# Matter Accretion onto a Conformal Gravity Black Hole

G. Abbas1 and A. Ditta1
1Department of mathematics, The Islamia University of Bahawalpur,
Bahawalpur-63100, Pakistan ghulamabbas<EMAIL_ADDRESS>
###### Abstract

The accretion of test fluids flowing onto a black hole is investigated.
Particularly, by adopting a dynamical Hamiltonian approach, we are capable to
find the critical points for various cases of black hole in conformal gravity.
In these cases, we have analyzed the general solutions of accretion employing
the isothermal equations of state. The steady state and spherically symmetric
accretion of different test fluids onto the conformal gravity black hole has
been considered. Further, we have classified these flows in the context of
equations of state and the cases of conformal gravity black hole. The new
behavior of polytropic fluid accretion is also discussed in all three cases of
black hole. Black hole mass accretion rate is the most important part of this
research in which we have investigated that the Schwarzschild black hole
produce a typical signature than the conformal gravity black hole and
Schwarzschild de-Sitter black hole. The critical fluid flow and the mass
accretion rate have been presented graphically by the impact parameters
$\beta$, $\gamma$, $k$ and these parameters have great significance.
Additionally, the maximum mass rate of accretion fall near the universal and
Killing horizons and minimum rate of accretion occurs in between these
regions. Finally, the results are compared with the different cases of black
hole available in the literature.

Keywords: Accretion; conformal gravity BH; Schwarzschild de-Sitter BH;
Schwarzschild BH.
PACS numbers:04.40.Dg,95.30.Sf,04.50.Gh

##  1 Introduction

Einstein’s theory of gravity was suggested at the end of $1915$ and is still
the standard framework for the account of the chrono-geometrical structure and
the gravitational fields of spacetime. Despite its certain success to account
for a comprehensive observational data [1], the theory is troubled by some
important issues that surely point out the entity of new physics. One of these
issues is the existence of spacetime singularities in the physically
applicable solutions of the Einstein equations. At a singularity, capability
of prediction is lost and the standard physics breaks down. It has been
suggested that the issue of spacetime singularities in the Einstein theory of
gravity can be resolved by the familiar theory of quantum gravity. Since, the
present understanding of physical laws cannot assume a singularity and there
are various efforts to solve the singularity issue [2]-[13]. One of the effort
to solve the singularity issue has been proposed in [10], where, singularity
free black hole (BH) solutions have been proposed in conformally invariant
theories of gravity.

Latter on, conformal (Weyl) gravity characterized by a pure Weyl squared
action has taken a large amount of curiosity as an alternate theory to
Einstein gravity. Every conformal class of the Einstein solution occurs
genuinely as a solution to the conformal gravity by the equation of motion and
the correspondence of conformal gravity. Mainly, in case of the Neumann
boundary condition, conformal gravity can single out an Einstein solution [14,
15]. Further, it has been investigated that unlike Einstein gravity, conformal
gravity is perturbatively renormalizable in four dimensions [16], on basis of
this aspect conformal gravity is more attractive alternative to quantum
gravity [17].

Another interesting aspect of conformal gravity appears from cosmology. It is
an admitted fact that Einstein5 gravity can explain the physics within the
scale of the Solar system satisfactorily, but there are still several unsolved
puzzles when tested to a larger scale, such as the disparity with the
observations of accelerating universe and galactic rotation curves.
Accordingly, anonymous entities, namely dark matter and dark energy, have been
proposed to remove the inconsistency issues. However, one may wonder certainly
wether it is feasible to modify the nature of gravity to illustrate the
physics at a larger scale, while maintaining the appropriate behaviors at the
scale of the Solar system. Subsequently conformal gravity permit more general
solutions than Einstein gravity. It can yield the effective potential
compatible with the observed phenomenon, which designate it as a fascinating
modified theory of gravity [18]-[21].

Furthermore, conformal theory and Einstein gravity may share similar metric
solutions, while thermodynamic quantities of the BH, such as the mass and
entropy depends on the action of the theory rather than the metric. The
thermodynamical feature of the conformal theory BH has been investigated in
the four-dimensional spacetime in Refs.[22]-[25]. In four-dimensions, there is
an important feature that any conformal Einstein metric is automatically a
solution of conformal gravity. In this respect, the light rays and test
particle orbits can promote the understanding of the physical properties of
conformal field equations. Likewise, the motion of light and fluid can be used
to classify the conformal spacetime and to highlight its features. The study
of geodesic motion provides some information about the spacetime
characteristics of the BHs [26]. The solutions of the geodesic equation of
motion with the same mathematical structure but different spacetime has been
discussed in [27, 28]. The Jacobi inversion issues have been solved [29, 30]
with Schwarzschild anti de-Sitter spacetime and next studied in [31, 32]. Some
authors investigated the physical meaningful study of conformal gravity theory
[33]-[35]. The work for on the geodesics study in this theory have been
discussed in [36]. Sultana et al. [37] have studied the applications of
geodesic solutions in the theory of conformal gravity.

The effects of electromagnetic fields around compact stars in conformal
gravity have been explored by Toshmatov et al. [38]. The observational test of
conformal gravity has been done by using X-ray observations of supermassive
BHs Ref.[39]. Also, Toshmatov et al. [40, 41] have discussed explicitly, the
energy conditions and scalar perturbations around BHs in conformal gravity.
Recently, Haydarov et al. [42] have studied the motion of magnetized particle
around BHs in conformal gravity near the external magnetic fields. In view of
the above advancements in conformal theory of gravity, there is a blank space
in the literature that no work has been done for the accretion onto BH in this
theory and we are capable to honestly fill this gap in the present paper. For
this, we plan to investigate spherically symmetric accretion onto BH in
conformal theory of gravity.

In astronomical observations, accretion causes to increase the mass of
gravitating bodies and also the angular momentum as well. The accretion is
used to explain the inflow matter towards the central gravitating bodies. It
has the significant status and a very special importance in the astronomy and
the cosmology. The most of the astrophysical bodies increase substantially in
mass due to accretion. The determination of mass accretion rate for
astrophysical bodies can provide a strong evidence for the presence of compact
objects. In general, the mass of gravitating bodies increases due to accretion
onto it, but when phantom fluid accretes onto a gravitating object then its
mass decreased [43, 44].

The accretion onto celestial objects was started by Hoyle and Lyttleton [45].
Later on, the study of Hoyle and Lyttleton was extended by Bondi and Hoyle
[46] for gaseous dust cloud falling onto a compact object. The transonic-
hydrodynamics accretion for adiabatic fluid onto gravitating object was
investigated by Bondi [47] in Newtonian gravity, he found the existence of
only one saddle-type sonic point in an adiabatic flow . The general
relativistic model of spherically symmetrically accretion onto Schwarzschild
BH was investigated by Michel [48] that has been further extended by many
authors [49]-[54].

It has been observed [56], that the theory of GR is facing a large number of
challenges that has led to the introduction of the dark matter and dark energy
with ordinary matter [57]. Accretion of BHs in the presence of dark energy
studied has been studied by Babichev et al [58]. They have examined the
behavior of different types of dark energy in the locality of Schwarzschild
and electrically charged BHs. Debnath [59] has studied the Babichev et al.
[58] and proposed a study of spherically symmetrically accretion onto some
generalized BHs. Moreover, accretion of phantom fluid onto BHs has been
formulated in [60]-[62].

