**2. Linearized perturbation equations of the problem**

We consider an infinite homogeneous, thermally conducting, radiating, porous plasma with FLR corrections in the presence of magnetic field *H* (0, 0, *H*). The perturbation in fluid pressure, density, temperature, velocity, magnetic field and heat-loss function are given as *δp*, *δρ, δT*, *u*(*δux*, *δuy*,*δuz*), *δH*(*δHx*, *δHy*, *δHz*), and *L* respectively. The perturbation circumstances is given as.

$$p = p\_0 + \delta p, \rho = \rho\_0 + \delta \rho, T = T\_0 + \delta T, \mathfrak{u} = \mathfrak{u}\_0 + \delta \mathfrak{u}, \mathcal{H} = \mathcal{H}\_0 + \delta \mathcal{H}, \text{and } L = L\_0 + L. \tag{1}$$

Suffix '0' represents the initial equilibrium state, which is independent of space and time. L (*ρ*,*T*) is the heat-loss function of the material limited of thermal conduction and is in general a cause of the local values of density and temperature Field [2]. The operator ð Þ *d=dt* is the substantial derivative given as ð Þ¼ *d=dt* ð Þ *<sup>∂</sup>=∂<sup>t</sup>* <sup>þ</sup> ð Þ <sup>1</sup>*=<sup>ε</sup> <sup>u</sup>:***<sup>∇</sup>** . With these effects the linearized perturbation equations of the problem are

$$\left(\frac{1}{\varepsilon}\right)\partial\_t\delta\mathfrak{u} = -\left(\frac{\nabla\delta p}{\rho}\right) - \left(\frac{\nabla\cdot\mathbf{P}}{\rho}\right) + \left(\frac{1}{4\pi\rho}\right)(\nabla\times\delta\mathbf{H})\times\mathbf{H} + 2(\mathfrak{u}\times\mathfrak{Q}),\tag{2}$$

$$
\boldsymbol{\varepsilon} \,\boldsymbol{\partial}\_t \delta \boldsymbol{\rho} + \rho \,\nabla \cdot \delta \boldsymbol{u} = \mathbf{0},\tag{3}
$$

$$
\rho \left(\frac{1}{\chi - 1}\right) \partial\_t \delta p - \left(\frac{\chi}{\chi - 1}\right) \left(\frac{p}{\rho}\right) \partial\_t \delta \rho + \,\_\rho \left[\delta \rho \left(\frac{\partial L}{\partial \rho}\right)\_T + \delta T \left(\frac{\partial L}{\partial T}\right)\_\rho\right] - \,\_\lambda \delta \nabla^2 \delta T = 0,\tag{4}
$$

$$
\left(\frac{\delta p}{p}\right) = \left(\frac{\delta T}{T}\right) + \left(\frac{\delta \rho}{\rho}\right),
\tag{5}
$$

$$
\partial\_t \delta H = \left(\frac{1}{\varepsilon}\right) \nabla \times (\mathfrak{u} \times \mathfrak{H}),
\tag{6}
$$

$$
\nabla \delta \mathbf{H} = \mathbf{0},
\tag{7}
$$

where ð Þ *<sup>∂</sup>L=∂<sup>T</sup> <sup>ρ</sup>*, ð Þ *<sup>∂</sup>L=∂<sup>ρ</sup> <sup>T</sup>* are the partial derivatives of temperature dependent heat-loss function *LT* and density dependent heat-loss function *L<sup>ρ</sup>* respectively. The components of pressure tensor **P**, considering the finite ion gyration radius for the magnetic field along z-axis as given by Roberts and Taylor [28] are

$$\begin{split} P\_{\rm xx} &= -\rho v\_0 \left[ \left( \delta \delta u\_\gamma / \delta \mathbf{x} \right) + \left( \delta \delta u\_\mathbf{x} / \delta \eta \right) \right], P\_{\mathcal{yy}} = \rho v\_0 \left[ \left( \delta \delta u\_\mathbf{y} / \delta \mathbf{x} \right) + \left( \delta \delta u\_\mathbf{x} / \delta \eta \right) \right], \\ P\_{\rm xy} &= P\_{\rm yx} = \rho v\_0 \left[ \left( \delta \delta u\_\mathbf{y} / \delta \mathbf{x} \right) - \left( \delta \delta u\_\mathbf{y} / \delta \eta \right) \right], \\ P\_{\rm xx} &= P\_{\rm xx} = -2\rho v\_0 \left[ \left( \delta \delta u\_\mathbf{y} / \delta \mathbf{z} \right) + \left( \delta \delta u\_\mathbf{z} / \delta \eta \right) \right], \\ P\_{\rm yx} &= P\_{\rm xy} = 2\rho v\_0 \left[ \left( \delta \delta u\_\mathbf{z} / \delta \mathbf{x} \right) + \left( \delta \delta u\_\mathbf{x} / \delta \mathbf{z} \right) \right], \ P\_{\rm xz} = 0. \end{split} \tag{8}$$

The parameter *υ*<sup>0</sup> has the dimensions of the kinematics viscosity and called as magnetic viscosity defined as *<sup>υ</sup>*<sup>0</sup> <sup>¼</sup> <sup>Ω</sup>*LR*<sup>2</sup> *<sup>L</sup>=*4, where *RL* is the ion-Larmor radius and Ω*<sup>L</sup>* is the ion gyration frequency.

