Questions tagged [navier-stokes]

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

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Can we derive Stokes' drag law from Navier-Stokes' equation?

So basically Stokes' law states that, "The drag force on a spherical body of radius $r$ and velocity $v$ is $$F_{d}=6\pi \eta rv.$$ My question: $(1)$ Can we derive Stokes' drag law from Navier-...
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No slip boundary condition in Hele-Shaw Cell

A Hele-Shaw cell can be used to visualise potential flow around a cylinder. See this image from Van Dyke Album of fluid motion with the z axis of the cylinder pointing out of the page: Potential ...
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Acceleration of fluid in a magnetohydrodynamics vortex

In the following video we see a mercury vortex done by magnetohydrodynamics: https://youtu.be/au4hbUm4mMo In the following manuscript they say that the fluid will accelerate according to (14) ${}^1$: ...
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Finding the stress tensor in the Stokes equation from free energy

I have found free energy for a system of particles. The free energy is a functional of a scalar field which is the area of the particles. So I have the following free energy $F(A,p)= \int dx dy \: f[A(...
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What is the hydrodynamic limit exactly and why is it called that?

Hydrodynamics is one of those words which are used everywhere in the literature, but I can not seem to find a clear definition! My idea (which could be wrong) is it is a continuum limit of a theory ...
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Pressure in stokes flow

In the stokes equation for an incompressible fluid, there is a pressure term that enforces the fluid to have $\nabla \cdot u=0$, where $u$ is the velocity field. The stokes equation reads: $$\eta \...
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What is the mathematical derivation for no diffusion term in the mass continuity equation of the Navier-Stokes/Euler equations?

In this post the fact that the mass continuity equation in a mixture of gases has no diffusion term, i.e., $$\frac{\partial\rho}{\partial t}+\nabla\cdot(\rho\vec{v})=0$$ has been discussed. ...
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Navier-Stokes equation viscosity term

I cannot understand how the term μ$\nabla^2(\vec u)$ is derived. Can anyone help?
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What would be the Stokes hypothesis in a 1D flow?

In this paper, the author derives the Navier-Stokes equation for a Newtonian fluid starting from the Cauchy equation: $$\rho \frac{D\mathbf V}{Dt} = \rho \mathbf{f} + \nabla\cdot\mathbf{T}$$ where $\...
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Can we use Darcys flow equation through porous media for pipe flow?

I am looking at flow in a pipe with no solid matrix. Therefore, the porosity is always equal to one. For such a set-up, I can use the Direct Numerical Simulations (DNS) on a 2D framework. However, a ...
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Calculation of velocity of water flowing along inclined plane by applying Navier-Stokes theorem

Assume that water flows along a inlined plane through a circular irrigation waterway. The inclined plane has its bottom length $B=50 m$, and has height $H=10 m$. So the angle of inclination $\theta$ ...
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Convective derivative N-S

This is probably an easy answer, but I've not been able to find it yet - Why in some formulations of the N-S equations (for example here https://www.grc.nasa.gov/www/k-12/airplane/nseqs.html), is the $...
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In Reynolds transport theorem, is the integral region simply-connected?

Reynolds transport theorem for a material element (parcels of fluids or solids which no material enters or leaves) reads $^1$ $$ (1):\frac{d}{dt}\left(\int_{\Omega(t)} \mathbf{f}\,dV\right) = \int_{\...
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Derivation of the slender body theory for Stokes flows

I am trying to get through the Keller and Rubinow (1976) paper on the slender body theory for Stokes flows. However the paper does not seem to explain the inner and outer expansions in detail and just ...
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Is it possible to derive Navier-Stokes equations of fluid mechanics from the Standard Model?

