Questions tagged [navier-stokes]

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

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2answers
230 views

Near a hard-wall, in which direction does a rigid spherical particle move when a positive torque is exerted upon it?

Consider a small rigid spherical particle of radius $a$ immersed in a viscous incompressible Newtonian fluid of shear viscosity $\eta$ in close vicinity to a hard-wall with stick (no-slip) boundary ...
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1answer
560 views

Is it possible to change Pressure without changing Density?

I am still trying to understand Navier-Stokes equations. I understand the equations in general, BUT there is one aspect I still cannot digest: The equations assume that density is constant (okay, I ...
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1answer
1k views

Fluid Density Vs Pressure (incompressible fluid)?

I am trying to understand Navier-Stokes fluid equations and I have two questions (these are very primitive questions, I am still trying to understand basic concepts) 1) all books say "assume that ...
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1answer
121 views

Different ways of expressing the Euler equations [closed]

In the Euler equations, there is a term of the form $(\textbf{U}\cdot \nabla) \textbf{U}$ I've seen this expression replaced with the vector identity $(\textbf{U}\cdot \nabla) \textbf{U} = \nabla(\...
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1answer
811 views

Navier-Stokes equations of motion in the Lagrangian description

In general, the Navier-Stokes equations of motion are derived in the Eulerian description. I tried to find the Navier-Stokes in the Lagrangian description but was not very successful. I would be glad ...
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Conservation condition in rotating cylindrical coordinates [closed]

Is this correct? Cylindrical coordinates $(r, \theta,z,t)$: $$\frac{\partial \rho}{\partial t} +\frac{1}{r}\frac{\partial}{\partial r} (r\rho V_r)+\frac{1}{r}\frac{\partial}{\partial \theta} (\rho ...
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3answers
5k views

Can you explain Navier-Stokes equations to a layman? [closed]

Can you explain Navier-Stokes equations to a layman? Could someone explain this famous and important equation with "plain words"? If my question is too broad for an answer, I will also be very ...
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2answers
246 views

Are the diffusion terms conservative?

Generally the diffusion terms are of the form $$D = \dfrac{\partial}{\partial x} \left(\mu \dfrac{\partial u}{\partial x} \right) .$$ Is this this term conservative or nonconservative?
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Why is the pressure gradient zero at a wall?

It's accepted to impose a zero pressure gradient normal to a wall when solving the Navier-Stokes equation. Is there any mathematical reasoning for that? Which pressure (static pressure, total pressure....
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1answer
714 views

Conservative form of Navier-stokes equation in cylindrical coordinates

Is it possible to write the conservative form of Navier-stokes equation in cylindrical coordinates? Almost all texts I have referred (Frank M. White, Kundu & Cohen,G.Batchelor) have it in non-...
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1answer
292 views

Navier Stokes - Euler's Approximation

I am working on a question from my practice exam. We are asked if the following equation is a valid expression of Euler's equation - an approximation to Navier Stokes for high Reynold's number. $$\...
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4answers
838 views

Navier Stokes Approximation For Low Reynold's number

The generic form of the Navier Stokes equation is (assuming incompressibility and Newtonian fluid): $$\rho\cfrac{Dv}{Dt} = -\nabla P + \mu\nabla^2v + \rho g$$ This equation can be rearranged as ...
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1answer
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Applying Navier Stokes to Fluid on Inclined Plane

I am trying to solve the well known problem of identifying the velocity profile of a fluid on an inclined plane. The assumptions we know are that the flow is Newtonian, incompressible, with a low ...
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Deriving Bernoulli's equation from Navier-Stokes equation [closed]

I can prove the Bernoulli equation in an elementary way (that is, using the conservation of energy law). But I was asked to derive the Bernoulli equation from the Navier-Stokes equations, and I have ...
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1answer
228 views

What is the physical explanation as to why the kinematic boundary condition must hold at the free surface of a wave?

The kinematic boundary condition at the surface of a water wave is given as: "a particle on the free surface remains there always". This is then written as the material derivative of the free surface ...
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318 views

Pressure generation due to opposing flows

If two fluids are flowing at unequal velocities towards each other in a circular pipe, will a pressure be generated at the intersection? If yes, what will the direction of this pressure generated and ...
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1answer
843 views

Flow of viscous fluid between two walls inclined at an angle

I suppose this is a basic fluid mechanics problem but I have one thing I do not understand. I am about to solve a problem with steady, incompressible, parallel, laminar flow of viscous fluid falling ...
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1answer
104 views

What branch of mathematics describes fluid motion?

