show the most important inviscid flow equations, for exmaple bernoullis
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PK Bot’s answer:
Inviscid flow equations are used when viscous forces are negligible in some meaningfully large fraction of the flow domain. The inviscid approximation is always singular, meaning it is always incorrect somewhere in the domain (at a solid wall and very close to solid walls). Here are some of the most important inviscid flow equations:
Bernoulli Equation a.k.a Euler equation in streamwise direction
The Bernoulli equation is a statement of conservation of energy (or, equivalently, conservation of streamwise momentum) for a fluid element moving along a streamline. For steady, incompressible, inviscid flow, it is given by:
p + \frac{1}{2} \rho U^2 + \rho g z = B
where:
- p is the pressure.
- \rho is the fluid density.
- U is the fluid velocity.
- g is the gravitational acceleration.
- z is the elevation above a reference point.
and B is the Bernoulli constant, which is uniform along a streamline.
Euler equation in normal direction
In streamline coordinates, the normal component of the Euler equations relates the curvature of streamlines to gradients of the hydrostatic constant:
\frac{\partial k}{\partial n} = -\frac{\rho U^2}{R}
where:
- k = p + \rho g z is the hydrostatic constant.
- n is the normal coordinate to the streamline.
- R is the radius of curvature of the streamline.
Euler equation in binormal direction
In streamline coordinates, the binormal component of the Euler equations says that the hydrostatic constant is conserved along the binormal direction:
\frac{\partial k}{\partial \ell} = 0
where:
- k = p + \rho g z is the hydrostatic constant.
- \ell is the binormal coordinate.
Steady Euler Equations
In general the steady Euler equations describe the motion of an inviscid fluid. For steady flow, they are expressed as:
\rho \boldsymbol{\vec{u}} \cdot \nabla \boldsymbol{\vec{u}} = -\nabla p + \rho \boldsymbol{\vec{g}}
We usually find the formulations in streamline coordinates most useful for analytical work, and the general formulation most useful for computational work.
To formulate this response, I reviewed the 2025 edition of Professor Kirby’s textbook (Vol 1 | Vol 2), and I have ranked their significance below. If they are marked as relevant, I encourage you to take a look at them:
Textbook sections:
- Appendix F.3: Volumetric Viscous Force Part 1 (HIGHLY RELEVANT)
- Chapter 5 Engineering Models: Steady Euler Equations (HIGHLY RELEVANT)
- Chapter 5.4: Euler Equation: Streamwise Direction i.e. Bernoulli Equation Part 1 (HIGHLY RELEVANT)
- Chapter 5.1: Inviscid Incompressible Flow (HIGHLY RELEVANT)
- Appendix D.2: Rederivation of Bernoulli (HIGHLY RELEVANT)
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