We know that the Standard Model is a theory about almost everything (except gravity). So it should be the basis of fluid mechanics, which is a macroscopic theory from experiences. So is it possible ...
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Question regarding vector calculus [closed]

Question regarding vector calculus: Mathematically, what properties does this vector field have, in order for its line integral to be equal to the area enclosed by that curve?
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Physical Interpretation of Large-Time Decay Estimates of Solutions to Navier-Stokes

It is well known (see for example Hoff-Zumbrun (1995)) that solutions to the compressible Navier-Stokes equation converge in $L^p$ spaces to the heat kernel. Formally, to keep things simple, we can ...
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Dynamics of ideal gas during free expansion

I would like to compute (even numerically) the time evolution of the density profile of an ideal gas which undergoes free expansion. For concreteness, let's imagine an infinite cylinder with the axis ...
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Tensor gradient order of terms

I am working on some algebraic manipulation with the compressible Navier-Stokes equations, specifically this form, screenshotted from Wikipedia: I'm confused by the gradient operator being applied to ...
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Fluid Momentum Equation

I have a equation regarding the fluid equation of momentum. What exactly is being accelerated/acted on? 

My understanding is this: The momentum is equation (at least in my textbook) is derived by ...
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"Contradiction" Between 2D Steady-State Continuity and Navier-Stokes Equations

I am looking for some clarification on the incompressible, 2D, steady-state, cartesian Navier-Stokes equations for flow through a straight cylinder (flow aligned with the $x$-axis, so $u_y =0$), with ...
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Meaning of the transpose of a gradient

Sometimes I encounter PDE's with a term like this $\nabla \cdot c(\nabla \textbf{v} + (\nabla \textbf{v})^T)$ An example are the Navier-Stokes equations. Oftentimes this can be further simplified to $...
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How to derive gravitational potential from Navier-Stokes equation?

Starting from the Navier-Stokes equation I want to be able to derive the gravitational potential using the Poisson equation but am unsure how to do it in spherical polar coordinates. This is what I ...
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Boltzmann Transport Equation existence and smoothness - Is it proved?

Currently, Navier-Stokes Equation, its solution's existence and smoothness is not well established, making the problem as one of famous Millennium Prize Problems. On the other hand, I noticed that ...
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How does turbulence arise from Navier-Stokes?

I would like to know how turbulence arises from the standard Navier-Stokes equations, both mathematically and also physically. At least I suspect this is the case as many of the "vanilla" ...
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Impact of total stress tensor's definition on a continuum's linear momentum equation derivation and result

I am reading the book: "The Finite Volume Method in Computational Fluid Dynamics An Advanced Introduction with OpenFOAM® and Matlab®" - DOI 10.1007/978-3-319-16874-6 The section of concern ...
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Why Fourier transform for linearized NS?

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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Why do we subtract the “transport velocity” from the "domain velocity" in ALE?

Why do we subtract the “transport velocity” from the "domain velocity" in Arbitrary Lagrange Eulerian framework when writing the Navier Stokes equation? Book: Cardiovascular mathematics ...
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About derivation of Navier Stokes Equation

Following the steps of derivation, everything is clear just for one small argument which is: Why is the divergence of the transpose of gradient equal to gradient of the divergence, and why does it ...
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Speed of water at bottom of an inclined plane

I have a question. We are given an inclined plane with height $h$ and hypotenuse length $d$. And water is falling down through a hose on the slope. The hose has a inner diameter $l$ and thickness $b$. ...
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How identify from velocity values, if a flow is viscous or inviscid?

So in my Fluid Mechanics classes we need to know how to identify viscous flows from inviscid ones, using Matlab, and only with velocity values. How can I know de difference? An example is pictured in ...
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Physical Meaning and/or Justification to $\lambda = 0$ Case for Compressible Navier-Stokes

I am looking for information on the coefficient $\lambda$ in the following formulation of the barotropic compressible Navier-Stokes system: \begin{align} & \partial_t \rho + \text{div}(\rho u) = 0,...
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How to derive Bernoulli's equation in steady irrotational flow?

Condsider an incompressible, inviscid, irrotational fluid with constant density $\rho$. Let $\overrightarrow u$ be its velocity field, $p$ its pressure field and $\overrightarrow F$ be an external ...
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What is the molecular underpinning of pressure in fluid mechanics?

The Navier-Stokes equation tells us that there are two ways of transferring momentum to an infinitesimal volume of fluid through the action surface forces: (1) pressure and (2) viscous stress. On a ...
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Conservative form of the vector diffusion equation

For some reason I am unable to find a source on the internet about this. I think I have an answer, but I want to be doubly sure about this. All I could find (here), is that for an incompressible fluid,...
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Is momentum flux scalar or vector?