We seek plain wave solution of the form

$$\exp\left(i\sigma t + ik\_{\mathbf{x}}\pi + ik\_{\mathbf{z}}z\right),\tag{9}$$

where *σ* is the frequency of harmonic disturbance, *kx* and *kz* are the wave numbers of the perturbations along x and z axes. Such that

$$k^2 = k\_x^2 + k\_x^2 \tag{10}$$

The components of Eq. (6) may be given

$$\delta \mathcal{H}\_{\mathbf{x}} = (\mathrm{i}\mathrm{H}/\varepsilon ao)\mathrm{k}\_{\mathbf{z}} \delta \mathfrak{u}\_{\mathbf{x}}, \quad \delta \mathcal{H}\_{\mathbf{y}} = (\mathrm{i}\mathrm{H}/\varepsilon ao)\mathrm{k}\_{\mathbf{z}} \delta \mathfrak{u}\_{\mathbf{y}}, \quad \delta \mathcal{H}\_{\mathbf{z}} = -(\mathrm{i}\mathrm{H}/\varepsilon ao)\mathrm{k}\_{\mathbf{x}} \delta \mathfrak{u}\_{\mathbf{x}}.\tag{11}$$

where *iσ* ¼ *ω*. Using Eqs. (4), (5) and (9) we write

$$\delta p = \frac{\left\{ (\boldsymbol{\chi} - \mathbf{1}) \left[ T \mathcal{L}\_T - \rho \mathcal{L}\_\rho + \left( \frac{\lambda k^2 T}{\rho} \right) \right] + a \, \boldsymbol{c}^2 \right\}}{\left\{ (\boldsymbol{\chi} - \mathbf{1}) \left[ \left( \frac{T \rho}{p} \right) \mathcal{L}\_T + \left( \frac{\lambda k^2 T}{p} \right) \right] + a \right\} \delta \rho}, \tag{12}$$

Using Eqs. (3)–(11) in Eq. (2), we may engrave the subsequent algebraic equations for the constituents of Eq. (2)

$$
\delta\mathfrak{d}\_{\mathbf{x}}\left[\boldsymbol{\varrho} + \left(\boldsymbol{V}^{2}\boldsymbol{k}^{2}/\boldsymbol{\omega}\right)\right] + \delta\mathfrak{d}\_{\mathcal{V}}\left[\varepsilon\nu\_{0}\left(\boldsymbol{k}\_{\mathbf{x}}^{2} + 2\boldsymbol{k}\_{\mathbf{z}}^{2}\right) - 2\varepsilon\mathfrak{Q}\_{\mathbf{z}}\right] + \varepsilon\left(i\boldsymbol{k}\_{\mathbf{x}}/\boldsymbol{k}^{2}\right)\mathfrak{Q}\_{\mathbf{T}}^{2}\,\mathrm{s} = \mathbf{0},\tag{13}
$$

$$-\delta u\_x \left[ \varepsilon \nu\_0 \left( k\_x^2 + 2k\_z^2 \right) - 2\Omega\_z \right] + \delta u\_y \left[ \alpha + \left( V^2 k\_z^2 / \alpha \right) \right] - \delta u\_z \left[ 2\varepsilon \left( \nu\_0 k\_x k\_z + \Omega\_x \right) \right] = 0,\tag{14}$$

$$
\left[\delta\mu\_{\mathcal{Y}}\left[\mathcal{Q}\epsilon(\nu\_{0}k\_{\text{x}}k\_{\text{x}}+\mathcal{Q}\_{\text{x}})\right]+\delta\mu\_{\text{x}}\alpha+\epsilon\left(ik\_{\text{x}}/k^{2}\right)\mathcal{Q}\_{\text{T}}^{2}\text{ s}=\mathbf{0}.\tag{15}
$$

