2. Powerful tool For Effective study and to Understand Flow Devices…… P M V Subbarao Professor Mechanical Engineering Department I I T Delhi Tensor Notation for Viscous Fluid Flow

3. Tensors : Vectors of Second Order Second order vectors are more complex constructs. The three projections of this tensor, onto coordinate axes are obtained by inner product. These projections are vectors (not scalars!). In array form the three components of a tensor  are vectors denoted by

4. where each vector has three components therefore  is written in the matrix form

5. Tensor Product The tensor product is a product of two or more vectors, where the unit vectors are not subject to scalar or vector operation. Consider the following tensor operation: The result of this purely mathematical operation is a second order tensor with nine components: The operation with any tensor such as the above second order one acquires a physical meaning if it is multiplied with a vector (or another tensor).

6. Einstein's Notation for creation of a tensor A tensor can be written as dyadic product. Dyadic product of two vectors is a tensor such that Dyadic product has the following properties

7. Differential Operator  The spatial differential operator,  (nabla, del) which has a vector character. In Cartesian coordinate system, the operator nabla is defined as: Gradient of a scalar, T

8. Tensor Product of  and v This operation is called the gradient of the velocity vector v. Its result is a second tensor. Using the index notation, the gradient of the vector v is written as:

9. The components of Gradient of A vector

10. The material acceleration The material acceleration is:

11.  applied to a product of two or more vectors Using the Leibnitz's chain rule of differentiation:

12. Scalars, Vectors and Tensors