Questions tagged [navier-stokes]

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

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253 views

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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461 views

Origin of pressure gradient in Navier-Stokes integral

I am not understanding the origin of the second term on RHS of momentum conservation equation (cf. the Wiki page), $$ \frac{\partial}{\partial t}\int_V\rho\mathbf u\,dV=-\oint_S\left(\rho\mathbf u\...
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Velocity profile of a viscously damped wave

For a test case, I want to determine the velocity profile of a viscously damped standing wave. By linearizing the density ($\rho=\rho_0+\rho'$) and velocity ($ux=ux'$), the continuity and Navier-...
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No diffusion term in conservation of mass in Navier-Stokes equations?

I have followed derivations of the Navier-Stokes equations and I can see how the various terms arise in the "main equation", the momentum conservation equation. However I don't understand why the ...
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1answer
499 views

Does a whirlpool(vortex in water) continue in air(vortex in air),and when does a vortex stop?

First part: The question is both about the continuity of the water vortex(whirlpool) to vortex in air in time and in space. About continuity in time,does the vortex of the water slowly produce a ...
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Why can't the Navier Stokes equations be derived from first principle physics?

At the 109th UCLA Faculty Research lecture, Seth Putterman gave a talk on Sonoluminescence. During the lecture he emphasized that "The Navier Stokes equations cannot be derived from first principles [...
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2answers
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Incompressible Navier-Stokes pressure solve in simulations

I am a complete newcomer when it comes to fluid simulations. I'm currently working through some tutorials to understand the idea of of the discretized Navier-Stokes equations for numerical simulations....
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1answer
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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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1answer
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Exact solution to a 2D/3D Poiseuille flow in a channel

Hi everyone and thank you in advance for any help. I am struggling to find an analytical solution to either a 2D or 3D Poiseuille flow in a rectangular duct. All I can find is 1D example. Can someone ...
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2answers
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Why is the solution to the Blasius boundary layer problem self-similar?

In every course or textbook that I encountered so far, the authors transform the Navier-Stokes equations of the Blasius boundary layer problem into the Blasius ODE. The problem with many of those ...
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Could Navier-Stokes equation be derived directly from Boltzmann equation?

I know how to derive Navier-Stokes equations from Boltzmann equation in case where bulk and viscosity coefficients are set to zero. I need only multiply it on momentum and to integrate it over ...
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Fluid flow: Force acting on the fluid and the Navier-Stokes equation

Consider a one dimensional fluid flow in a rectangular tube. Typical streams are the poiseuille streams. Consider the case in wich we apply a force on the fluid. The Navier-Stokes equation (for ...
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0answers
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Why hydrodynamic turbulence without heat terms matters?

A lot of research is made on turbulence in "pure" Navier-Stokes equations (NSE). There is a notion of energy cascade when energy comes from larger scales to lower scales and than dissipate. However ...
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Solving inhomogeneous Stokes equation

I want to solve the Stokes inhomogeneous equation, i.e. $$\nabla^2 \vec v -\nabla P = \vec f(r,\theta)$$ $$\nabla\cdot\vec v=0$$ where $\vec f$ is irrotational, i.e. $\partial_y f_x - \partial_x ...
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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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Bulk and dynamic viscosity in the atmosphere

I'm studying the physics of the atmosphere but I'm struggling with the matter of viscosity (Navier-Stokes equation) for gravito-acoustic waves. From Landau-Lifschitz : $$ (T)_{ij} = -p\delta_{ij} + \...
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1answer
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Why hasn't an exact solution to the Navier-Stokes equations been found?

I once tried to read the Millennium Problems statement about the Navier-Stokes equations, decided it was beyond me, and left it at that. But now I am studying general relativity and through Physics ...
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1answer
685 views

Can I use Runge-Kutta to solve these equations?

