# Frictionless and inviscid

In the book Fluid Mechanics by Robert A. Granger, there is a study question, 4.10, asking

"How can frictionless real fluids exist and inviscid fluids not exist?"

Please note that this is no homework-question, but is part of a self-study. I have tried to google around, but could not find any answers. I have seen several webpages stating that frictionless is the same as inviscid, for instance

http://wwwcourses.sens.buffalo.edu/mae335/files/assignment/Lecture_Notes_11_03.pdf

• Inviscid is non viscous? Jun 25 '15 at 12:27
• What about superfluids? They are inviscid fluids, surely, albeit due to quantum effects rather than normal 'fluid mechanics' effects. Jun 25 '15 at 12:53
• @HritikNarayan, yes Jun 25 '15 at 12:58
• I suspect the author is not considering exotic fluids and the like, but rather ordinary fluids. I suspect also the answer is in some clever tautology and a matter of definition, considering the space of all fluids. There are real fluids and ideal fluids. Some real fluids can be considered as ideal fluids but ideal fluids in a strict sense are not real. All real fluids have viscosity --so the second half of the statement is true. The first part --- I am stuck Jun 25 '15 at 15:14
• @docscience I agree -- I think the author is trying to make the reader understand the difference between mathematical assumptions/approximations and physical properties. Jun 25 '15 at 15:22

I think this is a pretty ridiculous question (not yours, the textbook authors'). But my guess would be that the distinction is made between inviscid fluids which are governed by the Euler equations:

$$\frac{\partial u_i}{\partial t} + \frac{\partial u_i u_j}{\partial x_i} = -\frac{1}{\rho} \frac{\partial p}{\partial x_i}$$

and frictionless fluids which are governed by the Navier-Stokes equations:

$$\frac{\partial u_i}{\partial t} + \frac{\partial u_i u_j}{\partial x_i} = -\frac{1}{\rho}\frac{\partial p}{\partial x_i} + \nu \frac{\partial^2 u_i}{\partial x_i \partial x_j}$$

where $\nu = 0$.

In other words, inviscid is a mathematical assumption where we assume the viscous term does not exist in the equation at all (which changes the mathematical nature of the equation) and frictionless is a physical property of the fluid where the viscosity is zero (which has the exact same effect as being inviscid).

Poor question, but I think they want you to understand the difference between mathematical assumptions and physical properties.

• I think you have it backwards. Wouldn't $\mu = 0$ for an inviscid flow and the $\nabla ^{2} \textbf{v} = 0$ for a frictionless flow? Oct 11 '16 at 3:08