Taking divergence of Eq. (2) and using Eqs. (3)–(11), we obtain as

$$
\delta u\_x \left[ ik\_x \left( \frac{V^2 k^2}{\varepsilon \alpha} \right) \right] + \delta u\_y \left[ i v\_0 k\_x \left( k\_x^2 + 4 k\_x^2 \right) + 2i (k\_x \Omega\_\mathbf{x} - k\_x \Omega\_\mathbf{z}) \right] - \mathbf{s} \left[ \alpha^2 + \Omega\_T^2 \right] = \mathbf{0}, \tag{16}
$$

we have made following substitutions *<sup>α</sup>* <sup>¼</sup> ð Þ *<sup>γ</sup>* � <sup>1</sup> *TLT* � *<sup>ρ</sup>L<sup>ρ</sup>* <sup>þ</sup> *<sup>λ</sup>k*<sup>2</sup> *T ρ* h i � � , *<sup>β</sup>* <sup>¼</sup> ð Þ *<sup>γ</sup>* � <sup>1</sup> *<sup>T</sup>ρLT p* � � <sup>þ</sup> *<sup>λ</sup>k*<sup>2</sup> *T p* h i � � , *<sup>s</sup>* <sup>¼</sup> *δρ <sup>ρ</sup>* , *Ω*<sup>2</sup> *<sup>T</sup>* <sup>¼</sup> *<sup>Ω</sup>*<sup>2</sup> *<sup>I</sup>* <sup>þ</sup>*ω Ω*<sup>2</sup> ð Þ*<sup>J</sup>* ð Þ *ω*þ*β* � �, *<sup>Ω</sup>*<sup>2</sup> *<sup>J</sup>* <sup>¼</sup> *<sup>c</sup>*<sup>2</sup>*k*<sup>2</sup> , *Ω*<sup>2</sup> *<sup>I</sup>* <sup>¼</sup> *<sup>k</sup>*<sup>2</sup> *<sup>α</sup>*, *<sup>V</sup>*<sup>2</sup> <sup>¼</sup> *H*2 <sup>4</sup>*πρ* � �, *<sup>c</sup>* <sup>¼</sup> ð Þ *<sup>γ</sup>p=<sup>ρ</sup>* <sup>1</sup>*=*<sup>2</sup> is the adiabatic velocity of sound in the medium.

## **3. Dispersion relation**

The nontrivial solution of the determinant of the matrix gained from Eqs. (13)– (16) with *δux*, *δuy*, *δuz*, *s* having various coefficients that should disappear is to give the subsequent dispersion relation

*Transverse Thermal Instability of Radiative Plasma with FLR Corrections for Star… DOI: http://dx.doi.org/10.5772/intechopen.99924*

*<sup>ω</sup>*<sup>2</sup> <sup>þ</sup> *<sup>Ω</sup>*<sup>2</sup> *T* � � *<sup>ω</sup>* <sup>þ</sup> *<sup>V</sup>*<sup>2</sup> *k*2 *<sup>=</sup><sup>ω</sup>* � � � � *ω ω* <sup>þ</sup> *<sup>V</sup>*<sup>2</sup> *k*2 *<sup>z</sup>=<sup>ω</sup>* � � � � <sup>þ</sup> <sup>4</sup>*ε*<sup>2</sup> ð Þ *<sup>υ</sup>*0*kxkz* <sup>þ</sup> *<sup>Ω</sup><sup>x</sup>* <sup>2</sup> n o � <sup>2</sup>*ε*<sup>2</sup> *Ω*2 *T=k*<sup>2</sup> � � �ð Þ *<sup>υ</sup>*0*kxkz* <sup>þ</sup> *<sup>Ω</sup><sup>x</sup> <sup>ω</sup>* <sup>þ</sup> *<sup>V</sup>*<sup>2</sup> *k*2 *=ω* � � � � *υ*0*kxkz k*<sup>2</sup> *<sup>x</sup>* <sup>þ</sup> <sup>4</sup>*k*<sup>2</sup> *z* � � <sup>þ</sup> <sup>2</sup> *<sup>k</sup>*<sup>2</sup> *<sup>z</sup>Ω<sup>x</sup>* � *kxkzΩ<sup>z</sup>* � � � � <sup>þ</sup>*ω ευ*<sup>0</sup> *<sup>k</sup>*<sup>2</sup> *<sup>x</sup>* <sup>þ</sup> <sup>2</sup>*k*<sup>2</sup> *z* � � � <sup>2</sup>*εΩ<sup>z</sup>* � �<sup>2</sup> *<sup>ω</sup>*<sup>2</sup> <sup>þ</sup> *<sup>Ω</sup>*<sup>2</sup> *T* � � <sup>þ</sup> <sup>2</sup>*εkxkzV*<sup>2</sup> *=ω* � �*Ω*<sup>2</sup> *<sup>T</sup>*ð Þ *υ*0*kxkz* þ *Ω<sup>x</sup>* � *ευ*<sup>0</sup> *<sup>k</sup>*<sup>2</sup> *<sup>x</sup>* <sup>þ</sup> <sup>2</sup>*k*<sup>2</sup> *z* � � � <sup>2</sup>*εΩ<sup>z</sup>* � � � *ω εΩ*<sup>2</sup> *<sup>T</sup>=k*<sup>2</sup> � � *ευ*<sup>0</sup> *<sup>k</sup>*<sup>2</sup> *<sup>x</sup>* <sup>þ</sup> <sup>2</sup>*k*<sup>2</sup> *z* � � � <sup>2</sup>*εΩ<sup>z</sup>* � � � *<sup>υ</sup>*0*k*<sup>2</sup> *<sup>x</sup> <sup>k</sup>*<sup>2</sup> *<sup>x</sup>* <sup>þ</sup> <sup>4</sup>*k*<sup>2</sup> *z* � � �þ<sup>2</sup> *kxkzΩ<sup>x</sup>* � *<sup>k</sup>*<sup>2</sup> *<sup>x</sup>Ω<sup>z</sup>* � �� � *ω ω* <sup>þ</sup> *<sup>V</sup>*<sup>2</sup> *k*2 *<sup>z</sup>=<sup>ω</sup>* � � � � *<sup>V</sup>*<sup>2</sup> *k*2 *<sup>x</sup>=<sup>ω</sup>* � �*Ω*<sup>2</sup> *T* � <sup>4</sup>*ε*<sup>2</sup> *k*2 *xV*<sup>2</sup> *=ω* � �*Ω*<sup>2</sup> *<sup>T</sup>*ð Þ *<sup>υ</sup>*0*kxkz* <sup>þ</sup> *<sup>Ω</sup><sup>x</sup>* <sup>2</sup> <sup>¼</sup> <sup>0</sup>*:* (17)

