Could anyone give me a clear definition for a tensor, whose rank is 3

rank 0 = scalar Rank 1 = vector (scalar + direction) Rank 2 = (scalar + 2 directions) Rank 3 = ?????

Thanks

A tensor of rank 3 is a 3 by 3 matrix. The definition is different, but you do not need it.

The strict definition of a tensor is that it is an object that follows certain transformation rules. A tensor of rank R and order n (dimension) has n^R components (~numbers). The reason you group objects together and call them tensors is that these transformation laws are shared by many properties in nature.

I can't think of an example of a 3:rd rank tensor that describes an easily understood concept in nature, so it is a bit difficult to interpret it in more concrete terms, but there are plenty of lower rank examples as you point out:

0: temperature, density, ... (also called scalar properties)

1: velocity, vorticity, ... (also called vector properties)

2: stress, strain, inertia, ... (usually called tensor properties, implicitly meaning 2nd rank tensors).

The name "tensor" originates from "tension" (stress), but as always, when mathematicians get hold of things they turn it into something much more abstract and general. General curvilinear tensors of arbitrary order and rank is a bit complex and not very frequently used in fluid dynamics. *Cartesian* *3rd* order tensors are commonly used in fluid dynamics though. However, you can just view this as a compact and practical notation for scalar, vector and "2nd rank tensor" properties. To my knowledge there are no properties in fluid dynamics that are 3:rd rank tensors, so don't worry too much about it.

Your interpretation in terms of scalar + #directions sounds okay for 1st and 2nd rank tensor properties, but I don't know how to extend it to 3rd rank tensors in a meaningfull manner. Already thinking about stress (2nd rank tensor) as a vector (scalar + direction) assigned to every direction (plane) is becoming a bit abstract. I just view a 3rd rank tensor as a mathematical object obeing certain transformation laws.

Not sure about a clear definition (look in a maths book). A tensor of rank 3 is a quantity represented by the tensor product of 3 base vectors. For vectors A and B, the scalar product A.B is a scalar quantity, the vector product AxB is a vector and the tensor product AB is a tensor of rank 2 (often a cross with a circle around is used for the operator). Tensors of rank 3 typically arise by taking the gradient of tensors of rank 2 (eg stress).