jacob Ivanov's user avatar
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0 answers
26 views

Laplacian version of surface tension gradient

I have read that the surface tension gradient operator is $ \nabla_{s} = (I - nn). \nabla$ , where n is the unit normal to the surface given by : $$ n = (-\frac{\partial h}{\partial r},1) \frac{1}{\...
JayJonesy's user avatar
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0 answers
30 views

What is $u$? How to pass the limit?

In the article "Stability of Rarefaction Waves to the 1D Compressible Navier–Stokes Equations with Density-Dependent Viscosity" whose doi is 10.1080/03605302.2010.516785 $(\rho_\varepsilon,...
xyz's user avatar
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1 vote
0 answers
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Fluid mechanic derive [4.5-4.9] and [4.8-.11]

Hye, this is my first time to ask in this group. I am not familiar with fluid mechanics and currently self-learning for postgraduate study and facing difficulties in deriving both 4.9 and 4.11 as ...
amir ubaidah's user avatar
3 votes
2 answers
94 views

How does the concentration of two dyes change in a fluid ? is there an analytical solution?

I was thinking of how we can model two or more dye concentrations in the same fluid. is the answer just a system of two PDEs where the concentration becomes a vector to is there an interaction between ...
babzzz's user avatar
  • 31
1 vote
0 answers
25 views

Is the weak form on this book about a level set FSI problem wrong?

I'm trying to repeat a level set FSI problem on the book :Level Set Methods for Fluid-Structure Interaction, on the page 89, the provided freefem code define a weak form of the discretized equation ...
吴yuer's user avatar
  • 301
0 votes
1 answer
39 views

Navier-Stokes: clarification with quantifiers of the initial condition (4) in CMI official problem definition

I am a bit lost. The Clay Mathematics Institute has written an $n$ dimensional Navier-Stokes problem description here. On the first page, an initial condition $(4)$ is: $$\vert \partial_x^\alpha \, u°(...
someone's user avatar
  • 63
1 vote
1 answer
43 views

Solving for constants of integration continuously resulting in trivial statement 0=0

I am attempting to solve a fairly simply pair of ODEs that represent a flow field for a fluid dynamics problem. I am attempting to find the particle path but I am getting stuck trying to find x and y ...
en_croissant's user avatar
1 vote
0 answers
28 views

The symmetry of the isotropic cartesian tensor of Newtonian fluid

I'm trying to prove the symmetry of the isotropic tensor in the linear relation between the shear stress and strain rate for Newtonian fluid $$T_{ij}=\beta_{ijlm}\frac{\partial u_{l}}{\partial x_{m}}.$...
Mohamed Obeid's user avatar
0 votes
0 answers
41 views

How to get the limit?

In the article "Stability of Rarefaction Waves to the 1D Compressible Navier–Stokes Equations with Density-Dependent Viscosity" (DOI: 10.1080/03605302.2010.516785), in page 619, it get the ...
xyz's user avatar
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