In this research, Poissy flow instability is analyzed using Orr-Summerfield equation. First, the Over-Summerfield equation is obtained, then the eigenvalues of this equation will be obtained by using the spectral method and using the Chebysheff polynomial. Using special values, the thumb diagram of this flow is drawn.
Note that the stability or not of a flow is the only thing that can be said that a person can prove that a physical state is unstable, without being able to determine the stable state from which this unstable state was created. In viscous flow, it can be proved that the laminar flow will become unstable at Reynolds numbers greater than a certain Reynolds number. But this is all we have: the analysis reveals something about turbulent flow, the proper steady state at high Reynolds numbers. All analyzes of stability against small disturbances follow a set of general principles that have seven steps and we have examined these seven steps.
Continuity and Navier-Stokes equations for two variables \( V \) and \( P \):
(1) \( \nabla \cdot V = 0 \)
(2) \( \frac{D V}{D t} = -\frac{1}{\rho} \nabla P + \nu \nabla^2 V \)
Summerfield's equation is obtained as equation (3):
(3) \( (U - c)(v'' - \alpha^2 u) - U'' v + i \frac{\nu}{\alpha}(v''' - 2 \alpha^2 v'' + \alpha^4 v) = 0 \)
The boundary condition for equation (3) is:
\( v(\pm h) = v'(\pm h) = 0 \) (Flow in channel)
\( v(0) = v'(0) = 0 \) (Boundary conditions)
\( v(\pm \infty) = v'(\pm \infty) = 0 \) (Streams without shear)
\( v(\infty) = v'(\infty) = 0 \)
To obtain the thumb diagram in the viscous state, it is done in such a way that for each \( \alpha \) and each Reynolds number, \( Re \), the value of \( c \) is obtained. If its imaginary value is greater than zero, it means the flow is unstable, and if it is smaller than zero, Steady flow. In order to obtain the stability limit, it is possible to change the Reynolds number in an \( \alpha \) range in order to determine the limit of stability and instability of the flow. If we repeat this work for other values of \( \alpha \), the thumb diagram is obtained.
To ensure the correctness of the solution method, the Orr-Sommerfeld equation has been solved for Poissy flow. According to Henningson's experimental results, the onset of instability occurs at Reynolds number \( Re = 72.5772 \) and \( \alpha = 1.02 \), which is consistent with the results of solving the equation. In this code, if the eigenvalues have a value greater than zero, instability occurs.
More results are in the attached file.