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Questions tagged [navier-stokes]

The Navier-Stokes equations describe fluid flows in continuum mechanics.

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Bernoulli principle (classic form) in irrotational viscous flow (a paradox?)

We derive the Euler equation for inviscid flow. Then, for irrotational flow, we use the Euler equation to get Bernoulli equation (classic form) and show it holds in the irrotational region (still ...
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How do I find the absolute maximum and minimum values of the Lamb-Oseen Vortex?

For an angular velocity function derived by Navier-Stokes, $$ \omega \left(r,t\right)=\frac{\omega _0R_0^2}{R\left(t\right)^2}exp\left(-\frac{r^2}{R\left(t\right)^2}\right)$$ from which the azimuthal ...
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Favre Averaged Navier-Stokes equations

Consider the Navier-Stokes (NS) equation \begin{equation} \frac{\partial (\rho u_i)}{\partial t} + \frac{\partial (\rho u_i u_j)}{\partial x_j} = - \frac{\partial p}{\partial x_i} + \frac{\partial}...
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How to conceptually explain diagonal elements of viscous stress tensor in a fluid?

How can the diagonal components of viscosity tensor be explained conceptually? In other words, how can viscosity affect the normal stress (pressure) on the cube element of the fluid? $$\sigma_{ii} = 2\...
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Recovering the classical Navier-Stokes equation from Landau's relativistic tensor [closed]

I'm trying to get to the classical compressible Navier-Stokes equation from the relativistic tensors in the Landau/Eckart frame. I understand these have their own problems about entropy. In several ...
Gar's user avatar
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Does Navier-Stokes equations get correct result for complex turbulent flow without turbulence model?

Does Navier equations (1822 formulation) get correct result for complex turbulent flow without turbulence model? Is this 1822 formulation?
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What is the relationship between a fluid vortex's expansion rate and its kinematic viscosity?

I've read through research papers with very specific radius formulas pertaining to 2D fluid vortexes, such as confined rotation with variations in pressure and density ($\rho(r,t)$ and $p(r,t)$), ...
Tayler Montgomery's user avatar
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Is this assumption about a vortex's angular velocity reasonable?

I am deriving a velocity flow function $\psi(r,t)$ that could be derived by (1) establishing the relation between two vortex area functions, $a(t)$ and $A(r)$, using the disk method of integration, ...
Tayler Montgomery's user avatar
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What are the differences between Navier and Stokes versions of equation?

What are the main differences between Navier(1822) and Stokes(1845) versions of equation? If I understood correctly, original equations formulated Euler, what did they invent that they deserved to ...
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Force-free condition from Cauchy stress

I have an outer Stokes problem, i.e. I want to solve (outisde of the ball $\{ r \leq l\}$): $$ \begin{cases} \Delta u - \nabla p=0\\ \nabla \cdot u=0\\ B.C. \text{ on } u \text{ in } \{r=l\}\\ u(x) \...
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2D Stokeslet: a reference?

Consider the Stokes equation $$ \begin{cases}\Delta u = \nabla p - F \\ \operatorname{div}(u)=0 \end{cases} \text{ in } \mathbb{R}^2,$$ with $u$ a velocity field, $p$ a pressure and $F$ a point force ...
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Finite differencing for velocity cross terms?

In writing down the Navier-Stokes equation I have encountered two equations as follows: $$ 4\frac{\partial^2 v_x}{\partial x^2} + \frac{\partial^2 v_x}{\partial y^2} + 3\frac{\partial^2 v_y}{\partial ...
Raj Upadhyay's user avatar
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Are there purely 1D instabilities in fluid dynamics?

If I have a purely 1D fluid flow governed by the 1D Navier-Stokes equations (let's assume compressible flow for more generality), are there any instabilities that can happen? It seems like you'd need ...
confusion'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 have cross-listed it on mathSE. Let $u$ be either a solution to either the Euler equations or Navier-Stokes equations over a domain $\Omega$. In fluid ...
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How did G. I. Taylor derive the PDE's in his landmark 1941 paper?

To gain a deeper understanding of the derivation of G. I. Taylor's foundational equation $R= S(γ)t^\frac{2} {5} E^\frac{1}{5} ρ_0^\frac{-1}{5} $, I'd be grateful if you could point me towards the ...
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Can frictional force be un-aligned with flow direction?

There is a well known solution to Navier-Stokes equations for atmospheric boundary layer according to Ekman (see for instance PalArya's book "Introduction to Micrometeorology" or Holton's &...
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Why shear stress vector must be parallel to flow in this example?