Edit: I'm going to give some more background and derivation to show how I got to these equations. I am basically following the derivation that is found in the appendix of the following paper: R.M. ...
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1answer
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Conservation Equations forming a Determinate Set

I was reading "molecular gas dynamics" by Bird and came across the the statement that For conservation equations to form a determinate set shear stress and heat flux must be expressed in terms of ...
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1answer
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Divergence of Cauchy Stress Tensor [closed]

On the wikipedia page for the Cauchy Momementum Equation, it's stated that the equation can be written as $$\rho \frac{D\,\textbf{v}}{D\,t} = \nabla \cdot \sigma + \textbf{f}$$ Where $\sigma$ is ...
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2answers
772 views

Stress Force - Understanding Cauchy Stress Tensor

I've been trying to understand the derivation for the Cauchy Momentum Equation for so long now, and there is one part that every derivation glides over very quickly with practically no explanation (I'...
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3answers
393 views

How do the flow equations relate to the actual situation?

This question might seem silly but I'll try to make it clear. It's a question (I think) about partial differential equations systems in general, but since currently I'm studying fluid mechanics I'll ...
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4answers
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Index notation with Navier-Stokes equations

This is an index-notation question rather then the NS one: For incompressible flow and Newtonian fluid, the continuity equation is denoted with: $$\frac{\partial u_i}{\partial x_i} = 0, $$ which ...
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1answer
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Difference between a “source dipole” and a “force dipole”

I know quite well what a dipole is and in general what multipole moments are (in the context of, for instance, electrodynamics). What I find myself confused by is something called a "force dipole" in ...
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3answers
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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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1answer
386 views

Is there a normalized form of the Euler equation discretized with finite volumes?

I want to calculate a flux on my fpga using the Euler equations with the finite volume method. Unfortunately the values of the state variables differ a lot. For example the pressure has a value of ...
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1answer
272 views

Shallow water wave question from Acheson's book

I am learning Fluid mechanics by reading Acheson's book entitled "Elementary Fluid Dynamics". Below is from problem 3.1. Consider the Euler equation for an ideal fluid in the irrotational case. We ...
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2answers
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Analytical solution of transient barometric formula for fluid in one dimension

Consider a column of fluid of length $L$, with initial density $\rho_0$ and initial velocity ($u_0 =0$) everywhere. Now at time $t=0$ gravity is switched on. No-slip boundary conditions are assumed at ...
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2answers
689 views

Question on using Leibniz formula to derive thin-film equation from Navier-Stokes

I actually posted this to math.stackexchange.com a few months ago but never got any answers. I am trying to work through the derivation in this paper by Petr Vita, which derives a thin-film ...
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2answers
408 views

How can a gas support tensile stresses?

In working through a rigorous derivation of the compressible Navier-Stokes equations, I find that the momentum flux in the X-direction should be driven not only by the normal pressure gradient $\frac{\...
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1answer
113 views

Calculate Euler equations of fluid dynamics without division?

I'm working on the calculation of the Euler equations with the finite volume method. Unfortunately I'm not allowed to do a division. So I'm wondering if there's a form which does not need a division. ...
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1answer
257 views

Existence and uniqueness of solutions to $\nabla_a T^{ab}=0$ in general (or special) relativity

The equation in the title of this question can be a relativistic analogue of the Navier-Stokes equation (in the sense that, in the low-velocity limit, it reduces to Euler's equation when $T^{ab}$ is ...
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Euler equation with single state variables [duplicate]

I want to simulate a flow using the Euler equations. For this reason I'm wondering if there's a modified version of the Euler equations. At the moment my formula looks like this: $$ \frac{\partial}{\...
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1answer
752 views

What is the physical application of Navier-Stokes existence and smoothness?