The BHs, planets and stars are formed due to accretion during the evolutionary
phases of dusty plasma clouds. The existence of supermassive BHs in the center
of galaxies intimate that these BHs could have been developed through the
accretion. The accretion onto higher dimensional BHs [63] and string cloud
model [64] has been investigated in detail and it has been predicted that the
matter accretion rate decreases with the increase of dimension of spacetime.
It has been shown [64] that accretion rate increases by the increase of the
string cloud parameter. Furthermore, many authors [65]-[68] have analyzed the
radial flows of perfect fluids and dark energy onto some BHs in modified
theories of gravity. Sharif and his collaborators [69]-[75] have studied the
phantom accretion onto charge de-Sitter and string BHs.

Another approach to the BH accretion comes from the modified theories of
gravities where, several authors have investigated the radial flows on static
BHs [76], quantum gravity corrections to accretion onto BH [77], accretion
onto a non-commutative BH [78] and accretion onto BH in scalar-tensor-vector
gravity [79]. The accretion of cyclic and heteroclinic fluid flows near $f(R)$
and $f(T)$ theories BHs have been explored in [80, 81]. The Hamiltonian
approach, has been used by several authors [82]-[87] to determine the maximum
accretion flows of perfect fluid.

The purpose of the present research is to study the spherically symmetrically
fluid accretion onto conformal gravity BH. We have analyzed the accretion
process by applying the Hamiltonian approach (as developed in [86, 87]). The
accretion of various forms of fluids (such as ultra-relativistic fluid,
radiation fluid, ultra-stiff fluid and sub-relativistic fluid) onto conformal
gravity BH would provide the possibility of testing the conformal theory in
strong gravity regimes. From the effects of conformal gravity BH parameters
$\gamma$, $k$ and $\beta$, we have investigated that the consequences of fluid
accretion in case of conformal gravity BH are more significant and metric of
the conformal gravity BH is more generalized as compared to BH metrics
available in literature in the context of accretion flows.

The setting of this research is as follows: The conformal gravity BH solution
with different values of parameters is presented in section 2. In section 3,
we have formulated the horizon structure of conformal gravity BH. Section 4 is
dedicated to calculate the conservation laws and general formulism for
accretion onto conformal gravity BH. The sound speed at the critical points is
formulated in section 5. Further, in the subsequent sections, we have applied
the Hamiltonian approach for the accretion with isothermal equation of state.
In section 6, we have analyzed the nature of flow around conformal gravity BH
for various cases of fluids such as ultra-stiff fluid, ultra-relativistic
fluid, radiation fluid and sub-relativistic fluid. The polytropic fluid
accretion has been discussed explicitly in section 7. The mass accretion rate
for various BHs is formulated in section 8. Finally, we concluded the results
of the paper in section 8. Here, we have used the signature of metric as
$(-,+,+,+)$ and the geometric units $G=c=1$.

##  2 Conformal Gravity Black Hole

According to conformally invariant theory of gravity, the action is invariant
with respect to general conformal transformations and coordinate
transformations. Let a regular quantity be singular in a reference frame but
not in alternative one due to a conformal transformation, the singularity is
not physical but exact an artefact of the reference frame. Here, we have a
conformal singularity, which is related to the preference of the conformal
factor but not an intrinsic singularity of spacetime. It is noted that one
cannot apply the same mathematical mechanism for studying spacetime
singularities in conformal gravity and Einstein gravity. For instance, the
Kretschmann scalar and the scalar curvature are not invariant with respect to
conformal transformations, so they are not linked with any intrinsic property
in conformal gravity spacetime. Therefore, the general action of conformal
theory can be developed by on the basis of following four points.

  * •

It is a completely covariant advancement theory of General Relativity.

  * •

It is an additional symmetry principle or local conformal invariance and the
existence of the symmetry principle prevents the Einstein Hilbert action and
cosmological term in the action.

  * •

The conformal transformation of the theory is
$g_{\mu\nu}\rightarrow\Omega^{2}(x)g_{\mu\nu}$.

  * •

The conformal gravity action is defined in terms of Weyl tensor
$C_{\eta\lambda\mu\nu}$ and a coupling constant $\alpha_{g}$, which is a
dimensionless constant and this allows the conformal gravity theory is a
quantum theory of gravity. The action of conformal gravity contributes to
fourth order of equation of motion in the presence of ghosts. The fourth order
equations of motion involve more constants of integration and also solutions
contain more parameters.

The action and the field equations of metric are given by

$\displaystyle S_{CG}$ $\displaystyle=$ $\displaystyle-\alpha_{g}\int
d^{4}x(-g)^{1/2}C_{\eta\lambda\mu\nu}C^{\eta\lambda\mu\nu},$ (1)
$\displaystyle C_{\eta\lambda\mu\nu}$ $\displaystyle=$ $\displaystyle
R_{\eta\lambda\mu\nu}-\frac{1}{2}(g_{\eta\mu}R_{\lambda\nu}-g_{\eta\nu}R_{\lambda\mu}+g_{\lambda\nu}R_{\eta\mu}-g_{\lambda\mu}R_{\eta\nu})+\frac{R}{6}(g_{\eta\mu}g_{\lambda\nu}-g_{\eta\nu}g_{\lambda\mu}),$
(2)

where $\alpha_{g}$ is the pure dimensionless constant and
$C_{\eta\lambda\mu\nu}$ represents the Weyl tensor. The following
gravitational field equations are achieved by varying the action (1) in the
presence of $W_{\mu\nu}$ and the energy momentum tensor $T_{\mu\nu}$ given by

$\displaystyle 2\alpha_{g}W_{\mu\nu}=$ $\displaystyle\frac{1}{2}T_{\mu\nu}.$
(3) $\displaystyle W_{\mu\nu}=$
$\displaystyle\frac{1}{3}\nabla_{\mu}\nabla_{\nu}R-\nabla_{\lambda}\nabla^{\lambda}R_{\mu\nu}+\frac{1}{6}(R^{2}+\nabla_{\lambda}\nabla^{\lambda}R$
$\displaystyle-3R_{\eta\lambda}R^{\eta\lambda})g_{\mu\nu}+2R^{\eta\lambda}R_{\mu\eta\nu\lambda}-\frac{2}{3}RR_{\mu\nu}.$

Therefore, the exact vacuum solution is given by the metric

$\displaystyle ds^{2}=$
$\displaystyle-\left(1-\frac{\beta(2-3\beta\gamma)}{r}-3\beta\gamma+\gamma
r-kr^{2}\right)dt^{2}+$
$\displaystyle\left(1-\frac{\beta(2-3\beta\gamma)}{r}-3\beta\gamma+\gamma
r-kr^{2}\right)^{-1}dr^{2}+r^{2}(d\theta^{2}+\sin^{2}\theta d\phi^{2}).$

The above metric can be written as

$\displaystyle ds^{2}$
$\displaystyle=-A(r)dt^{2}+\frac{1}{A(r)}dr^{2}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2}),$ (7) $\displaystyle A(r)=$ $\displaystyle
1-\frac{\beta(2-3\beta\gamma)}{r}-3\beta\gamma+\gamma r-kr^{2}.$ (8)

Here $A(r)>0$, while $\beta$, $\gamma$ and $k$ are constants of integration.
The choice $\gamma=0$ yields the Schwarzschild-de Sitter solution and
$\gamma=k=0$ the Schwarzschild solution.

If all three parameters of BH $\gamma$, $\beta$ and $k$ are not equal to zero,
the BH can be termed as case 1, which is a general case. If we impose
$\gamma=0$, $A(r)=1-\frac{2\beta}{r}-kr^{2}$ (Schwarzschild-de Sitter
solution), which is the case 2. We obtained the Schwarzschild solution
$A(r)=1-\frac{2\beta}{r}$ for $\gamma=k=0$, which is the case 3. In all cases
$\beta$ is considered as mass of BH and $k$ behaves as cosmological constant
[18, 19].

