Lecture 20: Ideal Fluids and Euler Equations
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Overview
Steven M. Wise derives the ideal-fluid model from incompressibility and the spherical Cauchy stress, obtaining constant density, zero velocity divergence, and the Euler momentum equation without viscosity. He checks material frame indifference and the mechanical energy inequality, then develops flow-line definitions, potential-flow conditions, vorticity preservation, and the Bernoulli equation for conservative body forces.
Key takeaways
- For an ideal fluid with uniform initial density, mass conservation and incompressibility together imply that density stays constant and div v = 0 at every time.
- The ideal-fluid Cauchy stress S = -pI supports only normal traction, so the Euler momentum equation has no viscosity or shear-stress term.
- Pressure is determined only up to an arbitrary function of time because adding f(t) does not change grad p in the Euler equations.
- Ideal-fluid stress does zero mechanical power: S:L = -p div v = 0, satisfying the mechanical energy inequality exactly.
- In a simply connected domain, curl-free velocity is equivalent to a velocity potential; with a conservative body force, initially zero vorticity remains zero.
- For irrotational ideal-fluid motion under conservative body forces, ∂φ/∂t + ½|v|² + p/ρ + β is spatially constant, though it may vary with time.
Chapters
- Chapter 6 assumes temperature is constant in space and time, setting aside heat-convection effects.
- The Eulerian model has 16 field components but only 10 balance equations, so six constitutive relations are needed to close it.
- Candidate fluid models must satisfy material frame indifference and the mechanical energy inequality associated with the second law.
- An ideal fluid starts with uniform reference mass density and remains incompressible during motion.
- Its Cauchy stress is spherical: S = -pI, where pressure p acts as a multiplier enforcing incompressibility.
- The stress produces normal traction, T = -pn, but no shear traction; its spherical form also satisfies angular-momentum balance.
- Local mass conservation is expressed as the material density rate plus density times div v equaling zero.
- Incompressibility gives div v = 0, so the material rate of density is zero.
- Because the initial density is uniform, density remains the same positive constant everywhere and at every time.
- Substituting S = -pI into linear momentum balance gives constant-density acceleration equal to -grad p plus density times body force.
- The ideal-fluid system couples three velocity components with pressure through four equations: momentum balance and div v = 0.
- Adding any spatially uniform function f(t) to pressure leaves its gradient unchanged, so the Euler equations determine pressure only up to a time-dependent offset.
- The stress is decomposed into an active part and a reactive part: the ideal-fluid active stress is zero, while pressure supplies the reactive stress needed for incompressibility.
- The material constraint is det F = 1, and it is unchanged under a superposed rotation Q because det Q = 1.
- The zero active stress transforms as Q0Qᵀ = 0, while rotational invariance of the constraint supports material frame indifference.
- The rate-of-deformation tensor is the symmetric part of grad v, and ideal-fluid stress is S = -pI.
- The stress power S:L reduces to -p div v, which is identically zero under incompressibility.
- Since the mechanical energy inequality requires nonnegative stress power, the ideal-fluid model satisfies it with equality.
- The setup uses a fixed fluid region D, constant positive density, body force per unit mass, and prescribed initial velocity.
- The no-penetration condition requires v·n = 0 at the boundary, while allowing tangential motion unlike a no-slip condition.
- Compatible initial data must satisfy div v₀ = 0 and v₀·n = 0; the pressure still has its time-dependent additive freedom.
- Once velocity is known, the motion map follows by integrating dφ(X,t)/dt = v(φ(X,t),t) for each fixed material point X.
- A pathline records the trajectory through time of one particle starting from a specified initial position.
- In a fixed container, the velocity solution can be used to reconstruct particle positions as a post-processing step.
- A streamline is obtained by integrating curves through a velocity field frozen at one chosen time.
- A streakline traces fluid particles that have passed through a fixed spatial injection point, such as continuously released dye.
- Pathlines, streamlines, and streaklines generally differ in time-dependent flow, but coincide when the velocity field is steady.
- The velocity gradient splits into the symmetric rate-of-deformation tensor and the skew spin tensor.
- Motion is irrotational when the spin tensor vanishes, equivalently when curl v, the vorticity, is zero.
- Stokes' theorem then gives zero circulation around every admissible closed curve bounding a surface.
- In a simply connected region, curl v = 0 if and only if there is a scalar velocity potential φ with v = grad φ.
- If v is the gradient of a sufficiently smooth potential, equality of mixed second partial derivatives makes its curl zero.
- The same gradient-potential idea applies to conservative body forces, including the gravitational potential near Earth's surface.
- Spatial acceleration is the material derivative of velocity: the local time derivative plus the advective contribution.
- The skew part of grad v connects the spin tensor to curl v and separates rotational effects from the velocity-gradient terms.
- For irrotational flow, the spin and vorticity terms vanish, simplifying the acceleration used in the momentum equation.
- A conservative body force is written B = -grad β, so its curl is zero; pressure forces also have zero curl because they are gradients.
- Taking the curl of ideal-fluid momentum balance yields a vorticity evolution equation without contributions from pressure or the conservative body force.
- The stated result is that zero initial vorticity remains zero, so initially irrotational motion stays potential flow; the derivation is deferred.
- For irrotational motion, velocity can be written v = grad φ; assume also that the body force is conservative with potential β.
- The momentum equation implies that ∂φ/∂t + ½|v|² + p/ρ + β is a function of time only.
- This is Bernoulli's relation for an ideal fluid, with the proof scheduled for the following lecture.
Transcript excerpt
0:05 Okay, welcome back to continuum mechanics chapter 6. This is our first uh of probably four or five lectures in chapter six. That means then we'll have a couple lectures in solid mechanics and I hope to get back to thermal fluid mechanics after that because I think the the uh various versions of the flu of navier sto equations that involve heat convection are quite interesting and useful for you guys useful for maybe to to have seen. We're going to start talking about isothermal fluids in this chapter. So this is uh in this chapter we assume that the temperature is always fixed um constant in in time and space and this puts some constraints on our
0:58 modeling. Um it's kind of actually the more um standard uh place of introduction to continuum mechanics. In fact, when I learned continuum mechanics, I didn't know that there was a non-isothermal version. Um, so this is kind of more the standard introduction you get to continuum mechanics, the isothermal introduction, that is to say, where you start talking about things like Navier Stokes and the Koshy Navier equations, but no temperature variation at all. All right. So uh what is it about thermal isothermal uh fluid models that we need to know? So in the oilarian description the big picture is that there are going to be 16 total fields or
1:46 components of fields that we need to solve for and therefore a closed system would have to have six at least 16 equations. And then to add on top of that, we'll also need things like initial conditions and boundary conditions as well. The problem is for these uh 16 equations, we're only going to have 10 equations to balance them. That means we need some sort of way of closing the
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Summary, takeaways, and chapters were generated by AI from the video's transcript and may contain errors. The video belongs to its creator, Steven M. Wise.