# Differential Balances in Turbomachinery

## Abstract

In this and the following chapter, we present the conservation laws of fluid mechanics that are necessary to understand the basics of flow physics in turbomachinery from a unified point of view. The main subject of this chapter is the differential treatment of the conservation laws of fluid mechanics, namely conservation law of mass, linear momentum, angular momentum, and energy. These subjects are treated comprehensively in a recent book by Schobeiri [1]. In many engineering applications, such as in turbomachinery, the fluid particles change the frame of reference from a stationary frame followed by a rotating one. The absolute frame of reference is rigidly connected with the stationary parts, such as casings, inlets, and exits of a turbine, a compressor, a stationary gas turbine or a jet engine, whereas the relative frame is attached to the rotating shaft, thereby turning with certain angular velocity about the machine axis. By changing the frame of reference from an absolute frame to a relative one, certain flow quantities remain unchanged, such as normal stress tensor, shear stress tensor, and deformation tensor. These quantities are indifferent with regard to a change of frame of reference. However, there are other quantities that undergo changes when moving from a *stationary frame* to a rotating *one*. Velocity, acceleration, and rotation tensor are a few. We first apply these laws to the stationary or absolute frame of reference, then to the rotating one.

The differential analysis is of primary significance to all engineering applications such as compressor, turbine, combustion chamber, inlet, and exit diffuser, where a detailed knowledge of flow quantities, such as velocity, pressure, temperature, entropy, and force distributions, are required. A complete set of independent conservation laws exhibits a system of partial differential equations that describes the motion of a fluid particle. Once this differential equation system is defined, its solution delivers the detailed information about the flow quantities within the computational domain with given initial and boundary conditions.

## Keywords

Fluid Particle Deformation Tensor Christoffel Symbol Inviscid Flow Flow Quantity## Preview

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## References

- 1.Schobeiri, M.T.: Fluid mechanics. A Graduate Level Textbook. Springer, New York (2010)MATHGoogle Scholar
- 2.Truesdell, C., Noll, W.: Handbuch der Physik. Editor S. Flügge. Springer, Berlin (1965)Google Scholar
- 3.Truesdell, C.: J. of Rational Mech. Analysis 1, 125 (1952)MathSciNetMATHGoogle Scholar