Determining the amount of entropy generated in an incompressible fluid flow moving around a rotating disk assignment

FORMULATION OF THE PROBLEM

The mathematical formulation of the problem of determining the amount of entropy generated in an incom- pressible fluid flow moving around a rotating disk is presented below.
                                  1.1. Flow and Thermal Fields
Consider a three-dimensional steady laminar flow of an incompressible fluid over an infinite rotating disk. Uniform injection is applied at the disk surface. The disk rotates about its axis with a constant angular velocity, which causes its non-coaxial rotation about an axis originating at (ro. a) (Fig. 1).

Based on these assumptions, the continuity and momentum equations in cylindrical coordinates (r. 6,=) have the form

where we, us, and u, are the radial, tangential, and axial components of velocity, respectively, p is the density, p is the pressure, and p is the dynamic viscosity. Taking the effect of viscous dissipation into account, we can write the energy equation as

Here c, is the specific heat, T is the temperature, & is the thermal conductivity, and represents viscous dissipation. The boundary conditions of the problem can be written as follows:

-as z ? ?, the velocity components u,, te, and u, are finite, and T = T. Here f? is the angular velocity of the disk, and s is the axial component of velocity; the subscript w refers to the disk surface (wall), and the subscript o is used to indicate the ambient medium. However, unlike the classical von Kármán's axisymmetric similarity solution, the present study implies the dependence on the coordinate 0, thus, enabling a non-axisymmetric flow to develop. Further, it is assumed that the axial component of the velocity vector u, is constant. It should be noticed that this assumption is identically satisfied by the third equation of momentum, provided that the pressure p is not dependent on .
We introduce the dimensionless parameters

(n) is the dimensionless axial coordinate, and is the kinematic viscosity). The expressions for the dimensionless velocity components and pressure fields can be formulated as [5]

where ro, o, and p1 are constants, while Re = r2/v and Res raft/ are the Reynolds numbers. In view of the dimensionless variables introduced above, the continuity equation and the equations of motion are written as [4, 5]

The boundary conditions have the following form:

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