7.1 Capillary instability
In the absence of the magnetic field, we assume that the streaming fluid is acted upon only by the capillary force. In such a case, the dispersion relation of this model is given from the relation (69) in the form
{(\sigma +imW+ikU)}^{2}=\frac{T}{\rho {R}_{0}^{3}}\frac{x{I}_{1}(x)}{{I}_{0}(x)}(1{m}^{2}{x}^{2}).
(75)
By using the fact, for each nonzero real value of x and m\ge 0, that
the analytical and numerical discussions of the relation (76) reveal the following results.
In the computer for different values of M and different cases of {U}^{\ast} and {W}^{\ast}.
In the most important sausage mode m=0.
The dimensionless dispersion relation is
\frac{{(\sigma +imW+ikU)}^{2}}{4\pi G\rho}=\frac{x{I}_{m}^{\prime}(x)}{{I}_{m}(x)}[{I}_{m}(x){K}_{m}(x)\frac{1}{2}]+M\left[(1{m}^{2}{x}^{2})\frac{x{I}_{m}^{\mathrm{\prime}}(x)}{{I}_{m}(x)}\right],
(78)
where
M=\left[\frac{T}{(4\pi G{\rho}^{2}{R}_{0}^{3})}\right],\phantom{\rule{2em}{0ex}}{U}^{\ast}=\left[\frac{ikU}{{(4\pi G\rho )}^{1/2}}\right],\phantom{\rule{2em}{0ex}}{W}^{\ast}=\left[\frac{imW}{{(4\pi G\rho )}^{1/2}}\right].
The numerical data associated with \sigma /{(4\pi G\rho )}^{\frac{1}{2}} correspond to the unstable states, while those associated with \omega /{(4\pi G\rho )}^{\frac{1}{2}} correspond to the stable domains. It has been found that there are many features of interest in this numerical analysis as we see in the following.

(i)
For M=0.5,1.0,1.5,2.5,3.0,3.5, see Figure 2.
Corresponding to {U}^{\ast}={W}^{\ast}=0.2. It has been found that the unstable domains are
while the neighboring stable domains are
where the equalities correspond to the marginal stability states.

(ii)
For M=0.5,1.0,1.5,2.5,3.0,3.5, see Figure 3.
Corresponding to {U}^{\ast}={W}^{\ast}=0.5. It has been found that the unstable domains are
while the neighboring stable domains are
where the equalities correspond to the marginal stability states.

(iii)
For M=3.6,4.0,4.5,5.0,5.5,6.0, see Figure 4.
Corresponding to {U}^{\ast}={W}^{\ast}=0.2. It has been found that stable domains are 0\le x<\mathrm{\infty}.

(iv)
For M=3.6,4.0,4.5,5.0,5.5,6.0, see Figure 5.
Corresponding to {U}^{\ast}={W}^{\ast}=0.5. It has been found that stable domains are 0\le x<\mathrm{\infty}.
We conclude that the streaming full fluid cylinder has stable and unstable domain for M less than 3.5 and stable domain only for M greater than this value whatever the values of velocities are. Increasing the value of M, the unstable domain is decreasing. The effect of changing velocities cases on the capillarity effect is such small that it may be considered as no effect.
7.2 Selfgravitating instability
Consider only the selfgravitating force effect, and then the dispersion relation of the model is given from equation (69) as follows:
{(\sigma +imW+ikU)}^{2}=4\pi G\rho \frac{x{I}_{m}^{\prime}(x)}{{I}_{m}(x)}[{I}_{m}(x){K}_{m}(x)\frac{1}{2}].
(79)
Consider the inequalities (77) and (78) and, for each nonzero real value of x, that
the analytical and numerical discussion of the relation (79) reveal the following.
For U=0, W=0, it has been found that the model is gravitationally unstable in the domain (0<x<1.0667 for m=0 mode) while it is stable in the domains (1.0667\le x\le \mathrm{\infty} for m=0 mode) and (0\le x\le \mathrm{\infty} for m\ge 1 modes).
For U\ne 0, W\ne 0, it has been found that the axial flow has a strong destabilizing influence. That effect does not rely on the kind of perturbation and it is so for all short and long wavelengths. Therefore, the streaming has the effect of increasing the axisymmetric stable domain 1.0667\le x\le \mathrm{\infty} and the nonaxisymmetric domains 0<x<\mathrm{\infty}.
We conclude that the streaming selfgravitating fluid cylinder is unstable not only for the axisymmetric mode m=0, but also for nonaxisymmetric modes m\ge 1.
7.3 Magnetogravitodynamic stability
This is the case in which the streaming fluid cylinder is acted upon by the combined effects of the selfgravitating and magnetic forces. It is difficult to determine exactly in analytical ways the (un) stable domains in such a general case. However, we could determine them via the numerical discussions. Also, by means of such discussion, we may find out the effects of the magnetic field on the selfgravitating force. This could be carried out by calculating the dimensionless dispersion relation
\frac{{(\sigma +imW+ikU)}^{2}}{4\pi G\rho}=\frac{x{I}_{m}^{\prime}(x)}{{I}_{m}(x)}[{I}_{m}(x){K}_{m}(x)\frac{1}{2}]+\gamma [{x}^{2}+{\alpha}^{2}\frac{{x}^{2}{I}_{m}^{\prime}(x){K}_{m}(x)}{{I}_{m}(x){K}_{m}^{\prime}(x)}]
(81)
in the computer for different values of
\gamma (=(\mu /16{\pi}^{2}G){({H}_{0}/\rho {R}_{0})}^{2})\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{U}^{\ast}=\left[\frac{ikU}{{(4\pi G\rho )}^{1/2}}\right]
in the most important sausage mode m=0.
The numerical data associated with \sigma /{(4\pi G\rho )}^{\frac{1}{2}} correspond to the unstable states, while those associated with \omega /{(4\pi G\rho )}^{\frac{1}{2}} correspond to the stable domains. It has been found that there are many features of interest in this numerical analysis as we see in the following.