In the book by S. Pal Arya "Introduction to Micrometeorology" there is a chapter about Laminar Ekman Layers. I refer to the following example: Variables are: U, V wind velocity in x, y ...
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Where did the idea for the approach to solving this boundary layer problem come from?

I found the derivation of Blasius-Equations here: A free stream velocity hits a flat plate and the goal is to derive boundary layer behavior. Everything in this tutorial is clear - but the most ...
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Derivation of Lighthill Equation

I'm trying to understand the meaning of Lighthill equation. Based on my text book (sound and source of sound, Dowling and Williams 1983), it is derived from combining mass conservation and momentum ...
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Can angular momentum be transported between scales in turbulent flows?

Consider a turbulent rotating flow. You are interested in its average features so you use the Reynolds-averaged Navier-Stokes equation. Now, conservation of angular momentum implies the viscous ...
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Energy of a compressible viscous fluid

I'm trying to derive an expression for the total energy of a compressible viscous fluid. I know that the energy per unit volume of a compressible inviscid fluid is given by the sum of its kinetic ...
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Non-dimensionalization of the Navier-Stokes equations

In 2D simulations using Large Eddy Simulation (LES) methodology, Favre averaging is commonly applied to the variables involved in the Navier-Stokes equations, resulting in: \begin{align}\label{aq} ...
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Stresslet vs hydrodynamic force

I would like to understand all types of forces exerted on a small particle in a Stokes flow. According to this, page 9, "stresslet" and the hydrodynamic force were defined as two different ...
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Large number of infinitesimal particles in an incompressible flow

Consider an expansion channel, where isothermal incompressible flow enters the domain from the smaller channel. Assume I inject infinitesimal particles (with same density as the flow) into the domain ...
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What is exactly particle inertia in a fluid-particle mixture?

In a mixture of a cylindrical particle and the carrier fluid, what exactly particle inertia refer to? (let's neglect gravitation force and Brownian motion). When the size of particles are small (...
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Forces exerted from fluid on a particle in Stokes flow

Assume we have some small cylindrical particles near a specific point in a laminar flow, such that the particle Reynolds number is around unity. Neglecting mechanical interactions between particles, ...
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How does a fluid element have vorticity if the stress tensor is symmetric

Considering incompressible, Newtonian fluid as an example. The stress tensor is dependent on the symmetric part of the strain rate tensor, which is also symmetric. What I don't understand is, with a ...
Ishan Tandon's user avatar
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Diffusive finite-size effects of fluids in periodic boundary conditions

Consider a fluid of given density $\rho [\frac{kg}{cm^{3}}]$ in thermal equilibrium at a given temperature $T [K]$. If at time $t_{0}$ we apply a force $f(\vec{r}, t) = f_{0}\delta(t-t_{0})\delta(\vec{...
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In which cases should one use the Navier-Stokes equations for modelling of incompresible fluid dynamics?

I have been doing some exercises in incompressible fluid dynamics, particularly with Navier-Stokes equations, and realized there was an exercise that involved a tubular pipe, that could use ...
Eduardo Kuri's user avatar
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Rate of change of momentum of a fluid

Consider a volume G that is the volume containing the same number of fluid particles, that evolves with time The total momentum of the collective volume is obviously $$\vec{P} = \iiint_{G} \rho \vec{V}...
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How to mathematically derive the pressure contribution to drag force over a with viscous contribution determined?

I've looked in a lot of books on how it would be possible to derive the drag force over a sphere using PDEs. I've also learned in my fluid dynamics class how to derive the drag force of a rotating ...
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How to mathematically derive the pressure contribution to drag force over a sphere after the viscous stress is determined? [duplicate]

I've looked in a lot of books on how it would be possible to derive the drag force over a sphere using PDEs. I've also learned in my fluid dynamics class how to derive the drag force of a rotating ...
Benri's user avatar
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1 answer
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Finite Differencing with Incompressible Navier-Stokes Equations (Only Advection)

I'm trying to improve the advection method in a 2D-windfield. The Navier-Stokes Equations (NSE) are currently used for the influence of pressure, viscosity,... I am just focusing on the convective ...
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Where to find the article(s) saying that 3D Navier-Stokes solutions exist in case of a small initial velocity?

I've already been searching for here and on the internet, maybe not enough. PS: this result is stated in the Clay Mathematics Institute N-S problem definition, page 2: https://www.claymath.org/wp-...
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Cylindrical incompressible Navier-Stokes, are the derivatives commuative?