##  3 Horizon structure

In this section, we study the horizon structure of the conformal gravity
theory BH. For the metric function $A(r)$ given in Eq. (8), the effect of the
parameters $\beta$, $\gamma$ and $k$ is significent. It is well known that the
conformal gravity BH appears to describe a massive body fixed in a conformally
flat space. The conformally flatness of the spherically symmetric space is
characterized by the absolute influence of the massive body. As aforementioned
by the analysis of Weyl tensor, the shattering of conformally flatness is
evident at infinity. In view of the above arguments, the conformal theory BH
has the following structure:

  * •

When one take the mass function $\beta=0$, the conformally flat solution is
obtained [18].

  * •

When $\beta\neq 0$, the coefficient of the $\frac{1}{r}$ term disappear as $r$
approaches to infinity. The non-vanishing contribution exists due to the
constant term $-3\beta\gamma$.

  * •

For the line element, the influence of the Newtonian term $\frac{1}{r}$, which
should reign at small lengths with $\gamma r$ term entity the further dominant
one at larger lengths [18].

Figure 1: The horizon structure of conformal BH displays the behavior of
$A(r)$ versus $r$. Left panel for various values of $\gamma$ and others
parameters are taken as fixed $k=-1$, $\beta=0.96$. In the right panel,
$\beta$ horizons mean, we take various values of $\beta$. In the bottom panel,
$k$ horizons mean, we take various values of $k$. (Figure color online).

Figure 2: The horizon structure of Schwarzschild-de Sitter BH (left panel) and
Schwarzschild BH (right panel) display the behavior of $A(r)$ versus $r$. The
horizon curves of Schwarzschild-de Sitter BH are obtained for
$\beta=0.10,0.19,0.29$ and other parameters are taken as fixed $k=-0.0001$,
$\gamma=0$. The horizon curves of Schwarzschild BH are obtained for
$\beta=0,0.15,0.19,0.25$ and others parameters are taken as fixed $k=0$,
$\gamma=0$. (Figure color online).

In the radial distance $g_{rr}=0$ or $A(r)=0$, one would have an event
horizon. Figure 1 (left panel) has the following key points:

  * •

We have two horizons (green curve) for $\gamma=0.70$.

  * •

We have one horizon (red curve) for $\gamma=0.84$.

  * •

We have no horizons (black curve) for $\gamma=0.95$.

Figure 1 (right panel) has the following key points:

  * •

We have two horizons (green curve) for $\beta=0.73$.

  * •

We have one horizon (red curve) for $\beta=0.86$.

  * •

We have no horizons (black curve) for $\beta=0.96$.

Figure 1 (bottom panel) has the following key points:

  * •

We have two horizons (green curve) for $k=-0.0001$.

  * •

We have one horizon (red curve) for $k=-0.17$.

  * •

We have no horizons (black curve) for $k=-1$.

Figure 2 left panel presents the Schwarzschild-de Sitter BH horizon structure
for $\gamma=0$, there exists only one event horizon with the variations of
$\beta$. According to the right panel of Fig. 2, one gets Schwarzschild
solution for $\gamma=0$ and $k=0$, there is only one event horizon. When we
take $\beta=0$, the Schwarzschild solution is flat (see black curve). It is
noted that when all parameters appearing from conformal gravity are set equal
to zero then the solution is conformally flat [18].

##  4 Spherically Symmetrically Accretion

The study of spherical accretion onto BHs describes the movement of fluid near
the event horizon. We assume two important laws that fully characterized the
spherically accretion of perfect isotropic fluid. One is the law of
conservation of mass and the other is the law of conservation of energy. We
begin by the equation of continuity defined by Rezzolla and Zanotti [88] in
the following form

$\nabla_{\mu}J^{\mu}=0,$ (9)

where $J^{\mu}=nu^{\mu}$ and $n$ is the proper baryon number density and
$u^{\mu}=\frac{dx^{\mu}}{d\tau}$ is the velocity-fluid. Now, we introduce the
matter the energy-momentum tensor as
$T^{\mu\nu}=(\varrho+p)u^{\mu}u^{\nu}+pg^{\mu\nu}$, with $\varrho$ as the
energy density, $p$ is the pressure and
$u^{\mu}=\frac{dx^{\mu}}{d\tau}=(u^{0},u^{1},0,0)$ is $4$-velocity of the
particle. The conservation of energy momentum for perfect fluid is given by

$\nabla_{\mu}T^{\mu\nu}=0.$ (10)

The fluid follows the normalization condition $u^{\mu}u_{\mu}=-1$, which leads
to

$u^{0}=\frac{\sqrt{A(r)+u^{2}}}{A(r)},$ (11)

also

$u_{0}=-\sqrt{A(r)+u^{2}}.$ (12)

Using the equatorial plane $(\theta=\pi/2)$, the mass conservation Eq.(9) can
be written as

$\frac{1}{r^{2}}\frac{d}{dr}(r^{2}nu)=0,$ (13)

after integrating, it gives

$r^{2}nu=c_{1},$ (14)

where, $c_{1}$ is an integration constant. For accretion, the fluid velocity
is $u^{r}<0$ and therefore $c_{1}<0$ in above equation. We define the enthalpy
as $h(\varrho,p,n)=\frac{\varrho+p}{n}$. For smooth flow, Eq. (10) can be
written as

$nu^{\mu}\nabla_{\mu}(hu^{\nu})+g^{\mu\nu}\partial_{\mu}p=0.$ (15)

Moreover, if the entropy of a moving fluid along a streamline is constant,
then the fluid flow should be an isentropic [80]. So, above equation reduces
to

$u^{\mu}\nabla_{\mu}(hu^{\nu})+\partial_{\nu}h=0.$ (16)

The zeroth component of above equation yields

$\partial_{r}(hu_{0})=0,$ (17)

after integrating above equation gives

$h\sqrt{A(r)+(u)^{2}}=c_{2},$ (18)

where $c_{2}$ is constant of integration. Thus, Eqs. (14) and (18) are
important for the critical flow of fluid onto the considered BH.

##  5 Hamiltonian dynamical approach

###  5.1 Sound Speed at Sonic Points

The sonic point (critical point) is a point, where the velocity of the moving
gas must be equal to the local sound speed. In view of this definition, the
maximum accretion rate occurs, when fluid passes through the critical point.
Here, we are interested to calculate the critical points of the flow and the
sound speed at these points, for this, we assume the barotropic fluid with
constant enthalpy that is $h=h(n)$. Therefore, the equation of state for this
flow becomes [89]

$\frac{dh}{h}=a^{2}\frac{dn}{n},$ (19)

where $a$ represents the local sound speed. Also, above equation gives $\ln
h=a^{2}\ln n$. Using Eqs.(11), (18) and (19), we get

$\left[\left(\frac{u}{u_{0}}\right)^{2}-a^{2}\right](\ln
u)_{,r}=\frac{1}{r(u_{0})^{2}}\left[2a^{2}(u_{0})^{2}-\frac{1}{2}rA^{\prime}(r)\right].$
(20)

Now, for critical points, both sides of the above equation must be equal to
zero. So, the sound speed at the critical point becomes

$a^{2}_{c}=\left(\frac{u_{c}}{u_{0c}}\right)^{2},$ (21)

where the quantities, $a_{c}$ , $r_{c}$ and $u_{c}$ designate the sound speed,
distance and velocity of the fluid at the critical point, respectively.
Therefore, another result of Eq. (20) at the critical point is given by

$2a^{2}_{c}(u_{0c})^{2}-\frac{1}{2}r_{c}A^{\prime}_{rc}=0.$ (22)

From Eqs.(21) and (22), one can get the radial velocity at the critical
points, which is given by

$(u_{c})^{2}=\frac{1}{4}r_{c}A^{\prime}_{rc}.$ (23)

Using Eqs. (12), (22) and (23), we obtain

$r_{c}A^{\prime}_{rc}=4a^{2}_{c}[A(r_{c})+(u_{c})^{2}].$ (24)

Thus, the sound speed is given by

$a^{2}_{c}=\frac{r_{c}A^{\prime}_{rc}}{r_{c}A^{\prime}_{r_{c}}+4A(r_{c})}.$
(25)

Thus, one can obtain the critical points as $(r_{c},\pm u_{c})$ using Eqs.
(23) and (25), if one has the value of sound speed.