The dispersion relation (17) demonstrates the jointed influence of rotation, FLR corrections, radiative heat-loss function, thermal conductivity and porosity on the thermal instability of homogeneous plasma flowing through porous medium. The above dispersion relation is long and to learn the consequence of all parameter we now diminish the dispersion relation (17) for transverse mode of transmission.

## **4. Conversation of the dispersion relation**

### **4.1 Transverse mode of transmission (K⊥B)**

In this situation the perturbations are in use to be vertical to the path of the magnetic field ð Þ *i:e: kx* ¼ *k*, *kz* ¼ 0 *:* The dispersion relation (17) reduces to

$$\begin{aligned} & \left( \omega^2 + \Omega\_T^2 \right) \left\{ \left[ \alpha + \left( V^2 k^2 / \alpha \right) \right] \left( \alpha^2 + 4 \varepsilon^2 \Omega\_\mathbf{x}^2 \right) + \alpha \left( \varepsilon \nu\_0 k^2 - 2 \varepsilon \Omega\_\mathbf{z} \right)^2 \right\} \\ & - \Omega\_T^2 \left\{ \alpha \left[ \left( V^2 k^2 / \alpha \right) + \varepsilon^2 \left( \nu\_0 k^2 - 2 \Omega\_\mathbf{z} \right)^2 \right] + 4 \varepsilon^2 \Omega\_\mathbf{x}^2 \left( V^2 k^2 / \alpha \right) \right\} \\ & = \mathbf{0}. \end{aligned} \tag{18}$$

This dispersion relation (18) provides the control of rotation, FLR corrections, radiative heat-loss function thermal conductivity and porosity on thermal unsteadiness of plasma for transverse mode of transmission. Now we discuss the dispersion relation (18) for rotation axis parallel and vertical to the magnetic field.

### *4.1.1 Axis of rotation along the magnetic field (Ω||B)*

For the case of axis of rotation along the magnetic field, we put *Ω<sup>x</sup>* ¼ 0 and *Ω<sup>z</sup>* ¼ *Ω* in dispersion relation (18) which reduces to

$$\alpha^3 \left\{ \alpha \left[ \alpha + \left( V^2 k^2 / \alpha \right) \right] + \varepsilon^2 \left( \nu\_0 k^2 - 2 \mathfrak{Q} \right)^2 + \left( \mathfrak{Q}\_l^2 + \alpha \mathfrak{Q}\_l^2 \right) / (\omega + \beta) \right\} = \mathbf{0}. \tag{19}$$

Eq. (19) has two independent factors. The first factor of Eq. (19) gives *<sup>ω</sup>*<sup>3</sup> <sup>¼</sup> 0, which is a marginal stable mode. The second factor of Eq. (19) gives the following dispersion relation on alternating the values of *Ω*<sup>2</sup> *<sup>I</sup>* , *Ω*<sup>2</sup> *<sup>J</sup>* , *α* and *β*.

