# 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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### 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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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? $\rho\... 0 votes 1 answer 58 views ### Why do we subtract the “transport velocity” from the "domain velocity" in ALE? Why do we subtract the “transport velocity” from the "domain velocity" in Arbitrary Lagrange Eulerian framework when writing the Navier Stokes equation? Book: Cardiovascular mathematics ... 0 votes 1 answer 57 views ### About derivation of Navier Stokes Equation Following the steps of derivation, everything is clear just for one small argument which is: Why is the divergence of the transpose of gradient equal to gradient of the divergence, and why does it ... 1 vote 0 answers 41 views ### Speed of water at bottom of an inclined plane I have a question. We are given an inclined plane with height$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$. ... 0 votes 0 answers 23 views ### How identify from velocity values, if a flow is viscous or inviscid? So in my Fluid Mechanics classes we need to know how to identify viscous flows from inviscid ones, using Matlab, and only with velocity values. How can I know de difference? An example is pictured in ... 2 votes 2 answers 90 views ### Physical Meaning and/or Justification to$\lambda = 0$Case for Compressible Navier-Stokes I am looking for information on the coefficient$\lambdain the following formulation of the barotropic compressible Navier-Stokes system: \begin{align} & \partial_t \rho + \text{div}(\rho u) = 0,... 2 votes 1 answer 89 views ### 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$be an external ... 0 votes 0 answers 23 views ### What is the molecular underpinning of pressure in fluid mechanics? The Navier-Stokes equation tells us that there are two ways of transferring momentum to an infinitesimal volume of fluid through the action surface forces: (1) pressure and (2) viscous stress. On a ... 0 votes 0 answers 43 views ### Conservative form of the vector diffusion equation For some reason I am unable to find a source on the internet about this. I think I have an answer, but I want to be doubly sure about this. All I could find (here), is that for an incompressible fluid,... 0 votes 2 answers 72 views ### Is momentum flux scalar or vector? I'm trying to derive Navier Stokes equation and stacked the linear momentum equation below. The second term is momentum flux, but it seems scalar value for me because it is vector times vector. How ... 2 votes 0 answers 46 views ### Does the Boltzmann equation reduce to Navier-Stokes equation? [duplicate] Since the derivation of the Boltzmann Equation uses the molecular chaos assumption, it seems to me that it should not be valid for dense systems such as fluids. Now, according to Chapman-Enskog theory,... 1 vote 1 answer 57 views ### Terminal velocity of particle in medium with non-uniform velocity assume that a particle of radius$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 $$\mathbf{v}... 0 votes 1 answer 17 views ### Can you calculate the sedimentation rate of a coarse chemical suspension under gravity using a centrifuge? If the sedimentation rate can be determined for a coarse chemical suspension or a suspension containing large particles (i.e., particles with radii between 100 to 200 \mu m) in a medium like ... 0 votes 1 answer 59 views ### Pressure gradient term at low Reynolds number I was going over the derivation of the dimensionless Navier-Stokes equation, which is explained in this answer. By introducing dimensionless variables in the NS equation, one gets$$ \frac{D\mathbf u}{... 1 vote 2 answers 79 views ### 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 $$\frac{Dm}{Dt}=\frac{\partial (\rho \text{Vol})}{\... 2 votes 2 answers 120 views ### What does \mu \nabla^{2} \vec V mean in the Navier-Stokes equations?$$\rho\frac{D \vec V}{Dt}=-\nabla p+ \mu \nabla^{2} \vec V+\rho g$$In the Navier-Stokes equations there's this term \mu \nabla^{2} \vec V . I don't really understand what this means. What is the ... 7 votes 4 answers 382 views ### How does hot air rise? If a balloon is filled with hot air, it is rising due to buoyancy: the mass of the hot air inside the balloon is lower than the mass of the same volume of the cold air outside the balloon cavity. ... 0 votes 0 answers 16 views ### Show how to obtain the mass accretion rate Ṁ = -2πRvRΣ from Navier-Stokes mass conservation ∂ρ/∂t + ∇ · (ρv) = 0 using cylindrical coordinates Consider a steady, cylindrical, slim axisymmetric magnetized disk of average density ρ, surface density Σ and semi-thickness H, around a Neutron star. Within a slim disk, rotational velocity vR... 0 votes 0 answers 139 views ### How can the conservation/Navier Stokes equations (mass, momentum,energy) be modified to model two phase flow in a porous media? Previously I have seen the derivation of the energy conservation equations for simulation of single phase flow in a porous media (a packed bed). These are the energy equations for the solid and fluid ... 1 vote 0 answers 41 views ### Marangoni Effect and the Navier-Stokes Equations A quite well-known phenomena, oftenly treated in recreational physics, is the Marangoni Effect. Roughly speaking, we have a flux caused by a gradient of surface tension. You can look up YouTube, and ... 0 votes 1 answer 40 views ### How to find the reasons for change of linear momentum? I would like to have an equation that says "The spatial change of the linear momentum vector is equal to the sum of these terms...", or written as an equation:$$\mathrm{grad}\left(\rho \... 0 votes 0 answers 17 views ### 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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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 ...