###  5.2 Isothermal Test Fluids

At very high speed, the fluid does not transfer its the heat to the
surrounding, so it is the the adiabatic situation , we introduce
$a^{2}=\frac{dp}{d\varrho}$ to solve analytically the equations of motion of
the fluid. For the analytical solutions, we define an important equation of
state $p=\omega\varrho$ with the energy density $\varrho$ and equation of the
state parameter $\omega$. Here, $0<\omega\leq 1$ as constrained in [66]. After
combining $p=\omega\varrho$ with $a^{2}=\frac{dp}{d\varrho}$, one gets,
$a^{2}=\omega$. Applying the first law of thermodynamics

$\frac{d\varrho}{dn}=\frac{\varrho+p}{n}=h.$ (26)

Integrating the above equation from the critical point to any point inside the
fluid, we get

$n=n_{c}\exp\left(\int^{\varrho}_{\varrho_{c}}\frac{d\varrho^{\prime}}{\varrho^{\prime}+p(\varrho^{\prime})}\right).$
(27)

With the help of $p=\omega\varrho$, the above Eq. (27) gives

$n=n_{c}\left(\frac{\varrho}{\varrho_{c}}\right)^{\frac{1}{\omega+1}}.$ (28)

Using enthalpy $h(\varrho,p,n)=\frac{\varrho+p}{n}$ in the above relation, we
obtain

$\displaystyle
h=\frac{(\omega+1)\varrho_{c}}{n_{c}}\left(\frac{n}{n_{c}}\right)^{\omega}.$
(29)

The following relation with constant of integration By using Eqs. (29) and
(18), we get

$n^{\omega}\sqrt{A(r)+(u)^{2}}=c_{3}.$ (30)

where $c_{3}=\frac{C_{2}n^{1-\omega}_{c}}{(\omega+1)\varrho_{c}}.$ Combining
Eqs.(26) and (11), we get

$\sqrt{A(r)+(u)^{2}}=C_{3}r^{2\omega}(u)^{\omega}.$ (31)

The Hamiltonian can be defined as [80, 81]

$H=\frac{A^{1-\omega}}{(1-v^{2})^{1-\omega}v^{2\omega}r^{4\omega}}.$ (32)

where $v\equiv\frac{dr}{fdt}$, is the three-dimensional speed for the radial
motion of particle in the equatorial plane. Thus, we have

$v^{2}=\left(\frac{u}{fu^{0}}\right)^{2}=\frac{u^{2}}{u^{2}_{0}}=\frac{u^{2}}{f+u^{2}}.$
(33)

Further, by using (23) and (24) the critical points can be obtained as follows

$\displaystyle(u_{c})^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{4}r_{c}A^{\prime}_{c},$ (34) $\displaystyle(u_{c})^{2}$
$\displaystyle=$
$\displaystyle\omega\left(\frac{1}{4}r_{c}A^{\prime}_{c}+A_{c}\right).$ (35)

Consequently, for the classification of fluid flow, the generalized expression
(32) can be solved numerically by choosing the any value of $\omega$
satisfying $0<\omega\leq 1$. In the coming subsections, we have assumed the
four kinds of fluid such as, ultra-stiff fluid, ultra-relativistic fluid,
radiation fluid and sub-relativistic fluid.

##  6 Analysis of Various Cases of Black Holes

###  6.1 Case 1.

The first and general case of the line element is a conformal BH, which is
given by Eq.(8). There are four subcases of case 1:

  1. 6.1.1

Hamiltonian for ultra-stiff fluid ($\omega=1$):
In this case, we have $p=\varrho$, from the equation of state and critical
point and event horizon are equal, that is $r_{h}=r_{c}$, with the condition
$A_{c}=0$, which can be easily obtained by using Eqs. (34) and (35). The
Hamiltonian (32) for this type of fluid takes the form

$H=\frac{1}{v^{2}{r_{c}}^{4}}.$ (36)

  2. 6.1.2

Hamiltonian for ultra-relativistic fluid ($\omega=1/2$):
In ultra-relativistic fluids, the relation between energy density ($\varrho$)
and pressure ($p$) is $p=\varrho/2$, with $\omega=1/2$ in the equation of
state, it means that the energy density is grater than the pressure in this
case. From Eqs. (34) and (35), we get $r_{c}A^{\prime}_{c}-4A_{c}=0$, which
gives

$2k{r_{c}}^{3}-3\gamma{r_{c}}^{2}+4(3\beta\gamma-1){r_{c}}+5\beta(2-3\beta\gamma)=0.$
(37)

The real solution of above equation is

$\displaystyle{{r_{c}}=\frac{\gamma}{2k}-\frac{24k-72k\beta\gamma+9\gamma^{2}}{{3\times
2^{2/3}kQ}}+{\frac{Q}{6\times 2^{1/3}k}}},$ (38)

where

$\displaystyle
Q=\Big{(}-1080k^{2}\beta+216k\gamma+1620k^{2}\beta^{2}\gamma-648k\beta\gamma^{2}+54\gamma^{3}$
$\displaystyle+\sqrt{4\left(-24k+72k\beta\gamma-9\gamma^{2}\right)^{3}+\left(-1080k^{2}\beta+216k\gamma+1620k^{2}\beta^{2}\gamma-648k\beta\gamma^{2}+54\gamma^{3}\right)^{2}}\Big{)}^{1/3}.$

Using $r_{c}$ from the above expression, we get $v_{c}$ from Eq.(34) and have
two critical points as $(r_{c},\pm{v_{c}})$. The Hamiltonian (32) reduces into
the form:

$H=\frac{\sqrt{A}}{{r_{c}}^{2}v\sqrt{1-v^{2}}}.$ (39)

The graphical behavior can be seen between $v$ and $r_{c}$ with particular
choice of $H=H_{c}$.

