Questions tagged [navier-stokes]

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

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Fluid simulation - why gradient of pressure is the term $\delta q$ in the projection step?

I'm learning stable fluids by reading Stam's paper. In the paper, it says, according to Helmholtz-Hodge Decomposition, I vector field $w$ could be decomposed to a divergence-free field and a curl-...
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Physical Meaning of Divergence of Convective Velocity Term

When taking the divergence of the convective velocity term, I get the following: \begin{align} \nabla\cdot\left[\mathbf u\cdot\nabla\mathbf u\right]&=\frac{\partial}{\partial x_i}\left[u_j\frac{\...
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Second-order covariant derivative in index notation [closed]

So I'm having problems finding the second order covariant derivitive in index notation. My teacher said to just find the covariant derivative of a covariant derivative, so I first started with finding ...
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What does $(\mathbf{u}\cdot\nabla)\mathbf{u}$ mean in the Navier-Stokes equation?

I am studying the Navier-Stokes equations and I have the equation in the form: $$\rho \dfrac{\partial{\mathbf{u}}}{\partial{t}} + \rho (\mathbf{u}\cdot\nabla)\mathbf{u} - \mu\nabla^2\mathbf{u} + \...
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Validity of the Navier Stokes equations for turbulent flows

The derivation of the Navier-Stokes equation presupposes that the pressure, $p$, and velocity, $u_i$, are infinitely differentiable, so that the forces in each face of the fluid element can be ...
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399 views

Will a proof of the existence and smoothness of the Navier-Stokes equations contribute to a GUT? [closed]

Note: throughout the course of the question, the word "describe" will often be used to suggest that a mathematical equation can "describe" what it happening in a physical context. One of the long ...
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139 views

What does $ \vert \partial^{\alpha} v_o(x) \vert $ mean in the Navier-Stokes initial velocity condition?

The initial condition $\displaystyle \mathbf{v}_0(x)$ is assumed to be a smooth and divergence-free function such that, for every multi-index $\displaystyle \alpha$ and any $\displaystyle K>0$, ...
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What is the importance for the pressure/velocity at infinity be constant in fluid dynamics?

I am studying fluid dynamics on my own and it is commom to see this assumption. I am asking this here because I didn't find any satisfactory answer. For example, I am studying a problem that is as ...
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Rationale behind the linearised Navier-Stokes equation

Some applications of fluid dynamics consider the linearised Navier-stokes equation where the advection term $(\vec{u}\cdot\vec{\nabla})\vec{u}$ is dropped. I am trying to build a convincing argument ...
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Solve unsteady state Bernoulli equation for flow in a pipe

I am an engineer studying an unsteady-state flow through a pipe. The Pipeline has been cleanly cut into two halves, without deforming the cylindrical form of the pipe, exposing the contents to ...
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How were the Navier-Stokes equations found in the first place if we can't solve them?

I was reading up on the Clay Institute's Millenium prizes in mathematics. And I noticed the Navier-Stokes equations were described as minimally understood. As far as I was taught in physics a few ...
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694 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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184 views

What causes fluid rotation in the Navier--Stokes equation?

It is known that rotation in the flow results from the viscous terms in the Navier-Stokes (N-S) equation. However, when deriving the N-S equation from the general principle of linear momentum in ...
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1answer
835 views

Stream function formulation for Navier-Stokes equations in 2D - boundary conditions

I'm particularly interested in 2D incompressible version of Navier-Stokes equations that describe flow in some domain of interest: $$\begin{aligned}\frac{\partial v_x}{\partial t} + \vec{v} \cdot \...
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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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Sound speed on Navier-Stokes/Euler equations

I'd like to simulate strong shock (i.e., Rankine Hugoniot conditions) on inviscid condition, using the non-conservative Euler equations and I don't know if I should use the relation, $$ c^2=\left(\...
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What does it mean that a substance can be smelled from far away?

I thought about this question in the middle of this video. Ok, Thioacetone takes the price for the World's smelliest chemical, I can accept it (why not?), but what about You can smell one drop of ...
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Is there an Okubo-Weiss function for axisymmetric flows

I have recently learned about the Okubo-Weiss criterion in a ($x,y$) and a ($r,\theta$) coordinate system. Is it possible to recast this criterium in an ($z,r$) system, that is for axisymmetric ...
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Weak Formulation of Fluid Equation

I am currently trying to obtain the weak formulation of a system of differential equations to program in Freefem++ a solver for the growth of plaque in human arteries. I am, however, failing ...
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Do the Shallow Water Equations produce 2d vorticity/eddies? Why/Why not?

My understanding so far: given a small flow moving forward within a larger stationary body of water, the water ahead would pile up, creating hydrostatic pressure in all 2d directions (and thus ...
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Boundary conditions for Navier-Stokes equations

The Navier-Stokes equations in combination with the continuity equation for incompressible Newtonian fluids is stated as $$\dfrac{\partial \mathbf{u}}{\partial t}+\left(\mathbf{u}\cdot \nabla\right)\...
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Solution of Navier-Stokes in a triangular pipe

I was doing some reading about Navier-Stokes equations and stumbled upon this PDF from a course at Clarkson University: http://webspace.clarkson.edu/projects/crcd/me537/downloads/...
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Life at low Reynolds number (E. M. Purcell)

I am reading the talk given by E. M. Purcell (Nobel Prize winning physicist) named "Life at low Reynolds number". There are two things that I cannot understand. Purcell tells that the Stokes-Equation ...
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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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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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575 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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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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Navier-Stokes equations through a orifice [closed]

I have reactor that looks as follows : In the first part, my reactants flow in. The reaction is started and generates heat as a result. It then expands slightly into the second part where after it ...
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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
827 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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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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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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How to derive the Karman-Howarth-Monin relation for anisotropic turbulence?

I find the derivation of the Karman-Howarth-Monin relation in the book Turbulence from Frisch (1995) a bit to short. Can someone point me to a more detailed derivation of that relation, if possible in ...
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What is the mystery of turbulence?

One of the great unsolved problems in physics is turbulence but I'm not too clear what the mystery is. Does it mean that the Navier-Stokes equations don't have any turbulent phenomena even if we solve ...
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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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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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503 views

What is the theoretical justification for a fluid flow's being irrotational?

I am not a fluid dynamicist, and I really just began thinking about this problem as my curiousity drew me into building an answer for the question What really allows airplanes to fly?. It is very ...
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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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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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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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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
845 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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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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Is there an analytical solution for fluid flow in a square duct?

I couldn't find one but assumed it must exist. Tried to find it on the back of an envelope, but got to an ugly differential equation I can't solve. I'm assuming a square duct of infinite length, ...
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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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Reynolds Average Navier Stokes equations and turbulence scale

To obtain the time average of an unsteady term like $\frac{\partial u_{i}}{\partial t}$ by definition we perform the following: \begin{align} \overline{\frac{\partial u_{i}}{\partial t}} &= \...
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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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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 ...