Recently, mathematician Mukhtarbay Otelbaev published a paper Existence of a strong solution of the Navier-Stokes equations, in which he claim that he solved one of the Millennium Problems: Existence ...
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1answer
948 views

Help with Modeling a Liquid Vortex. (Related to General Fusion)

I want to model liquid lead swirling in a sphere. This is connected to General Fusion’s fusion machine. A 55 million dollar, Jeff Bezos funded, 60 person company trying to change the world with ...
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1answer
585 views

Explicit form of the entropy production in hydrodynamics

I'm trying to understand how hydrodynamics arise from a precise, mathematical formulation of thermodynamics, learning mostly from Landau's "Hydrodynamics". So Landau starts from formulating the ...
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1answer
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Torque on a rotational cylinder in viscous fluid [closed]

I've been stuck on what I'm pretty sure is a simple part of a larger question. It's a cylinder (radius a) spinning in a viscous fluid. It's rotating at rate $\Omega$ .During this question we get that ...
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0answers
64 views

Has this boundary condition been used in fluid flow?

I would like to know whether anyone has seen a boundary condition used in a fluid flow problem, of the following type. Suppose viscous incompressible fluid is to the left of a plane $x_1=a$, so the ...
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3answers
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What is the body force in the Navier Stokes Equations?

The incompressible Navier Stokes equations are: $\rho(\frac{\partial v_i}{\partial t} + v_j\frac{\partial v_i}{\partial x_j}) = -\frac{\partial p}{\partial x_i} + \mu\frac{\partial^2 u_i}{\partial ...
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3answers
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Physical interpretation of the change of diffusion term in navier stokes equations

In the Navier-Stokes Equations, there is one term accounting for convective flow and one term for diffusive flow. At high flow rates, the diffusive term becomes much smaller compared to convective ...
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1answer
320 views

What is the apparent viscosity in shear thinning turbulent flow through a pipe? [duplicate]

The explanation of shear rate in laminar flow is straightforward: We imagine small layers of fluid that glide on each other. Now, in turbulent flow, this does not work as there are no layers. If I ...
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1answer
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What's the shear rate in a turbulent flow?

The explanation of shear rate in laminar flow is straightforward: We imagine small layers of fluid that glide on each other. Now, in turbulent flow, this does not work as there are no layers. I'm not ...
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1answer
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Momentum Equations for Micropolar Fluid

I am looking for the derivation of Momentum Equations for Micropolar Fluid $$\rho\frac{D\vec V}{Dt}=-\nabla p+(\mu+k_1^*)\nabla^2\vec V+k_1^*(\nabla\times\vec N^*)+\vec J\times\vec B ,\\ \rho j^*\...
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2answers
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What are the assumptions of the Navier-Stokes equations?

I wanted to model a real life problem using the Navier-Stokes equations and was wondering what the assumptions made by the same are so that I could better relate my entities with a 'fluid' and make or ...
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3answers
494 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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0answers
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From Navier–Stokes equations to Euler, Bernoulli, etc [closed]

How can one go from the 3D compressible Navier-Stokes equations to the simpler Euler equations, Bernoulli's equation and other fluid dynamic equations?
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1answer
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The Euler equations as a RNG fixed point

In this paper at the at the beginning of the last paragraph on p.2 it is said, that the Euler equations, which are an infinite Reynolds number limit of the Navier-Stokes equations, arise as an RNG ...
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1answer
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Difference between Eulerian and Lagrangian formulation of Fluid Dynamics

Difference between Eulerian and Lagrangian formulation of Fluid Dynamics. I am completely new to fluid mechanics. Until now definition $F = ma$ was sufficient for me to solve any rigid body problems ...
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1answer
297 views

Isn't this wikipedia equation of navier-stokes actually wrong? [closed]

There is a wikipedia page about NS Existance and Smoothness It seems to me that the Navier Stokes equations is wrong? (because in one side of equal sign unit is $\frac {m}{s^2}$ but in other side it ...
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3answers
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What do mathematicians mean by Navier Stokes existence and smoothness problem?

I still don't know what mathematicians mean by Navier-Stokes existence and smoothness. Since there is a reward for proving it, it seems important to them. (in past several months I've read online ...