  3. 6.1.3

Hamiltonian for radiation fluid ($\omega=1/3$):
For radiation fluids, the equation of state takes the form $p=\varrho/3)$.
Using Eqs. (34) and (35), we have $r_{c}A^{\prime}_{c}-2A_{c}=0$, which leads
to

$\gamma{r_{c}}^{2}-(3\beta\gamma-1){r_{c}}-3\beta(2-3\beta\gamma)=0.$ (40)

The critical solutions are

$r_{c_{\pm}}=\frac{2(3\beta\gamma-1)\pm 2}{2\gamma}.$ (41)

The Hamiltonian (32) in this case is given by

$H=\frac{A^{\frac{2}{3}}}{{r_{c}}^{\frac{4}{3}}v^{\frac{2}{3}}(1-v^{2})^{\frac{2}{3}}}.$
(42)

  4. 6.1.4

Hamiltonian for sub-relativistic fluid ($\omega=1/4$):
For such fluids, the equation of state obeys the form $p=\varrho/4$. This form
shows the energy density exceeds than the isotropic pressure. Using Eqs. (34)
and (35), we get $4A_{c}-3r_{c}A^{\prime}_{c}=0$, which reduces to

$2k{r_{c}}^{3}+\gamma{r_{c}}^{2}-4(3\beta\gamma-1){r_{c}}-7\beta(2-3\beta\gamma)=0.$
(43)

$\displaystyle{r_{c}}=\frac{\gamma}{3k}-\frac{2^{1/3}\left(-3k+9k\beta\gamma-\gamma^{2}\right)}{3kS}+\frac{S}{3\
2^{1/3}k}$ (44)

$S=\Big{(}P+\sqrt{4\left(-3k+9k\beta\gamma-\gamma^{2}\right)^{3}+P^{2}}\Big{)}^{1/3}$
$P=-54k^{2}\beta+9k\gamma+81k^{2}\beta^{2}\gamma-27k\beta\gamma^{2}+2\gamma^{3}$
After determining the critical points, the Hamiltonian is given by

$H=\frac{A^{\frac{3}{4}}}{{r_{c}}v^{\frac{1}{2}}(1-v^{2})^{\frac{3}{4}}}.$
(45)

###  6.2 Case 2

The second case of the line element is known as Schwarzschild-de Sitter BH,
the Hamiltonian and critical points for such BH can be obtained by taking
$\gamma=0$ in Case 1.

###  6.3 Case 3

The third and final case of the line element is called Schwarzschild solution,
the Hamiltonian and critical points for such BH can be obtained by taking
$\gamma=k=0$ in Case 1.

Figure 3: The left panel (conformal gravity BH) displays the behavior of
equation ($36$) with conformal parameters $\gamma=0.1$, $k=-1$, $\beta=1$. The
critical parameters are chosen as $r_{c}\approx 0.9738$, $v_{c}=1$,
$H_{c}\approx 1.112$. The right panel (Schwarzschild de-Sitter BH) displays
the behavior of Eq.($36$) with Schwarzschild de-Sitter BH parameters
$\gamma=0$, $k=-1$, $\beta=1$. The critical parameters are chosen as
$r_{c}\approx 1.001$, $v_{c}=1$, $H_{c}\approx 1.001$. The bottom panel
(Schwarzschild BH) displays the behavior of Eq.($36$) with Schwarzschild BH
parameters $\gamma=0$, $k=0$, $\beta=1$. The critical parameters are chosen as
$r_{c}=2$, $v_{c}=1$, $H_{c}\approx 0.0625$. The representation of colors for
$H=H_{c}\rightarrow$ orange, $H>H_{c}\rightarrow$ magenta and yellow (Figure
color online).

###  6.4 Visualization of results for all cases

This section elaborates the results of ultra-stiff fluids ($\omega=1$), ultra-
relativistic fluids ($\omega=\frac{1}{2}$), radiation fluids
($\omega=\frac{1}{3}$) and sub-relativistic fluids ($\omega=\frac{1}{4}$) for
conformal gravity BH (Case 1), Schwarzschild de-Sitter BH (Case 2) and
Schwarzschild BH (Case 3).

  1. 1.

Ultra-stiff fluids ($\omega=1$):
Figure 3 signifies the velocity $v$ of moving fluid versus the radius $r$
corresponding to various cases of BH. The fluid motion occurs in the two
regions, upper curves show the region $v>0$ whereas the lower curves for the
region $v<0$ for the three cases of BH. It is noted here that the critical
point is always equal to the critical horizon for ultra-stiff fluid. Moreover,
the critical points are closer in conformal gravity BH and Schwarzschild de-
Sitter BH as compare to the Schwarzschild BH. It is observed that the critical
points are close to the singularity in conformal gravity BH and Schwarzschild
de-Sitter BH than the Schwarzschild BH.

  2. 2.

Ultra-relativistic fluids ($\omega=\frac{1}{2}$):
We see the velocity essence of moving fluid $v$ versus the radius $r$ for
aforementioned all three cases of conformal gravity BH by putting the
corresponding values of $A(r)$ in Fig. 5. The critical values of horizon,
radius and velocity $(r_{h},r_{c},v_{c})$ are nearly equal to (0.974, 1.297,
0.707107), (1.3333, 1.3282, 0.707107) and (2.0025, 2.524, 0.707107) for the
cases $1$, $2$ and $3$, respectively. For $H=H_{c}=1.3030$ (Case 1),
$H=H_{c}=1.27174$ (Case 2) and $H=H_{c}=0.1431$ (Case 3), the behavior of
curves is seen through the CPs $(r_{c},\pm{v_{c}})$. It is shown that the
fluid outflow starts from the horizon and induces by the high pressure. The
curves behaviors shown in Fig. 4 are not all physical. For increasing radius
$r$, the region must be $v>0$ (positive), while for decreasing radius the
region must be $v<0$ (negative). The flow in the yellow, magenta, orange and
purple curves is unphysical. The fluid flow increases as $v>0$ and decreases
the radius, so there is neither fluid outflow nor an accretion. Only the red
curves display the supersonic accretion in the region $v>v_{c}$ and subsonic
accretion in the region $v<v_{c}$. It is also noted here that the fluid
elements are closer to Case 1 and Case 2 instead of Case 3, respectively.
Consequently, it is viewed that the CPs are closer for Case 1 and Case 2 as
compare to Case 3, respectively. It is revealed here that the fluid
experiences the particle emission or fluid outflow for $v>0$ while fluid
accretes for $v<0$. Specially, in Case 3 (Schwarzschild case), only the orange
curve represents the unphysical while the other four color curves show the
physical behavior which is clear as compare to Case 1 and Case 2. Further,
Fig. 4 shows the following four key points.

     * •

We notice, the subsonic/supersonic accretion occurs in the ranges $-v_{c}<v<0$
and $-1<v<-v_{c}$, whereas supersonic/subsonic fluid outflows for $v_{c}<v<1$
and $0<v<v_{c}$, respectively.

     * •

The emission of particles for $v>v_{c}$ and thus purely supersonic accretion
for $v<-v_{c}$.

     * •

The subsonic outflow followed by the subsonic accretion with $v_{c}>v>-v_{c}$.

     * •

The upper plot shows the supersonic outflow followed by subsonic motion, while
the lower plot shows the subsonic accretion followed by supersonic accretion.

Figure 4: For the accretion flow, the left panel (conformal gravity BH)
displays the behavior of Eq.($39$) with conformal parameters $\gamma=0.1$,
$k=-1$, $\beta=1$. The critical parameters involve $r_{c}\approx 1.29701$,
$v_{c}=0.70716$, $H_{c}\approx 1.30306$. Right panel (Schwarzschild de-Sitter
BH) displays the behavior of ($39$) with Schwarzschild de-Sitter BH parameters
$\gamma=0$, $k=-1$, $\beta=1$. The values of the critical parameters are
$r_{c}\approx 1.32827$, $v_{c}=0.70716$, $H_{c}\approx 1.27174$. Bottom panel
(Schwarzschild BH) displays the behavior of Eq.($39$) with Schwarzschild BH
parameters $\gamma=0$, $k=0$, $\beta=1$. The values of the critical parameters
are $r_{c}=2.5$, $v_{c}=0.70716$, $H_{c}\approx 0.143108$. The representation
of colors in $H=H_{c}\rightarrow$ orange, $H>H_{c}\rightarrow$ magenta and
yellow, $H<H_{c}\rightarrow$ purple and red (figure color online).

