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

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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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Is a 3D or 2D Poisson's equation separable or non sperable? [migrated]

Can someone please explain to me if a 2/3D Poisson's equation is separable or non separable? Thank you
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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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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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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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Acoustic/Gravity waves subject to constant wind

I'm trying to model an acoustic/gravity wave (atmospehric gravity waves) propagation through an idealized atmosphere but I'm struggling understanding the results I'm suppose to get. The atmospheric ...
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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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What is the relationship between the Boltzmann Transport Equation and the Navier-Stokes Equation?

What is the relationship between the Boltzmann transport equation and the Navier-Stokes equation? Can the Navier-Stokes equation be derived from a moment of the Boltzmann transport equation?
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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 ...
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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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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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203 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: ...
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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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244 views

Divergence of Cauchy Stress Tensor

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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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 ...
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3answers
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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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3answers
276 views

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
102 views

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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177 views

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 ...
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Need to differentiate Pressure from normal stress in Navier stokes equation

Can someone please explain why pressure term is differentiated from the normal stress term while deriving the Navier-Stokes equation? $$ \frac{d Fx}{dV} = -\frac{\partial P}{\partial x} + ...
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Derivation of the equation of motion of a viscous fluid by Landau & Lifshitz

I am trying to follow Landau and Lifshitz, from the Volume 6 (Fluid Mechanics) of the Course of Theoretical Physics, on their derivation of the momentum equation for a newtonian viscous fluid, but I ...
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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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129 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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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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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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115 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 ...
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st. venant shallow water equations for pipe flow

As far as I know from the fluid dynamics class, St. Venant equations (the shallow water equations) are derived by depth-integrating the Navier-Stokes equations. This depth-integrating is done with the ...
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60 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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Existence and uniqueness of solutions to $\nabla^a T_{ab}$ 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: $$ ...
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1answer
197 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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239 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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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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489 views

Torque on a rotational cylinder in viscous fluid

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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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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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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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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156 views

How can I model computationally or experimentally the flow in my aquarium tank?

I have a 60-gallon aquarium tank and I have always wondered about the flow of water in the tank. Let's represent the flow of water in terms of a direction field representing velocity, so if we see the ...
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184 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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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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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 ...
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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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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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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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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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201 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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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 ...
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307 views

Navier-Stokes equations: conservation of momentum

The first Navier-Stokes equation (conservation of mass) says: $\vec \nabla \cdot \vec v=0$ For a stationary flow, the l.h.s of the second equation is (conservation of momentum): $\rho \frac{D\vec ...
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Acceleration of a steady line vortex

In a question, I have to find the acceleration of a fluid parcel in a steady line vortex. I am given that $u_\theta=\frac{A_0}{r}$. So for a steady line vortex, the parcels are following circular ...