$$\begin{aligned} &\alpha^3 + \left\{ \left[ (\mathbf{y} - \mathbf{1}) \left( (T\rho L\_T/p) + (\lambda k^2 T/p) \right) \right] \right\} \alpha^2 + \left[ 4\epsilon^2 \Omega^2 + \epsilon^2 v\_0^2 k^4 \right. \\ &+ V^2 k^2 + \epsilon^2 k^2 - 4\epsilon^2 \Omega v\_0 k^2 \left] \alpha + (\mathbf{y} - \mathbf{1}) \left[ (T\rho L\_T/p) + (\lambda k^2 T/p) \right] \right. \\ &\times \left( 4\epsilon^2 \Omega^2 + \epsilon^2 v\_0^2 k^4 + V^2 k^2 - 4\epsilon^2 \Omega v\_0 k^2 \right) + k^2 (\mathbf{y} - \mathbf{1}) \\ &\times \left[ T\mathbf{L}\_T - \rho \mathbf{L}\_\rho + \left( \lambda k^2 T/\rho \right) \right] \right) \\ &= \mathbf{0}. \end{aligned} \tag{20}$$

### *Plasma Science and Technology*

This dispersion relation symbolizes the consequence of direct addition of rotation, FLR corrections, radiative heat-loss function, thermal conductivity and porosity on the thermal unsteadiness of the organization. When constant term of Eq. (20) is less than zero this allows at least one positive real root which converses to the unsteadiness of the organization. The situation of unsteadiness obtained from steady term of Eq. (20) is specified as

$$\left\{ k^2 \left[ \mathrm{TL}\_T - \rho \mathrm{L}\_\rho + \left( \frac{\lambda k^2 T}{\rho} \right) \right] + \left[ \left( \frac{T \rho L\_T}{p} \right) + \left( \frac{\lambda k^2 T}{p} \right) \right] \left( 4 \epsilon^2 \mathcal{Q}^2 + \epsilon^2 v\_0^2 k^4 + V^2 k^2 - 4 \epsilon^2 \mathcal{Q} v\_0 k^2 \right) \right\} < 0, \tag{21}$$

Eq. (21) symbolizes the modified form of thermal instability criterion by enclosure of rotation, FLR corrections, radiative heat-loss function and thermal conductivity. From Eq. (21) we conclude that rotation and FLR corrections stabilize the radiative instability.

In nonappearance of FLR corrections ð Þ *υ*<sup>0</sup> ¼ 0 Eq. (20) grows to be

$$\begin{aligned} &\alpha^3 + \left\{ (\mathbf{y} - \mathbf{1}) \left[ (T\rho L\_T/p) + \left( \lambda k^2 T/p \right) \right] \right\} \alpha^2 + \left[ 4\epsilon^2 \Omega^2 + V^2 k^2 + c^2 k^2 \right] \alpha \\ &+ \left\{ k^2 (\mathbf{y} - \mathbf{1}) \left[ T L\_T - \rho L\_\rho + \left( \lambda k^2 T/\rho \right) \right] + (\mathbf{y} - \mathbf{1}) \left[ (T\rho L\_T/p) + \left( \lambda k^2 T/p \right) \right] \right\} \\ &\times \left[ V^2 k^2 + 4e^2 \Omega^2 \right] \right) = \mathbf{0}. \end{aligned} \tag{22}$$

When constant term of Eq. (22) is less than zero this permits at least one positive real root which communicates to the instability of the system. The condition of instability attained from constant term of Eq. (22) is given as

$$\left\{k^{2}(\mathbf{y}-\mathbf{1})\left[\mathbf{T}\mathbf{L}\_{T}-\rho\mathbf{L}\_{\rho}+\left(\frac{\lambda k^{2}T}{\rho}\right)\right]+(\mathbf{y}-\mathbf{1})\left[\left(\frac{T\rho L\_{T}}{p}\right)+\left(\frac{\lambda k^{2}T}{p}\right)\right]\left[V^{2}k^{2}+4\epsilon^{2}\Omega^{2}\right]\right\}<0,\tag{23}$$

The above situation of instability is the changed form of thermal condition by addition of rotation and magnetic field strength. From Eq. (23) we bring to a close that rotation and magnetic field stabilize the radiative instability.