(i)
For \gamma =0.1,0.2,0.3,0.4,0.5,0.7, see Figure 6.
Corresponding to {U}^{\ast}={W}^{\ast}=0.2. It has been found that the unstable domains are
while the neighboring stable domains are
where the equalities correspond to the marginal stability states.

(ii)
For \gamma =0.1,0.2,0.3,0.4,0.5,0.7, see Figure 7.
Corresponding to {U}^{\ast}={W}^{\ast}=0.5. It has been found that the unstable domains are
while the neighboring stable domains are
where the equalities correspond to the marginal stability states.

(iii)
For \gamma =0.8,0.9,1.0,1.2,1.3,1.4, see Figure 8.
Corresponding to {U}^{\ast}={W}^{\ast}=0.2. It has been found that stable domains are 0\le x<\mathrm{\infty}.

(iv)
For \gamma =0.8,0.9,1.0,1.2,1.3,1.4, see Figure 9.
Corresponding to {U}^{\ast}={W}^{\ast}=0.5. It has been found that stable domains are 0\le x<\mathrm{\infty}.
We conclude that the streaming full fluid cylinder has stable and unstable domain for γ less than 0.8 and stable domain only. Increasing the value of magnetic field, the unstable domains are decreasing. The effect of changing velocities cases on magnetic effect is such small that it may be considered asno effect. If we compare these results with those of chapter two (only velocity in z direction), we observe that the existance of another velocity W in φ direction decreases the unstable domain.
7.4 Magnetogravitodynamic capillary stability
This is the general case in which the streaming fluid cylinder is acted upon by the combined effects of the selfgravitating, capillary, and magnetic forces. The dispersion relation is given in its general form by equation (69). It is difficult to determine exactly in analytical ways the (un) stable domains in such a general case. However, we could determine them via the numerical discussions. Also, by means of such discussion, we may find out the effects of capillary with a constant magnetic field on the selfgravitating force. This could be carried out by calculating the dimensionless dispersion relation
\begin{array}{rcl}\frac{{(\sigma +imW+ikU)}^{2}}{4\pi G\rho}& =& \frac{x{I}_{m}^{\prime}(x)}{{I}_{m}(x)}[{I}_{m}(x){K}_{m}(x)\frac{1}{2}]+M\left[(1{m}^{2}{x}^{2})\frac{x{I}_{m}^{\prime}(x)}{{I}_{m}(x)}\right]\\ +\gamma [{x}^{2}+{\alpha}^{2}\frac{{x}^{2}{I}_{m}^{\prime}(x){K}_{m}(x)}{{I}_{m}(x){K}_{m}^{\prime}(x)}]\end{array}
(82)
in the computer for different values of
M=\left[\frac{T}{(4\pi G{\rho}^{2}{R}_{0}^{3})}\right],\phantom{\rule{2em}{0ex}}\gamma (=(\mu /16{\pi}^{2}G){({H}_{0}/\rho {R}_{0})}^{2})\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{U}^{\ast}=\left[\frac{ikU}{{(4\pi G\rho )}^{1/2}}\right]
in the most important sausage mode m=0.
The numerical data associated with \sigma /{(4\pi G\rho )}^{\frac{1}{2}} correspond to the unstable states, while those associated with \omega /{(4\pi G\rho )}^{\frac{1}{2}} correspond to the stable domains. It has been found that there are many features of interest in this numerical analysis as we see in the following.

(i)
For M=0.5,1.0,1.5,2.5,3.0,3.5, see Figure 10.
Corresponding to \gamma =0.5 and {U}^{\ast}={W}^{\ast}=0.2. It has been found that the unstable domains are
while the neighboring stable domains are
where the equalities correspond to the marginal stability states.

(ii)
For M=0.5,1.0,1.5,2.5,3.0,3.5, see Figure 11.
Corresponding to \gamma =0.5 and {U}^{\ast}={W}^{\ast}=0.5. It has been found that the unstable domains are
while the neighboring stable domains are
where the equalities correspond to the marginal stability states.

(iii)
For M=0.5,1.0,1.5,2.5,3.0,3.5, see Figure 12.
Corresponding to \gamma =0.5 and {U}^{\ast}={W}^{\ast}=0.2. It has been found that stable domains are

(iv)
For M=0.5,1.0,1.5,2.5,3.0,3.5, see Figure 13.
Corresponding to \gamma =0.5 and {U}^{\ast}={W}^{\ast}=0.5. It has been found that stable domains are
We conclude that the streaming full fluid cylinder has stable and unstable domains for M less than 3.5 and stable domain only for M greater than this value whatever the values of velocities are. The effect of changing velocities cases on capillarity effect is such small that it may be considered as no effect. Increasing M with constant magnetic field increases the unstable domain.