# What is Incompressible: Definition and 67 Discussions

In fluid mechanics or more generally continuum mechanics, incompressible flow (isochoric flow) refers to a flow in which the material density is constant within a fluid parcel—an infinitesimal volume that moves with the flow velocity. An equivalent statement that implies incompressibility is that the divergence of the flow velocity is zero (see the derivation below, which illustrates why these conditions are equivalent).
Incompressible flow does not imply that the fluid itself is incompressible. It is shown in the derivation below that (under the right conditions) even compressible fluids can – to a good approximation – be modelled as an incompressible flow. Incompressible flow implies that the density remains constant within a parcel of fluid that moves with the flow velocity.

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1. ### What is the most incompressible elastomer?

Hi all! Usually, one would model a rubber as incompressible (##\nu \rightarrow \infty## or equivalently ##\kappa \rightarrow \infty##, where ##\nu## is Poisson ratio and ##\kappa## is bulk compressibility). However, I am trying to use rubber in an application where performance will improve the...
2. ### The Euler Equation and Incompressible Fluid Theorems

$$\frac{Du}{Dt} = -\frac{\nabla p}{\rho} - \nabla \chi$$I re-write the Euler equation for incompressible fluid using suffix notation $$\frac{\partial u_i}{\partial t} + u_j \frac{\partial u_i}{\partial x_j} + \frac{\partial}{\partial x_i} \left(\frac{p}{\rho} + \chi \right) = 0$$what theorems...
3. ### Three doubts in a paragraph on the equilibrium of Incompressible fluids

These are the images from Sommerfeld’s Lectures on Theoretical Physics, Vol 2 chapter 2, section 6, Equilibrium of Incompressible Fluids. Image 1 Image 2 Doubt 1 : What does it mean for a force to act on a fluid volume? Force acts on a point, force may act on a surface but I’m unable to...
4. ### Fluid dynamics and incompressible fluids

Hi sorry about the way I've posted I'm new to this site. Anyway basically I've been set this question which should be attached to this post, I have attempted to do this question but I'm having trouble in forming an equation in the first place. I'm unsure where to start, I understand I need to...
5. ### Airfoil drag prediction in incompressible and inviscid flow

What is the most elegant way of predicting drag on an airfoil in inviscid and incompressible flow? Or is this still an open problem? It seems like the consensus for airfoils in incompressible and inviscid flow is that they cannot produce drag; or at least, we are not able to predict it...
6. ### Compressible flow or incompressible flow equation?

Hello everyone.. I am using 2-dimension Navier Stokes equation, but I confused that my problem is compressible or incompressible flow form, because if I have initial pressure and temperature and velocity for x-axis only in one grid that they are so very high but the others grid are zero. Can I...
7. ### A What's the meaning of "Incompressible" for FQHE?

Someone says The fractional quantum Hall effect is an incompressible quantum liquid, what's the meaning?
8. ### Aerodynamics course order?

I'm going to be an aerospace/mechanical engineering student at UAH next year and I am coming up with my four year plan. Two courses, "fundamentals of aerodynamics" and "compressible aerodynamics", are not prerequisites for each other, and they are both required. Based on your experience with...
9. N

### Internal pressure due to vessel collapse

Hey All, I have a plastic vessel fully filled filled with an incompressible fluid (Water), at some time this vessel is impacted and crushed on one side (say 5% of the initial volume is lost). Only, I know the volume isn't lost, the fluid (being incompressible) will exert pressure on all...
10. ### I Why incompressible flow does not satisfy energy equation?

I encountered this statement on my lecture notes today, I don't understand why compressible flow needs to have another constraint of energy equation while incompressible flow only satisfies continuity and momentum equation. And how is this energy equation related to the speed of sound?
11. ### A How to Solve Isothermal Incompressible Navier-Stokes for Compressible Fluid?

I did a lot of googling but could not find a satisfying answer to my question, hence a post here. Question: How to solve (or close) the isothermal incompressible Navier-Stokes equations for an isothermal compressible fluid? Situation: We have a compressible fluid, for example a gas. The flow...
12. ### Constant density vs Incompressible

The continuity equation in fluid mechanics is: Do the condition of "constant-density fluid' and 'imcompressible flow' have the same effect on the continuity equation, in that the first two terms disappear? Or is there a difference between these assumptions?
13. ### I Spectral representation of an incompressible flow

Hi PH. Let ##u_i(\mathbf{x},t)## be the velocity field in a periodic box of linear size ##2\pi##. The spectral representation of ##u_i(\mathbf{x},t)## is then $$u_i(\mathbf{x},t) = \sum_{\mathbf{k}\in\mathbb{Z}^3}\hat{u}_i(\mathbf{k},t)e^{\iota k_jx_j}$$ where ι denotes the usual imaginary...
14. ### Theoretical Pipe Flow: Understanding Poiseuille's Law

Hey guys, I am doing an internship and I have had some thoughts about fluid flow that have come up and I am having trouble fully grasping some concepts due to no one being able to thoroughly explain any answer that they might come up with. So I have a crude understanding of some fluid dynamics...
15. ### I Is every conservative vector field incompressible?