I'm trying to derive Navier Stokes equation and stacked the linear momentum equation below. The second term is momentum flux, but it seems scalar value for me because it is vector times vector. How ...
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Does the Boltzmann equation reduce to Navier-Stokes equation? [duplicate]

Since the derivation of the Boltzmann Equation uses the molecular chaos assumption, it seems to me that it should not be valid for dense systems such as fluids. Now, according to Chapman-Enskog theory,...
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Terminal velocity of particle in medium with non-uniform velocity

assume that a particle of radius $R_p$ is moving under influence of gravity $g$ in a fluid medium of density $\rho_l$ and viscosity $\mu_l$. then the Stokes settling velocity is given as $$ \mathbf{v}...
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Can you calculate the sedimentation rate of a coarse chemical suspension under gravity using a centrifuge?

If the sedimentation rate can be determined for a coarse chemical suspension or a suspension containing large particles (i.e., particles with radii between $100$ to $200 \mu m$) in a medium like ...
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Pressure gradient term at low Reynolds number

I was going over the derivation of the dimensionless Navier-Stokes equation, which is explained in this answer. By introducing dimensionless variables in the NS equation, one gets $$ \frac{D\mathbf u}{...
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Conservation of mass from material derivative

Let the mass be $m=\rho \text{Vol}$, where $\text{Vol}$ is the volume of the domain and the velocity is $u$. Applying the material derivative, then $$\frac{Dm}{Dt}=\frac{\partial (\rho \text{Vol})}{\...
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What does $ \mu \nabla^{2} \vec V$ mean in the Navier-Stokes equations?

$$\rho\frac{D \vec V}{Dt}=-\nabla p+ \mu \nabla^{2} \vec V+\rho g$$ In the Navier-Stokes equations there's this term $ \mu \nabla^{2} \vec V $. I don't really understand what this means. What is the ...
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How does hot air rise?

If a balloon is filled with hot air, it is rising due to buoyancy: the mass of the hot air inside the balloon is lower than the mass of the same volume of the cold air outside the balloon cavity. ...
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Show how to obtain the mass accretion rate $Ṁ = -2πRvRΣ$ from Navier-Stokes mass conservation $∂ρ/∂t + ∇ · (ρv) = 0$ using cylindrical coordinates

Consider a steady, cylindrical, slim axisymmetric magnetized disk of average density $ρ$, surface density $Σ$ and semi-thickness $H$, around a Neutron star. Within a slim disk, rotational velocity $vR$...
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How can the conservation/Navier Stokes equations (mass, momentum,energy) be modified to model two phase flow in a porous media?

Previously I have seen the derivation of the energy conservation equations for simulation of single phase flow in a porous media (a packed bed). These are the energy equations for the solid and fluid ...
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Marangoni Effect and the Navier-Stokes Equations

A quite well-known phenomena, oftenly treated in recreational physics, is the Marangoni Effect. Roughly speaking, we have a flux caused by a gradient of surface tension. You can look up YouTube, and ...
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How to find the reasons for change of linear momentum?

I would like to have an equation that says "The spatial change of the linear momentum vector is equal to the sum of these terms...", or written as an equation: $$\mathrm{grad}\left(\rho \...
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Innert sphere immersed in a stokes flow near a point force

Let $B$ a solid ball of center $0 \in \mathbb{R}^3$ and radius $a$ immersed in a Stokes flow along with a point force of intensity $F$ outside of the ball. If $u_1$ denotes the speed of the fluid of ...
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Normalisation for pressure for low and high flow speed regimes: what is the explanation behind it?

In the book of Theoretical microfluidics by Bruus, at page 26 it is given that $$ p=\frac{\eta V_{0}}{L_{0}} \tilde{p}=P_{0} \tilde{p} $$ Note that a quantity often can be made dimensionless in more ...
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Fully developed flow in cylinder coordinates

I have a flow that is: steady (time derivative is zero) fully developed in-compressible (constant density) no slip at the wall axis-symetric (theta derivative is zero) The configuration of the pipe ...
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