Consequently, we observe that the starting point of the fluid outflow is at
horizon due to its very high pressure which influences to divergence and as a
result, the fluid with its own pressure flows back to spatial infinity [91].
We also observe from Fig. 4, the supersonic accretion (fluid outflow) followed
by subsonic accretion (fluid inflow) stops inside the horizon and it does not
give support for the claim that “the flow must be supersonic at the horizon”
[92]. It means that for conformal gravity BH, Schwarzschild de-Sitter BH and
Schwarzschild BH the flow of the fluid is neither supersonic nor transonic
near the horizon [93, 94]. These results agree with fine tunning and
instability issues in dynamical systems. The stability issue is connected to
the nature of the saddle points (CPs ($r_{c},v_{c}$) and ($r_{c},-v_{c}$)) of
the Hamiltonian system. The analysis of stability could be done by using
Lyapunov’s theorem or linearization of dynamical system [95, 96, 97] and their
variations [98]. Another stability issue is the outflow of the fluid starts in
the surrounding of horizon under the effect of pressure divergent. This
outflow is unstable because it follows a subsonic path passing through the
saddle point ($r_{c},v_{c}$) and becomes supersonic with speed approaches the
speed of the light. The point ($r=r_{h},v=0$) can be observed as attractor as
well as repeller where solution curves converge and diverge in the
cosmological point of view [92, 98].

Figure 5: In the physical structure of accretion, left panel (conformal
gravity BH) displays the behavior of ($42$) with conformal parameters
$\gamma=0.1$, $k=-1$, $\beta=1$. The critical parameters involve $r_{c}\approx
3$, $v_{c}=0.5773$, $H_{c}\approx 1.95007$. Right panel (Schwarzschild de-
Sitter BH) displays the behavior of ($42$) with Schwarzschild de-Sitter BH
parameters $\gamma=0$, $k=-1$, $\beta=1$. The critical parameters involve
$r_{c}\approx 3$, $v_{c}=0.5773$, $H_{c}\approx 1.93626$. Bottom panel
(Schwarzschild BH) displays the behavior of ($42$) with Schwarzschild BH
parameters $\gamma=0$, $k=0$, $\beta=1$. The critical parameters involve
$r_{c}=3$, $v_{c}=0.5773$, $H_{c}\approx 0.20998$. The representation of
colors in $H=H_{c}\rightarrow$ orange, $H>H_{c}\rightarrow$ magenta and
yellow, $H<H_{c}\rightarrow$ purple and red (figure color online).

Figure 6: In the physical structure of accretion, left panel (conformal
gravity BH) displays the behavior of ($44$) with conformal parameters
$\gamma=0.1$, $k=-1$, $\beta=1$. The critical parameters involve $r_{c}\approx
1.54267$, $v_{c}=0.5$, $H_{c}\approx 2.38717$. Right panel (Schwarzschild de-
Sitter BH) displays the behavior of ($44$) with Schwarzschild de-Sitter BH
parameters $\gamma=0$, $k=-1$, $\beta=1$. The critical parameters involve
$r_{c}\approx 2.25826$, $v_{c}=0.5$, $H_{c}\approx 2.6812$. Bottom panel
(Schwarzschild BH) displays the behavior of ($44$) with Schwarzschild BH
parameters $\gamma=0$, $k=0$, $\beta=1$. The critical parameters are taken as
$r_{c}=3.5$, $v_{c}=0.5$, $H_{c}\approx 0.26556$. The representation of colors
in $H=H_{c}\rightarrow$ orange, $H>H_{c}\rightarrow$ magenta and yellow,
$H<H_{c}\rightarrow$ purple and red (figure color online).

  3. 3.

Radiation fluids ($\omega=\frac{1}{3}$):
The rule played by the different parameters for the velocity profile $v$
versus radius $r$ is important in Fig. 5. The settlement of curves
corresponding to $H=H_{c}$ is colored Orange. The Magenta and Yellow curves
relate to $H<H_{c}$ and the Purple and Red plots to $H>H_{c}$. In Fig. 5, we
see the supersonic outflows of the fluid in the range $v_{c}<v<1$. In this
flow, we observe that the orange curve exactly passes through the critical
point ($r_{c}=3.0$) as compared to other curves. The critical points are very
close in Case 1 and Case 2 than Case 3. So, the orange, magenta and yellow
curves are purely supersonic outflows for $(v>v_{c})$ and these curves pass
through the critical speed Fig. 6 in Schwarzschild BH. One can see the similar
behavior for conformal gravity BH and Schwarzschild de-Sitter BH. The vertical
lines which are closer to the horizon are unphysical for $H>H_{c}$.

  4. 4.

Sub-relativistic fluids ($\omega=\frac{1}{4}$):
We have analyzed the accretion of the sub-relativistic fluid
$\omega=\frac{1}{4}$ for the conformal gravity BH. We have plotted the sub-
relativistic fluid motion versus the radius in Fig. 6. This figure shows that
all the solution curves are not passing through the critical velocity, which
confirms to the new solution in Schwarzschild BH. Since, the critical velocity
is located at $v_{c}\approx 0.5$ but the maximum speed in case of
Schwarzschild BH approaches to $v=0.06$. So, there is no accretion flow around
Schwarzschild BH for sub-relativistic fluid. We have observed the supersonic
accretion at $v>v_{c}$ followed by subsonic accretion at $0<v<v_{c}$ which
stop at the horizon for conformal gravity BH and Schwarzschild de-Sitter BH.
Furthermore, we have the supersonic accretion with $v<-v_{c}$ followed by the
subsonic accretion at $0<\nu<-\nu_{c}$.

##  7 Polytropic Fluids Accretion

The polytropic equation of state [80, 81, 82] is

$p=G(n)=\Gamma n^{\alpha},$ (46)

where $\Gamma$ and $\alpha$ are constants. One can consider the general
constraint $\alpha>1$ for an ordinary matter. The specific enthalpy can be
define [82] as

$h=m+\frac{\Gamma\alpha n^{\alpha-1}}{\alpha-1}.$ (47)

Three dimensional sound speed with the help of enthalpy is given by

$a^{2}=\frac{(\alpha-1)U}{m(\alpha-1)+U},$ (48)

where $U=\gamma\alpha n^{\alpha-1}$. Another useful result can be obtained
with the help of speed of sound, which is given by

$h=m\frac{\alpha-1}{\alpha-1-a^{2}},$ (49)

and therefore

$h=m\left(1+X\left(\frac{1-v^{2}}{r^{4}A(r)v^{2}}\right)^{(\alpha-1)/2}\right),$
(50)

where

$X=\frac{\Gamma\alpha
n_{c}^{\alpha-1}}{m(\alpha-1)}\left(\frac{r^{5}_{c}A^{\prime}(r_{c})}{4}\right)^{(\frac{\alpha-1}{2})}=constant>0,$
(51)

and $X>0$ is a constant. From the above result, it is clear that the constant
$X$ depends on the BH parameters and also on the test fluids. The final form
of Hamiltonian system can be obtained by putting Eq. (50) into (32), which is
given by

$H=\frac{A(r)}{1-v^{2}}\left[1+X\left(\frac{1-v^{2}}{r^{4}A(r)v^{2}}\right)^{(\alpha-1)/2}\right]^{2},$
(52)