So I have found that everyone conservative vector field is irrotational in a previous problem. Based on the relationship irrotational vector fields and incompressible vector fields have, div(curl*F)=0, does that also imply every conservative vector field is incompressible? Kindly, Shawn
16. ### Why should an ideal fluid be incompressible?

Hi, I read that an ideal fluid needs to be frictionless and incompressible. Please explain why, especially the incompressible part? Thanks
17. ### State postulate for incompressible substance

State postulate for a simple compressible system is completely specified by two independent intensive properties. But what about state postulate for a incompressible system. Why it is not so important?
18. ### Boundary layer equations for incompressible flow

Hello everyone. I posted this question in another forum and got no answers so I'll try and re-post it question here. I need to deliver a correct answer to the cited question below to my course responsible teacher. Below is also my own solution and thoughts on the problem but I don't know if I am...
19. ### Boundary layer equations for incompressible flow

Hi. I am new here so I please let me know if I should post this in another forum. I have been struggling for a while with the following homework problem: "State the boundary layer equations for incompressible flow over a solid, weakly curved boundary of a Newtonian fluid. What approximations...
20. ### Potential flow, inviscid flow, incompressible flow

I have a couple of questions regarding several key areas of fluid mechanics: 1. My first question deals with the Navier-Stokes equations. Does incompressible&irrotational flow imply inviscid flow? My answer is yes and here is my thought process. In the incompressible form of the Navier-stokes...
21. ### What is Incompressible material?

I sent a summary about ultrasound elastography to my teacher and somehow I wrote incompressible material in my paper. now he wants me to really explain him what is Incompressible material. I looked at internet about compressibility and Poisson ratio but doesn't still give a good understanding...
22. ### How long will it take for a gas tank to empty?

Hello all, I have an oxygen tank that is 10 L in volume, pressurized at 500 barr (pressure can be adjusted down to 1 barr out of the tube), and I want to 'bubble' the oxygen into a beaker of water. I will simply use a tube (diameter is around 0.5 cm) connected from the tank and with the other...

26. ### Specific heats for an incompressible substance

why specific heats at constant pressure and volume are the same for an incompressible substance?
27. ### Pressure change when volume changes for incompressible fluid

Homework Statement Hi guys, My first post here, I hope someone can help me out with a quick fluid mechanics problem. I'm looking to calculate the pressure increase inside a closed tube full of water when the tube is crushed and therefore its volume reduced. The tube is filled completely...
28. ### Compressible vs. incompressible substance

I'm confused about which phases are considered compressible and which are considered non-compressible. In classical thermodynamics, gases and liquids are considered compressible. This is what the state principle (2 independent properties define all other properties) applies to gases and...
29. ### Why incompressible fluid flow is advantageous in numerical computation

Hi, When we want to solve the Navier-Stokes equations coupled with the conservation of mass for incompressible fluids using the primitive-variable approach, we have to face to the problem that the equation for the continuity equation does not contain the pressure which leads to spurious...
30. ### Water Incompressible: The Impact on Metal Balls and Spheres

A book about liquid mechanics, says that water is considered incompressible , because the amount by which it compresses is too tiny to bother with. My question is , what would happen with a metal ball or sphere filled with water and put into extreme conditions , like in the very center of sun...
31. ### Steady-state incompressible Navier-Stokes discretization

Hi, I would like to solve the steady-state incompressible Navier-Stokes equations by a spectral method. When I saw the classic primitive-variable finite element discretization of the time-dependent incompressible N-S, it turned out that the coefficient matrix of the derivatives of the unknowns...
32. ### Exploring Pressure Energy in Incompressible Fluid Flow

Homework Statement Consider a tube of uniform cross section, let one end (a) of the tube is at height h1 and other end (b) is at height h2, where h2>h. Now an incompressible fluid passing through the tube the workdone to pass the liquid at end (a) P1V1 and workdone to pass the liquid at end...
33. ### Incompressible flow in a funnel

I am having a few issues reconciling Bernoulli's principle and the continuity equation for an incompressible flow in a horizontal funnel where there is significant difference in the area from the start to the end. More specifically, I want to work out exactly what the fluid is doing once it...
34. ### Fluid Mechanics: Incompressible flow and pressure

Most liquids can be assumed to be incompressible, since the Mach-number is much smaller than 1. That means that the density variations are negligible and from the relation between pressure p and density ρ, p=c_s^2 \rho we see that the pressure in constant as well. Now, say that I look at a...
35. ### Low Mach Numbers: Incompressible flow or fluid