I am currently trying to understand a paper by Eckhardt et al. (https://doi.org/10.1017/S0022112007005629). In it, a transformation is performed on the cylindrical incompressible Navier-Stokes ...
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Can direction of frictional force be opposite to velocity?

The frictional force in incompressible Navier-Stokes equation is given by $$\vec F = \nu \nabla^2\vec v$$ Lets assume simple 1D-flow along $x$, where $v$ depends on $z$: $$v_x = v(z)$$ Then the ...
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9 votes
2 answers
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What is the the Implication of Navier-Stokes Millenium Problem to Practice?

As a student of meteorology, I wonder why Navier-Stokes equations (NSE) are still not understood in terms of whether or not there are unique solutions. In atmospheric dynamics, NSE is used as a basic ...
MichaelW's user avatar
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1 vote
3 answers
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Viscous fluid under constant force?

For a one-dimensional fluid with viscosity $\eta$ subject to a homogenous acceleration $a$ in periodic boundary conditions, in my understanding the momentum equation is $$\rho\left(\frac{\partial u}{\...
YoussefMabrouk's user avatar
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1 answer
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Convergence to Stokes terminal velocity?

Consider a spherical particle immersed in a fluid at room temperature. Starting from some time $t_{0}$, I turn on a spatially homogenous force on the particle. After some time $t$ I expect the ...
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Physical meaning of the decoupling of the transverse velocity field in linearized Navier-Stokes equation

The Navier-Stokes equations: \begin{equation} \begin{split} \partial_t \rho &= -\vec \nabla \cdot (\rho \vec u) \\ \partial_t(\rho \vec u) + \vec \nabla \cdot (\rho \vec u \...
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Time derivative term in Navier Stokes equation for fluid in porous media

I was reading the research paper Homogenization of peristaltic flows in piezoelectric porous media and came across the hydrodynamic equation: $$\mu \nabla^2 v^f -\underline{ \rho_f (\dot{v}^f + w \...
user134613's user avatar
2 votes
2 answers
196 views

How does one calculate the viscous term in the integral form of navier stokes?

I learned about the integral form of the navier stokes equation and I am trying to find an explanation for this viscous term at the end but I've searched everywhere and I can't find anything. The ...
Mitsos YT's user avatar
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Modeling fluid flow in extremely small pores (or channels) compared to the component

I want to model a non-wetting fluid flowing into tiny pores of a component using Navier-Stokes. Just for you to visualize, say I have a 2D rectangular component of dimensions $L*W$, and within that ...
Mechanician's user avatar
3 votes
0 answers
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Questions regarding the derivation procedure of the Karman-Howarth-Monin relation

I was reading the book "Turbulence" written by Uriel Frisch (1995) and got stuck in following the proof of the Karman-Howarth-Monin relation. Although there is already a thread about it, I ...
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1 answer
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Non-dimensionalisation of Navier-Stokes Equations [closed]

I am trying to numerically simulate a flow in which a fluid with kinematic viscosity $\nu=0.00001167$ m$^2$/s is injected into a cylindrical tube of diameter 0.00745m at a velocity of 45.9 m/s. Thus, ...
Somestudent01's user avatar
1 vote
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Derive Navier-Stokes equation from the action principle (Euler-Lagrange's equation) [duplicate]

Navier-Stokes equations are the most general equations in fluid dynamics: We ususlly derive it by the consevation laws and $F=ma$. But how to derive it from the Action principle or equivalently Euler-...
user353731's user avatar
4 votes
1 answer
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Is there specific form of Navier-Stokes equation for which mass can cross bounding surface?

In my textbook, we learned that Navier-Stokes (NS) equations can be derived from Reynolds transport theorem where the control volume is assumed to be fixed. But when the control volume is moving, can ...
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Stress Tensor Normal Component (Newtonian Fluid) [duplicate]

I am trying to get an intuitive understanding of the normal components of the stress tensor for a Newtonian Fluid. The stress on the x face in the x direction is $$\sigma_{xx} = 2*\mu\frac{\partial u}{...
remusconnor's user avatar
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How to calculate pressure $p$ in the Navier-Stokes equation to simulate the time evolution of a fluid?

I wanted to simulate the motion of a fluid (continuously filling the entire space) in a given space, say $3$-dimensional Euclidean space. To calculate the dynamics of fluid motion, I used the Navier-...
Modular Discriminant's user avatar
6 votes
1 answer
514 views

Why are the Navier-Stokes equations inconsistent in this case?

Consider the case of a one-dimensional incompressible, non-viscous fluid flowing down a vertical pipe under the influence of gravity. Since we assume the flow is constant along the cross section of ...
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