I was wondering if there is a specific branch of mathematics that is used to describe fluid motion? I know that calculus is used to describe rates of change in fluid motion (i.e. how fast a container ...
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1answer
67 views

Hydrodynamics issue in Goldreich & Lynden-Bell (1965b)

I'm probably missing something stupid. In the paper mention above (see link, pg. 4) the hydrodynamical instability of the disk is reviewed with Navier-Stokes equation. Having the unperturbed ...
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1answer
680 views

Problems measuring viscosity of water with a Falling Ball Viscometer

I am measuring the viscosity in water using a falling ball viscometer. In my system, I believe that exists two forces: The force difference between the weight and buoyancy of the sphere: $F_g = m'g =...
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1answer
454 views

Do the incompressible Navier-Stokes Equations apply to incompressible fluids or incompressible flows?

As far as I understand an incompressible fluid is one where the density is constant and an incompressible flow is one where the material derivative of density is constant ($\frac{D\rho}{Dt}=0$). Both ...
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1answer
334 views

What are the Lorenz Equations used for?

In fluid dynamics I have come across two sets of equations, the Navier-Stokes equation and the Lorenz equations. From what I have read the Navier-Stokes equations always holds. So why do we need the ...
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Pressure recovery from vorticity with Stokes-flow [closed]

I am solving Stokes-flow for thermal convection in the form of $\nabla^2 \textbf{u}+\nabla p = \textbf{f}$ and $\nabla\cdot\textbf{u} = 0$. With all free-slip boundary conditions I can turn this into ...
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1answer
210 views

Does surface tension attenuate turbulence?

Rapids and several other "white water" systems could be typical examples of turbulent motion. To describe such flows the Navier-Stokes equations must be extended by terms describing the surface ...
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1answer
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Is the flow of a viscous fluid in free space under no pressure gradient always laminar?

Consider a (Newtonian) incompressible viscous fluid in three spatial dimensions, whose velocity field $\mathbb{v}=\mathbb{v}(x,y,z,t)$ moves according to the Navier-Stokes equations $$\tag{1}\label{...
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What is the velocity in the Navier-Stokes equation?

I have been looking at the Navier-Stokes equation, and can't seem to find anywhere a clear description of what velocity it represents. From what I have read it could be any of the following: The '...
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2answers
666 views

Non-dimensionalizing incompressible Navier-Stokes

I have a question about non-dimensionalization of the incompressible Navier-Stokes (NS) equations. My understanding is that the purpose of non-dimensionalization is to "collapse" solutions onto one ...
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1answer
139 views

Inertial Waves - Why neglecting the advecting term?

I'm trying to derive the dispersion relation for Inertial waves. In Cartesian coordinates: Inviscid and incompressible fluid is rotating uniformly with Angular Velocity: $\Omega = (0, 0, \Omega)$ ...
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1answer
283 views

Why is turbulence caused?

In high Reynolds numbers we have turbulent flow. This is because the inertial forces are much greater than the viscous forces. I understand inertial forces to be actually the fictional forces due to ...
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1answer
110 views

Finding boundary condition of stationary solid body

A fluid flows past a stationary solid body of arbitrary shape. Write down the boundary condition on the fluid velocity $\textbf u$ for an inviscid fluid and for a viscous fluid, at the solid surface. We ...
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1answer
410 views

Stokes-Einstein Relationship to find time taken to diffuse $x$ distance? [closed]

A molecule has a diffusion coefficient of 0.5 × 10-9 m2s-1. Calculate how long it would take on average for the molecule to diffuse 10 µm. So I have the above ...
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1answer
97 views

Transverse and longitudinal random forces

I am trying to read following article: http://arxiv.org/pdf/1410.1262v1.pdf According to the equation (2.10) and (2.11), the random force is defined as $ \langle f_i(x) \ f_j(x) \rangle = \delta(t-t'...
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1answer
614 views

Navier-Stokes Energy Equation

I've been assigned (for homework in a mathematical modelling course) the task of deriving the Navier-Stokes energy equation in one space dimension: Consider a fluid flowing through a cylindrical ...
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1answer
3k views

Wall Shear Stress

I have the solution of a Navier-Stokes simulation with an incompressible, Newtonian fluid with laminar flow. Now I compute the wall shear stress (vector) as $$\tau_n = \mu (\nabla u) n,$$ where $\mu$ ...
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0answers
273 views

Mathematical understanding of vortex solitons

I am wondering if anyone has ever come up with a mathematical description of something that (to me, and I am no expert) look like soliton vortexes. The example I can think of is if you create two ...
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1answer
736 views

Basis for Derivation of Stokes Friction Law for Spheres

When deriving Stokes law one uses the Navier Stokes equation with the assumptions: low Reynolds number stationary flow in compressible flow leading to this version of the N.S : $$\nabla p = \eta \...
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1answer
617 views

Navier Stokes : what about angular momentum?

