### Introductory Fluid Mechanics for Physicists and Mathematicians

Fluid properties can vary continuously from one volume element to another and are average values of the molecular properties. The continuum hypothesis can lead to inaccurate results in applications like supersonic speed flows, or molecular flows on nano scale. Those problems for which the continuum hypothesis fails, can be solved using statistical mechanics. To determine whether or not the continuum hypothesis applies, the Knudsen number , defined as the ratio of the molecular mean free path to the characteristic length scale , is evaluated. Problems with Knudsen numbers below 0.

The Navier—Stokes equations named after Claude-Louis Navier and George Gabriel Stokes are differential equations that describe the force balance at a given point within a fluid. Occasionally, body forces , such as the gravitational force or Lorentz force are added to the equations. Solutions of the Navier—Stokes equations for a given physical problem must be sought with the help of calculus.

In practical terms only the simplest cases can be solved exactly in this way. These cases generally involve non-turbulent, steady flow in which the Reynolds number is small. For more complex cases, especially those involving turbulence , such as global weather systems, aerodynamics, hydrodynamics and many more, solutions of the Navier—Stokes equations can currently only be found with the help of computers.

This branch of science is called computational fluid dynamics. In practice, an inviscid flow is an idealization , one that facilitates mathematical treatment. In fact, purely inviscid flows are only known to be realized in the case of superfluidity.

## Math Mathematical Fluid Dynamics

Otherwise, fluids are generally viscous , a property that is often most important within a boundary layer near a solid surface, [2] where the flow must match onto the no-slip condition at the solid. In some cases, the mathematics of a fluid mechanical system can be treated by assuming that the fluid outside of boundary layers is inviscid, and then matching its solution onto that for a thin laminar boundary layer. For fluid flow over a porous boundary, the fluid velocity can be discontinuous between the free fluid and the fluid in the porous media this is related to the Beavers and Joseph condition.

Further, it is useful at low subsonic speeds to assume that a gas is incompressible —that is, the density of the gas does not change even though the speed and static pressure change.

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• A Mathematical Introduction to Fluid Mechanics | Alexandre J. Chorin | Springer.
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A Newtonian fluid named after Isaac Newton is defined to be a fluid whose shear stress is linearly proportional to the velocity gradient in the direction perpendicular to the plane of shear. This definition means regardless of the forces acting on a fluid, it continues to flow. For example, water is a Newtonian fluid, because it continues to display fluid properties no matter how much it is stirred or mixed.

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A slightly less rigorous definition is that the drag of a small object being moved slowly through the fluid is proportional to the force applied to the object. Compare friction. Important fluids, like water as well as most gases, behave—to good approximation—as a Newtonian fluid under normal conditions on Earth. By contrast, stirring a non-Newtonian fluid can leave a "hole" behind. This will gradually fill up over time—this behaviour is seen in materials such as pudding, oobleck , or sand although sand isn't strictly a fluid.

Alternatively, stirring a non-Newtonian fluid can cause the viscosity to decrease, so the fluid appears "thinner" this is seen in non-drip paints. There are many types of non-Newtonian fluids, as they are defined to be something that fails to obey a particular property—for example, most fluids with long molecular chains can react in a non-Newtonian manner.

The constant of proportionality between the viscous stress tensor and the velocity gradient is known as the viscosity. A simple equation to describe incompressible Newtonian fluid behaviour is. For a Newtonian fluid, the viscosity, by definition, depends only on temperature and pressure , not on the forces acting upon it. If the fluid is incompressible the equation governing the viscous stress in Cartesian coordinates is.

If the fluid is not incompressible the general form for the viscous stress in a Newtonian fluid is. If a fluid does not obey this relation, it is termed a non-Newtonian fluid , of which there are several types. Non-Newtonian fluids can be either plastic, Bingham plastic, pseudoplastic, dilatant, thixotropic, rheopectic, viscoelastic.

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In some applications another rough broad division among fluids is made: ideal and non-ideal fluids. An Ideal fluid is non-viscous and offers no resistance whatsoever to a shearing force. An ideal fluid really does not exist, but in some calculations, the assumption is justifiable. One example of this is the flow far from solid surfaces. In many cases the viscous effects are concentrated near the solid boundaries such as in boundary layers while in regions of the flow field far away from the boundaries the viscous effects can be neglected and the fluid there is treated as it were inviscid ideal flow.

Main article: Newtonian fluid. Physics portal. Fluid mechanics at Wikipedia's sister projects.

Thank you for any recommendations! Please don't just post random books you found on Amazon I can do that myself.

I am interested in the opinion of people who have done some fluid dynamics, the more the better, and know about the field. Batchelor is a classic and is considered as the Bhagavad Gita of fluid dynamics. I have read this book as an undergrad and hence the knowledge required is just high school mathematics and physics. Recent texts, in my opinion, are unfortunately biased too much towards computational fluid dynamics, than explaining the mathematical and physical underpinning of the fluid dynamics.

I think it's well suited for a math student, as opposed to most books which have a more engineering flavor. You've probably found the books you need now.

## MEK1300 – Introduction to Fluid Mechanics

Anyhow 'The Introduction to computational fluid dynamics' by Versteeg is very good. I have enjoyed the following books: Elementary Fluid Dynamics by Acheson. The book by Kundu and Cohen, Fluid Dynamics is also extremely enjoyable. Besides, I have found the following lecture notes useful:. Dont ignore the CFD-oriented books. Good ones such as Godlewski and Raviart "Numerical approximation of hyperbolic systems of conservation laws" rely on solid theoretical underpinning. Another addition to your list might be M. Lighthill "An informal introduction to theoretical fluid dynamics".

Be aware that there are surprisingly large differences between compressible and incompressible flow. They sometimes look like almost different subjects. Sign up to join this community.