Now Eq. (20) can be written in the following form

$$\begin{aligned} \alpha^3 + c\_\ell \left\{ k\_T + \left( \frac{k^2}{k\_\lambda} \right) \right\} \alpha^2 + c\_\varepsilon^2 \left[ \frac{4\epsilon^2 \Omega^2}{c\_\varepsilon} + \frac{\epsilon^2 v\_0^2 k^4}{c\_\varepsilon} + \frac{V^2 k^2}{c\_\varepsilon} + k^2 - \frac{4\epsilon^2 \Omega v\_0 k^2}{c\_\varepsilon} \right] \alpha + c\_\varepsilon^3 \left[ k\_T + \left( \frac{k^2}{k\_\lambda} \right) \right] \\ \times \left[ \frac{4\epsilon^2 \Omega^2}{c\_\varepsilon} + \frac{\epsilon^2 v\_0^2 k^4}{c\_\varepsilon} + \frac{V^2 k^2}{c\_\varepsilon} + k^2 - \frac{4\epsilon^2 \Omega v\_0 k^2}{c\_\varepsilon} \right] + \left( \frac{c\_\varepsilon^3 k^2}{\gamma} \right) \left[ k\_T - k\_\rho + \left( \frac{k^2}{k\_\lambda} \right) \right] = 0. \end{aligned} \tag{24}$$

We have used

$$k\_{\rho} = \left[ (\boldsymbol{\chi} - \mathbf{1}) \rho \boldsymbol{L}\_{\rho} \right] / (\mathbf{R} \boldsymbol{c}\_{\boldsymbol{s}} \boldsymbol{T}), \ k\_{\mathrm{T}} = \left[ (\boldsymbol{\chi} - \mathbf{1}) \boldsymbol{L}\_{\boldsymbol{T}} \right] / (\mathbf{R} \boldsymbol{c}\_{\boldsymbol{s}}), \ k\_{\boldsymbol{\lambda}} = (\mathbf{R} \boldsymbol{c}\_{\boldsymbol{t}} \rho) / [(\boldsymbol{\chi} - \mathbf{1}) \boldsymbol{\lambda}], \quad \text{(25)}$$

To investigate the effect of viscosity, porosity, rotation and radiative heat-loss functions on the growth rate of thermal instability, we solve Eq. (24) numerically. Therefore Eq. (24) can be written in non-dimensional form with the help of following dimension-less quantities as given in Field [2]

$$
\alpha^\* = \alpha / k\_p \mathbf{c}\_\*, \ \mathfrak{Q}^\* = \Omega k\_\rho / \mathbf{c}\_\*, \ \ k^\* = \mathbf{k} / k\_\rho, \ \ k\_\lambda^\* = \mathbf{k}\_\rho / k\_\lambda, \ \ k\_T^\* = \mathbf{k}\_T / k\_\rho,\tag{26}
$$

*Transverse Thermal Instability of Radiative Plasma with FLR Corrections for Star… DOI: http://dx.doi.org/10.5772/intechopen.99924*

In astrophysical circumstances, instability of the organization is one of the most significant reasons of arrangement of entities. So we learn the consequences of medium porosity *ε*, rotation *Ω*<sup>∗</sup> , and FLR corrections *υ* <sup>∗</sup> <sup>0</sup> on the growth rate of unstable mode. Using Eq. (26), we write Eq. (24) in non-dimensional form as

$$\begin{aligned} &\alpha^{\*,3} + \left(k\_T^\* + k\_\lambda^{\*,2}\right)\alpha^{\*,2} + \left[4\epsilon^2\varOmega^{\*2} + \epsilon^2\nu\_0^{\*,2}k^{\*4} + V^{\*2}k^{\*2} + k^{\*2} - 4\epsilon^2\varOmega^{\*}\nu\_0^{\*}k^{\*2}\right]\alpha^{\*,2} \\ &+ \left(4\epsilon^2\varOmega^{\*2} + \epsilon^2\nu\_0^{\*,2}k^{\*4} + V^{\*2}k^{\*2} - 4\epsilon^2\varOmega^{\*}\nu\_0^{\*}k^{\*2}\right)\left(k\_T^\* + k\_\lambda^{\*,2}\right) \\ &+ \left(k^{\*,2}/\gamma\right)\left[k\_T^\* - \mathbbm{1} + k\_\lambda^{\*}k^{\*2}\right] = 0. \end{aligned} \tag{27}$$

Mathematical computations were executed to decide the roots of (*ω*<sup>∗</sup> ) as a function of wave number (*k* <sup>∗</sup> ) for moderately a few values of dissimilar parameters occupied captivating *γ* ¼ 5*=*3. Out of three modes, only one mode is unstable for which the computations are at presented in **Figures 1**–**5**, where the growth rate *ω*<sup>∗</sup> has been sketched versus the wave number *k* <sup>∗</sup> to display the reliance of the growth rate on the dissimilar substantial limitations such as porosity, rotation and FLR corrections. It is clear from **Figure 1** that the max out rate of the growth rate reduces with augment in the rate of medium porosity. Thus the consequence of medium porosity is stabilizing on the growth rate of the environment. From **Figure 2** we see that the growth rate diminishes with augment in the value of rotation. Thus it is bring to a close that rotation stabilizes the growth rate of the environment. One can examine from **Figure 3** that the growth rate diminishes with rising FLR corrections. Thus the effect of FLR corrections is stabilizing on the growth rate of the environment. From **Figure 4** it is clear that growth rate diminishes on raising the value of