I play with CFD for a while, and suddenly, a transcendantal question raises: :-) Navier Stokes is basically Newton applied on a continuum in Eulerian. For solids, we would consider linear, but also ...
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1answer
633 views

What fluid dynamics equations are like in zero gravity?

I don't know if this is a proper question. I am not so familiar with fluids. I am just curious about what Navier-Stokes equations for fluids will look like in zero gravity. Are they stay the same? If ...
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0answers
120 views

Why Velocity is referred as momentum? [closed]

In many text books the velocity is referred to as a linear momentum which is being convected. For example the table in the following page My old conception is that the momentum or more precisely ...
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0answers
482 views

Numerical model for a Air core Vortex, why it's still so limited? [closed]

The Question can be Formally presented as follows; How is the Numerical Model (CFD, Navier Stokes) of fluid dynamics for Vortexes at in it's limits? At this Publication from 2013 at page 48 is said;...
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368 views

assumptions about sound waves

When deriving the sound wave equation: $${1 \over c^2} {\partial^2 p' \over \partial t^2 }= \Delta^2 p' $$ by linearizing the Euler equation: $$\rho {d v \over dt }= - \nabla p $$ and the continuity ...
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3answers
131 views

Does this dimensioneless quantity have a name?

When studying creeping flows, a common choice for a characteristic pressure scale is $$p_0 = \frac{\mu_0 U_0}{L_0},$$ where $\mu_0$ is a reference dynamic viscosity, $U_0$ is a reference velocity and $...
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1answer
397 views

Fluid dynamics equations, number of variables and number of equations

Continuity and Navier-Stokes equation for fluid are, \begin{eqnarray} \frac{\partial \rho}{\partial t} + \nabla\cdot (\rho \mathbf{u}) &=& 0 \\ \rho\left(\frac{\partial \mathbf{u}}{\partial t} ...
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1answer
85 views

Reference values for viscosity and density in incompressible NSE

I come from a pure mathematics background, so I have very limited physics knowledge. I'm currently working out the non-dimensional form for the Navier-Stokes equations and have some questions. Where ...
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1answer
83 views

What does $\mathbb{R}^3$ and $\mathbb{T}^3$ look physically for the Navier-Stokes equation?

What does the Navier-Stokes equation solution according to the Clay Math Institute look like in real life? As in how do you visualize $\mathbb{R}^3$ and $\mathbb{T}^3$ without the math? I actually ...
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2answers
81 views

Is there a formulation for (self-)accelerating fluid flow through permeable medium?

I have a permeable system where is an accelerating fluid flow. Imagine a sponge that is squeezed. The fluid starts at rest, accelerates and flows out from the sponge. How to calculate the fluid speed? ...
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1answer
972 views

Stokes law in 2-dimensions

Stokes' law states that force on slow moving sphere (i.e. $Re\ll1$) in liquid is $$ F_d = 6 \pi \mu R V $$ In two dimensions we are in trouble (flow around disk in 2d or around cylinder in 3d), ...
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4answers
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How does the dissolution of salt affect the solution density?

Suppose you have a container of water as a solvent and you a certain amount of salt as a solute sitting at the bottom of the container that has yet to start dissolving. Supposing temperature and ...
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2answers
381 views

Why is the Reynolds number “the way it is?” Why is its order the way it is?

Why is the Reynolds number “the way it is?” Why is its order the way it is? I'm not sure if this is an appropriate question for this context, but I would like more intuition on this matter and so I'...
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1answer
131 views

What is this form of mass conservation equation?

I found the following equation of conservation of mass (continuity) in "Computational Fluid Dynamics Vol.III" by Hoffmann: $$ \frac{\partial \rho}{\partial t} + \frac{\partial}{\partial x}(\rho u)+ \...