# What is the exact meaning of “dot product of the pressure and viscous force coefficients on each face with the specified force vector”?

I studied a thesis about a axial turbine and but i didn't understood the bold part of following sentences. What is the exact meaning of "dot product of the pressure and viscous force coefficients on each face with the specified force vector"? This thesis is performed by using Fluent.

*The forces coefficients (CA and CT) on blade surface were calculated by summing the dot product of the pressure and viscous force coefficients on each face with the specified force vector.

• A little more context is needed... Could you provide the title of the thesis and a page number (and a link to the thesis, if possible)? The way it is worded, it sounds like there are coefficients which form a vector, not that pressure is a vector. – Paul Oct 29 '15 at 4:07
• I assume it's pressure force and viscous force, not just pressure. – Kaster Oct 29 '15 at 4:19

I am unsure of the context of you question but perhaps this will give some insight.

There is a pressure and viscous stress contribution to the force on a boundary. This force can be calculated from the Navier-Stokes equations by:

$$F=\int_{V}\left[\boldsymbol{\nabla}p+\mu\triangle\boldsymbol{v}\right]dV$$

Rewriting this slightly to:

$$F=\int_{V}\boldsymbol{\nabla}\cdot\left[p\boldsymbol{I}+\mu\boldsymbol{\nabla}\boldsymbol{v}\right]dV$$

We can transform this integral to a surface integral using the divergence theorem:

$$F=\int_{S}\left[p\boldsymbol{I}+\mu\boldsymbol{\nabla}\boldsymbol{v}\right]\cdot\boldsymbol{n}dS$$

where $n$ is the unit normal vector to the boundary.