Appendix C. Some Vector And Tensor Operations

TABLES 5.2–5.5 gave details of the most frequently used vector operations (gradient and Laplacian of a scalar, divergence and curl of a vector) in rectangular, cylindrical, and spherical coordinates. Here, we give a few more definitions for rectangular Cartesian coordinates, sufficient for the needs of this book.¹

¹ The reader who needs to learn more about vector and tensor operations, including those in cylindrical and spherical coordinate systems, is referred to Appendix A of the excellent text: R.B. Bird, W.E. Stewart, and E.N. Lightfoot, Transport Phenomena, 2nd ed., Wiley & Sons, New York (2002).

Dyadic Product of Two Vectors

The dyadic product uv of two vectors u and v is a tensor, such that it and its transpose (uv)^T are defined by:

If u is the nabla operator, we then have the following special dyadic products:

Such products are very useful in summarizing, for example, the strain-rate tensor that appears in Section 5.7:

The “Divergence” of a Tensor

Consider first the tensor:

The dot product of the nabla operator with a symmetric tensor, ∇· τ (in which τ_yx = τ_xy, etc.), is a vector with the following components in each of the three coordinate directions:

To obtain the first of these relations, it is as though we had considered the first row of the tensor to be a vector, and we had then taken its divergence, etc. The form ∇· τ enables us to summarize the three momentum balances in compact vector form for the case of variable viscosity, as in Example 5.9.

The “Laplacian” of a Vector

The product of the operator ∇² with a vector v, as in ∇²v, is a vector with the following components:

That is, we obtain three Laplacians of the individual components v_x, v_y, and v_z, one for each coordinate direction. As seen in Section 5.7, this enables a compact representation of the constant-viscosity terms in the Navier-Stokes equations.