**Figure 1.**

*Growth rate (ω*<sup>∗</sup> *) against wave number k* <sup>∗</sup> *for four values of parameter ε keeping the other parameters fixed KT* <sup>∗</sup> <sup>¼</sup> 1, *<sup>K</sup><sup>λ</sup>* <sup>∗</sup> <sup>¼</sup> 0, *<sup>V</sup>* <sup>∗</sup> <sup>¼</sup> 1, *<sup>ν</sup>*<sup>0</sup> <sup>∗</sup> <sup>¼</sup> 1, <sup>Ω</sup><sup>∗</sup> <sup>¼</sup> <sup>1</sup>*:*0*.*

**Figure 2.** *Growth rate (ω*<sup>∗</sup> *) against wave number k* <sup>∗</sup> *for four values of parameter* Ω<sup>∗</sup> *keeping the other parameters fixed K* <sup>∗</sup> *<sup>T</sup>* ¼ 1*:*0, *K<sup>λ</sup>* <sup>∗</sup> <sup>¼</sup> 0,*<sup>V</sup>* <sup>∗</sup> <sup>¼</sup> 1, *<sup>ν</sup>*<sup>0</sup> <sup>∗</sup> <sup>¼</sup> 1, *<sup>ε</sup>* <sup>¼</sup> <sup>1</sup>*:*0*.*

### **Figure 3.**

*Growth rate (ω*<sup>∗</sup> *) against wave number k* <sup>∗</sup> *for four values of parameter ν*<sup>0</sup> <sup>∗</sup> *keeping the other parameters fixed K* <sup>∗</sup> *<sup>T</sup>* ¼ 1*:*0, *K<sup>λ</sup>* <sup>∗</sup> <sup>¼</sup> 0,*<sup>V</sup>* <sup>∗</sup> <sup>¼</sup> 1, <sup>Ω</sup><sup>∗</sup> <sup>¼</sup> 1, *<sup>ε</sup>* <sup>¼</sup> <sup>1</sup>*:*0*.*

*K* <sup>∗</sup> *<sup>T</sup> :* So *K* <sup>∗</sup> *<sup>T</sup>* shows stabilizing effect on the growth rate of the environment. One can observe from **Figure 5** that as the value of *K*<sup>∗</sup> *<sup>λ</sup>* increases the growth rate of the environment decreases. So it is clear that *K* <sup>∗</sup> *<sup>λ</sup>* stabilize the growth rate of the environment.

*Transverse Thermal Instability of Radiative Plasma with FLR Corrections for Star… DOI: http://dx.doi.org/10.5772/intechopen.99924*

**Figure 4.**

*Growth rate (ω*<sup>∗</sup> *) against wave number k* <sup>∗</sup> *for four values of parameter K* <sup>∗</sup> *<sup>T</sup> keeping the other parameters fixed K* <sup>∗</sup> *<sup>λ</sup>* <sup>¼</sup> 1,*<sup>V</sup>* <sup>∗</sup> <sup>¼</sup> 1, <sup>Ω</sup><sup>∗</sup> <sup>¼</sup> 1, *<sup>ε</sup>* <sup>¼</sup> <sup>1</sup>*:*0*.*

### **Figure 5.**

*Growth rate (ω*<sup>∗</sup> *) against wave number for four values of parameter K*<sup>∗</sup> *<sup>λ</sup> keeping the other parameters fixed K*<sup>∗</sup> *<sup>T</sup>* <sup>¼</sup> 1,*<sup>V</sup>* <sup>∗</sup> <sup>¼</sup> 1, <sup>Ω</sup><sup>∗</sup> <sup>¼</sup> 1, *<sup>ε</sup>* <sup>¼</sup> <sup>1</sup>*:*0*.*

### *4.1.2 Axis of rotation vertical to the magnetic field (Ω*⊥*B)*

In the case of axis of rotation perpendicular to the magnetic field, we put *Ω<sup>x</sup>* ¼ *Ω* and *Ω<sup>z</sup>* ¼ 0 in the dispersion relation (18) which reduces to

$$\alpha \left\{ \alpha^2 \left( \alpha^2 + 4\epsilon^2 \mathcal{Q}^2 + \epsilon^2 \nu\_0^2 k^4 \right) + \left( \alpha^2 + 4\epsilon^2 \mathcal{Q}^2 \right) \left[ V^2 k^2 + \frac{\left( \mathcal{Q}\_I^2 + \alpha \mathcal{Q}\_I^2 \right)}{\left( \alpha + \beta \right)} \right] \right\} = 0. \tag{28}$$