Hi When we talk about a fluid moving at low Mach numbers, it is said to be incompressible. But does this mean that the flow is incompressible (i.e., material derivate is zero) or does it imply that the fluid itself is incompressible (constant density)? If anybody has a reference (book...
36. ### Incompressible fluid in horizontal test tube

Homework Statement An incompressible fluid with density ρ is in a horizontal test tube of inner cross-sectional area A. The test tube spins in a horizontal circle in an ultracentrifuge at an angular speed ω. Gravitational forces are negligible. Consider a volume element of the fluid of area...
37. ### Incompressible fluid acquiring momentum

Suppose we have the set up shown in the thumbnail, in which an incompressible fluid moves from left to right, trapped by converging streamlines. The rate of acquisition, F, of momentum by the fluid is surely F = \rho A_2 v_2^2 - \rho A_1 v_1^2 = \rho A_1 v_1^2 \left\{\frac{A_1}{A_2} -...
38. ### Divergence of velocity of incompressible fluid in uniform gravity

Hi! The velocity field as a function of poisition of an incompressible fluid in a uniform acceleration field, such as a waterfall accelerated by gravity can be found as follows: The position is \vec{x}. The velocity field is \vec{v} = \frac{d\vec{x}}{dt}. The constant acceleration field...
39. ### Incompressible vs constant density fluid flow

This may sound like a basic question, but it's just to get it clear: When describing fluid flows, does the term "incompressible" mean exactly the same thing as "constant density"? I was under the impression that if a fluid cannot be compressed, then its density must remain constant for any...
40. ### Compressible vs. Incompressible Flow Equations

As a private study I'm trying to figure out fluid dynamics applied to compressed air systems. Most of the material I am studying considers only incompressible fluid flow. From what I understand about the differences between compressible and incompressible flow in terms of the equations it only...

Advantages/disadvantages of "penalty formulation" for incompressible fluid flow Hi everyone: I'm interested in getting into some fluid flow modeling (e.g., advection-diffusion). I'm a newbie, so sorry in advance for any silly questions. Today I came across the "penalty formulation" and...
42. ### Shear stress on oscillating wall in viscous incompressible fluid

Homework Statement A layer of viscous incompressible fluid of thickness H lies on top of a solid wall that oscillates simple harmonically w/ angular frequency Ω. u(wall)=Acos(Ωt). Ignore the motion of air above the fluid layer and find the shear stress at the wall. (Shear stress on free...
43. ### Incompressible viscous flow

Hello, I'm looking at this problem which states: "A parallel air flow along a semi infinite flat plate has undisturbed parameters as follows: " Then they list the speed, temprature, viscousity and pressure. "Demonstrate that the boundary layer can treated by means of incompressible...
44. ### Is there really any such thing as an incompressible liquid?

Homework Statement Given- Pressure increases with depth. Homework Equations We know that pressure under a depth h is given by pgh. Where p=density of liquid, g= acceleration due to gravity. The Attempt at a Solution Let us take an example. There is an ocean of a liquid which is...
45. ### Density constant along streamline of incompressible fluid

Question: Show that for a steady flow with div.u=0, the density is constant along streamlines. I just don't see how to approach this question without Bernoulli's equation, I can see that the Lagrangian derivative of the density is zero but that doesn't specifically show that the density is...
46. ### (HELP )Compressible vs incompressible flow (QUESTION)

(HELP PLZ)Compressible vs incompressible flow!(QUESTION) Ok! I think I know the difference between the two, which is basically the density. My question is actually when is the equation of properties like (Static temperature, Total temperature, Total pressure, Static enthalpy, dynamic viscosity...
47. ### Calculating Stream Function of Incompressible Fluid from Velocity

Hi everybody, I have been looking for a way to calculate the stream function if I know the velocity \vec{v}=(vx,vy,0) of a bidimentional flux. From the formula given for \vec{v}, I know it is an incompressible fluid.
48. ### Bernoulli for incompressible inviscid flow

[b]1. Investigate and apply equations of stae; speed of sound; continuity; bernoulli for incompressible inviscid flow; shear forces and stresses; laminar flow; flow in boundary layer link to shear forces and viscosity- continuity equation [b]2. [b]3. I have answered 90% of the...
49. ### What is the incompressible fluid?

what is the incompressible fluid?
50. ### Solving Incompressible Flow Homework: Find f(r)

Homework Statement A velocity field is given by \vec {u} = f(r)\vec{x}, r = | \vec{x}| = \sqrt {x^2 + y^2 + z^2} written in rectangular cartesian coordinates, where f(r) is a scalar function. Find the most general form of f(r) so that \vec {u} represents an incompressible flow...