1\. Hamiltonian for conformal gravity BH

$\displaystyle H=$
$\displaystyle\frac{\Big{(}1-\frac{\beta(2-3\beta\gamma)}{r}-3\beta\gamma+\gamma
r-kr^{2}\Big{)}}{1-v^{2}}\Big{[}1$
$\displaystyle+X\Big{(}\frac{1-v^{2}}{r^{4}v^{2}\Big{(}1-\frac{\beta(2-3\beta\gamma)}{r}-3\beta\gamma+\gamma
r-kr^{2}\Big{)}}\Big{)}^{(\alpha-1)/2}\Big{]}^{2}.$

2\. Hamiltonian for Schwarzschild de-Sitter BH

$\displaystyle H=$
$\displaystyle\frac{\Big{(}1-\frac{2\beta}{r}-kr^{2}\Big{)}}{1-v^{2}}\Big{[}1+X\Big{(}\frac{1-v^{2}}{r^{4}v^{2}(1-\frac{2\beta}{r}-kr^{2})}\Big{)}^{(\alpha-1)/2}\Big{]}^{2}.$
(54)

3\. Hamiltonian for Schwarzschild BH

$\displaystyle H=$
$\displaystyle\frac{1-\frac{2\beta}{r}}{1-v^{2}}\Big{[}1+X\Big{(}\frac{1-v^{2}}{r^{4}v^{2}(1-\frac{2\beta}{r})}\Big{)}^{(\alpha-1)/2}\Big{]}^{2}.$
(55)

It is analyzed from the Hamiltonian results that $\frac{dA(r)}{dr}>0$ for all
$r$.

Adopting the technique in [80, 81, 82], one can get the following relation

$(\alpha-1-v^{2}_{c})\left(\frac{1-v^{2}_{c}}{r^{4}_{c}A(r_{c})v^{2}_{c}}\right)^{\frac{\alpha-1}{2}}=\frac{n_{c}}{2X}\left(r^{5}_{c}A^{\prime}(r_{c})\right)^{\frac{1}{2}v^{2}_{c}},$
(56)
$v^{2}_{c}=\frac{r_{c}A^{\prime}_{rc}}{r_{c}A^{\prime}_{r_{c}}+4A(r_{c})}.$
(57)

Figure 7: For the polytropic fluid accretion, left panel (conformal gravity
BH) displays the behavior of (7). Right panel (Schwarzschild de-Sitter BH)
displays the behavior of (54). Bottom panel (Schwarzschild BH) displays the
behavior of (55) (figure color online).

Figure 7 represents the contour plots for conformal BH (left panel),
Schwarzschild de-Sitter BH (right panel) and Schwarzschild BH (bottom panel)
with $n_{c}=0.15$, $X=5$, $\alpha=5/3$. We have presented the behavior of
matter by taking the sonic points $r_{c}\approx 1.9855$, $v_{c}\approx
0.56218$, $H=H_{c}\simeq 4.2876$ for conformal BH, $r_{c}\approx 3.7859$,
$v_{c}\approx 0.44216$, $H=H_{c}\simeq 2.1377$ for Schwarzschild de-Sitter BH
and $r_{c}\approx 5.2865$, $v_{c}\approx 0.24211$, $H=H_{c}\simeq 1.9374$ for
Schwarzschild BH. It is analyzed that the critical flow for conformal BH,
Schwarzschild de-Sitter BH and Schwarzschild BH starts from subsonic accretion
and then follows the supersonic accretion escaping the saddle point (sonic
point) and ends at the Killing horizon. The supersonic accretion begins from
the region of Killing horizon and ends at the subsonic accretion as $r$
approaches to infinity. It has been observed that the accretion behavior of
various BHs is different at critical points of polytropic test fluids case.
Also, it has been observed that at the trajectory of conformal BH, the
critical points are closer (see red curve), for Schwarzschild de-Sitter BH,
the critical points are also distant (see red curve) and for Schwarzschild BH,
the critical points are also distant (see red curve). In all these cases, the
trajectories do not pass through the saddle point (sonic point).

##  8 Black Hole’s Mass Accretion Rate

The mass accretion rate of BH is an important aspect in the study of
accretion, for this purpose, we have calculated the accretion rate
corresponding to $A(r)$. Specially, we have observed the effects of radius on
the accretion rate. Generally, mass accretion rate is the area times flux at
the boundary of BH and is denoted by $\dot{M}$, it evaluates the BH mass per
unit time. Here, we consider the general expression for the mass accretion
rate as $\dot{M}\mid_{rh}=4\pi r^{2}T^{r}_{t}\mid_{rh}$ [90], the energy
momentum tensor for perfect fluid can be as used. Since, the dynamical system
is conserved $\Delta_{\mu}J^{\mu}=0$ and $\nabla_{\nu}T^{\mu\nu}=0$. Thus, due
to this conserved system, Eqs. (14) and (18) give

$r^{2}u(\varrho+p)\sqrt{A(r)+(u)^{2}}=L_{0},$ (58)

where $L_{0}$ is the constant. Now, by taking the continuity equation
(relativistic energy flux) and the equation of state $p=p(\varrho)$, we get

$\frac{d\varrho}{\varrho+p}+\frac{du}{u}+\frac{2}{r}dr=0.$ (59)

After integrating, we obtain

$ru\exp\left[\int^{\varrho}_{\varrho\infty}\frac{d\varrho^{\prime}}{\varrho^{\prime}+p(\varrho^{\prime})}\right]=-L_{1},$
(60)

where $L_{1}$ is the constant of integration and $\varrho_{\infty}$ represents
the fluid density at infinity. Here, the minus is taken due to $u<0$. Dividing
Eq. (58) with (60), we get

$L_{3}=-\frac{L_{0}}{L_{1}}=(\varrho+p)\sqrt{A(r)+(u)^{2}}\exp\left[-\int^{\varrho}_{\varrho\infty}\frac{d\varrho^{\prime}}{\varrho^{\prime}+p(\varrho^{\prime})}\right],$
(61)

where $L_{3}$ is a constant. At infinity,
$L_{3}=\varrho_{\infty}+p(\varrho_{\infty})=-\frac{L_{0}}{L_{1}}$, with
$L_{0}=(\varrho+p)u^{0}ur^{2}=-L_{1}(\varrho_{\infty}+p(\varrho_{\infty}))$.
The problem is spherically symmetrically static at equatorial plane, so, the
mass flux equation $\nabla_{\mu}J^{\mu}=0$ takes the form

$r^{2}un=L_{2}.$ (62)

Where $L_{2}$ is an integration constant. Dividing the Eq. (58) with (62), we
get

$\frac{\varrho+p}{n}\sqrt{A(r)+(u)^{2}}=\frac{L_{0}}{L_{2}}\equiv L_{4},$ (63)

where $L_{4}=\frac{(\varrho_{\infty}+p_{\infty})}{n_{\infty}}$. Using Eq.
(58), the mass of BH takes the following form

$\dot{M}=-4\pi r^{2}u(\varrho+p)\sqrt{A(r)+(u)^{2}}=-4\pi L_{0}.$ (64)

Further, it takes the form

$\dot{M}=4\pi L_{1}(\varrho_{\infty}+p(\varrho_{\infty})).$ (65)

Above equation gives the valid result for any nature of fluids. Thus, we have

$\dot{M}=4\pi L_{1}(\varrho+p)|_{r=r_{h}},$ (66)

Assuming the isothermal equation of state $(p=\omega\varrho)$, which implies
that $(\varrho+p)=\varrho(1+\omega)$. Then, using the Eq. (60), it leads us to