This dispersion relation symbolizes the joint influence of FLR corrections, rotation, porosity magnetic field, radiative heat-loss function and thermal conductivity on the thermal instability of the considered organization. Eq. (28) has two independent factors. The first factor of Eq. (28) gives *ω* ¼ 0, which is a marginal stable mode. The second factor of Eq. (28) gives the following dispersion relation on replacing the values of *Ω*<sup>2</sup> *<sup>I</sup>* , *Ω*<sup>2</sup> *<sup>J</sup>* , and *β*.

In nonattendance of rotation ð Þ *Ω* ¼ 0 Eq. (28) becomes

$$\begin{aligned} &\alpha^3 + \left\{ (\gamma - 1) \left[ \left( \frac{T\rho L\_T}{p} \right) + \left( \frac{\lambda k^2 T}{p} \right) \right] \right\} \alpha^2 \\ &+ \left[ \epsilon^2 v\_0^2 k^4 + V^2 k^2 + c^2 k^2 \right] \alpha + \left\{ k^2 (\gamma - 1) \times \left[ \text{TL}\_T - \rho L\_\rho + \left( \frac{\lambda k^2 T}{\rho} \right) \right] \right] \\ &+ (\gamma - 1) \left[ \left( \frac{T\rho L\_T}{p} \right) + \left( \frac{\lambda k^2 T}{p} \right) \right] \left[ \epsilon^2 v\_0^2 k^4 + V^2 k^2 \right] \} \\ &= 0. \end{aligned} \tag{29}$$

The situation of instability acquired from constant term of Eq. (29) is given as

$$\left\{k^{2}(\boldsymbol{y}-\mathbf{1})\left[\boldsymbol{T}\mathbf{L}\_{T}-\rho\mathbf{L}\_{\rho}+\left(\frac{\lambda k^{2}T}{\rho}\right)\right]+(\boldsymbol{y}-\mathbf{1})\left[\left(\frac{T\rho L\_{T}}{p}\right)+\left(\frac{\lambda k^{2}T}{p}\right)\right]\left[\boldsymbol{\varepsilon}^{2}\boldsymbol{v}\_{0}^{2}\boldsymbol{k}^{4}+V^{2}\boldsymbol{k}^{2}\right]\right\}<\mathbf{0}.\tag{30}$$

In present case situation of instability and growth rate of instability both depend on FLR corrections and porosity.

## **5. Conclusions**

In the above present problem we have approved out the consequence of rotation, porosity and FLR corrections on the thermal instability of plasma counting the effects of radiative heat-loss function and thermal conductivity. The general dispersion relation is attained, which is customized due to the attendance of calculated physical limitations. This dispersion relation is condensed for transverse wave propagation to the route of magnetic field, which is additional argued for rotation axis parallel and vertical to the route of magnetic field.

In the situation of transverse wave propagation to the direction of magnetic field with axis of rotation along magnetic field we gained two modes. The first one is a marginal stable mode. The second one represents the thermal mode amended by rotation, porosity, FLR corrections and radiative heat-loss function. It is concluded that the condition of thermal unsteadiness is modified due to the attendance of porosity, rotation, FLR corrections, radiative heat-loss function, and thermal conductivity. For the case of non-FLR medium, it is found that the condition of radiative unsteadiness and expression of critical thermal wave number both are amended due to the occurrence of porosity, rotation, and it explains the stabilizing influence. It is found that for non-FLR the condition of radiative unsteadiness and expression

*Transverse Thermal Instability of Radiative Plasma with FLR Corrections for Star… DOI: http://dx.doi.org/10.5772/intechopen.99924*

of critical thermal wave number both are amended due to the occurrence of porosity, rotation and magnetic field. It is self-governing of FLR corrections.

In the case of axis of rotation vertical to the magnetic field for transverse wave propagation, we obtained two modes. The first one is a marginal stable mode. The second one symbolizes the impact of porosity, rotation, FLR corrections, radiative heat-loss function and thermal conductivity on thermal unsteadiness of plasma. It is concluded that the condition of unsteadiness is sovereign of porosity, rotation and FLR corrections and it depends on radiative heat-loss function and thermal conductivity. But the growth rate of the organization is exaggerated by the attendance of rotation, porosity and FLR corrections. For the case of non-rotating medium, it is found that condition of radiative unsteadiness is amended by the presence of FLR corrections, porosity and magnetic field, and it demonstrates stabilizing authority.