$\varrho=\left[-\frac{L_{1}}{r^{2}u}\right]^{1+\omega}.$ (67)

By the expression of $\varrho$ in Eq. (58) one can obtain the following
general equation

$(u)^{2}-\frac{L^{2}_{0}L^{-2(1+\omega)}_{1}}{(1+\omega)^{2}}r^{4\omega}(-u)^{2\omega}+A(r)=0.$
(68)

It can be solved for fluid velocity $u$ with any value of $\omega$. One can
calculate the energy density $\varrho$ by using $u$ with $\omega$.
Exact solution for ultra-stiff fluids $\omega=1$:
By assuming $\omega=1$ in Eq. (63) and (67), one can calculate the radial
velocity and energy-density of ultra-stiff fluids, that is

$u=\pm L_{1}^{2}\sqrt{\frac{A(r)}{L^{2}_{0}r^{4}-4L^{4}_{1}}},$ (69)

also the energy density is given by

$\varrho=\frac{(L^{2}_{0}r^{4}-4L^{4}_{1})}{4L^{2}_{1}r^{4}A(r)}.$ (70)

From Eqs. (66) and (70), the mass accretion rate of conformal gravity BH can
obtained in the following form

$\dot{M}=\frac{2\pi(L^{2}_{0}r^{4}-4L^{4}_{1})}{L_{1}r^{3}\left[3\beta^{2}\gamma-2\beta+(1-3\beta\gamma)r+\gamma
r^{2}-kr^{3}\right]}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(\rm{Case}\quad\textbf{1}).$
(71)

Similarly, following the same method, we can find the mass accretion rate for
Schwarzschild de-Sitter and Schwarzschild BHs

$\displaystyle\dot{M}$ $\displaystyle=$
$\displaystyle\frac{2\pi(L^{2}_{0}r^{4}-4L^{4}_{1})}{L_{1}r^{3}\left[-2\beta+r-kr^{3}\right]}.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(\rm{Case}\quad\textbf{2})$
(72) $\displaystyle\dot{M}$ $\displaystyle=$
$\displaystyle\frac{2\pi(L^{2}_{0}r^{4}-4L^{4}_{1})}{L_{1}r^{3}\left[-2\beta+r\right]}.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(\rm{Case}\quad\textbf{3})$
(73)

Figure 8: In this figure, left panel (conformal gravity BH) displays the
behavior of Eq. (71) with conformal parameters $\gamma=0.1$, $k=-1$,
$\beta=1$. Right panel (Schwarzschild de-Sitter BH) displays the behavior of
Eq. (72) for Schwarzschild de-Sitter BH parameters $\gamma=0$, $k=-1$,
$\beta=1$. Bottom panel (Schwarzschild BH) displays the behavior of Eq. (73)
for Schwarzschild BH parameters $\gamma=0$, $k=0$, $\beta=1$. Other constants
are taken as $L_{0}=0.90$, $L_{1}=0.5$ (figure color online).

In Fig. 8, we plot the mass accretion-rate $\dot{M}$ versus the radius $r$ for
aforemention BHs in ultra-stiff fluid, other parameters $\gamma$, $k$ and
$\beta$ are taken as fixed. In the left panel (conformal gravity BH) the
accretion rate is increased by decreasing the parameter $\gamma$. It has been
noted that the maximum accretion rate occurs for overlapping the critical
radius in the presence of different values of $\gamma$. For the left panel the
values of mass accretion rate are: $\dot{M}=0.15,0.235,0.29,0.36$ for
$\gamma=12.1,6.1,2.1,0.1$, we have the overlapping radius $r\approx 1.931$,
respectively. The same critical points are seen in the right panel
(Schwarzschild de-Sitter BH) in the presence of parameter $k$. The critical
values are: $\dot{M}=0.15,0.235,0.29,0.36$ for $k=-2.5,-2.0,-1.5,-1.0$, we
have the overlapping radius $r\approx 1.931$, respectively. The right panel
increases the mass of Schwarzschild de-Sitter BH by increasing the
cosmological constant parameter $k$. Now, the mass of Schwarzschild BH occurs
in the bottom panel where we observe the maximum accretion rate
$\dot{M}=15.5,16.5,17.5,18.5$ occur for different values of parameters

  * •

$\beta=0.7$, corresponding to $r\approx 2.536$

  * •

$\beta=0.8$, corresponding to $r\approx 2.936$

  * •

$\beta=0.9$, corresponding to $r\approx 3.156$

  * •

$\beta=1.0$, corresponding to $r\approx 3.956$

The mass accretion rate of Schwarzschild BH is increased by increasing the
mass function $(\beta)$. Hence, we conclude that the mass of Schwarzschild BH
is larger as compare to conformal gravity BH and Schwarzschild de-Sitter BH.
Also, it is concluded that the conformal parameters are critically important
for the maximum accretion rate in ultra-stiff fluids.

Exact solution for ultra-relativistic fluids $\omega=1/2$:
By assuming $\omega=1/2$ in Eqs. (63) and (67), we calculate the radial
velocity and the energy-density of the radiation fluids, that is

$\displaystyle u$ $\displaystyle=$
$\displaystyle\frac{2r^{2}L^{2}_{0}+\sqrt{4r^{2}L^{4}_{0}-81A(r)L^{6}_{1}}}{9L^{3}_{1}}.$
(74) $\displaystyle\varrho$ $\displaystyle=$ $\displaystyle
27\left(\frac{L^{4}_{1}}{r^{2}(2r^{2}L^{2}_{0}+\sqrt{4r^{2}L^{4}_{0}-81A(r)L^{6}_{1}})}\right)^{3/2}.$
(75) $\displaystyle\dot{M}$ $\displaystyle=$ $\displaystyle 216\pi
L_{1}\left(\frac{L^{4}_{1}}{r^{2}(2r^{2}L^{2}_{0}+\sqrt{4r^{2}L^{4}_{0}-81A(r)L^{6}_{1}})}\right)^{3/2}.$
(76)

1\. The mass accretion rate of conformal gravity BH is

$\displaystyle\dot{M}=$ $\displaystyle 216\pi L_{1}$
$\displaystyle\times\left(\frac{L^{4}_{1}}{r^{2}(2r^{2}L^{2}_{0}+\sqrt{4r^{2}L^{4}_{0}-81(1-\frac{\beta(2-3\beta\gamma)}{r}-3\beta\gamma+\gamma
r-kr^{2})L^{6}_{1}})}\right)^{3/2}.$

2\. The mass accretion rate of Schwarzschild de-Sitter BH is

$\displaystyle\dot{M}=$ $\displaystyle 216\pi L_{1}$
$\displaystyle\times\left(\frac{L^{4}_{1}}{r^{2}(2r^{2}L^{2}_{0}+\sqrt{4r^{2}L^{4}_{0}-81(1-\frac{2\beta}{r}-kr^{2})L^{6}_{1}})}\right)^{3/2}.$

3\. The mass accretion rate of Schwarzschild BH is

$\displaystyle\dot{M}=$ $\displaystyle 216\pi L_{1}$
$\displaystyle\times\left(\frac{L^{4}_{1}}{r^{2}(2r^{2}L^{2}_{0}+\sqrt{4r^{2}L^{4}_{0}-81(1-\frac{2\beta}{r})L^{6}_{1}})}\right)^{3/2}.$

In Fig. 9, we plot the mass accretion-rate ($\dot{M}$) versus the radius ($r$)
for aforemention BHs in ultra-relativistic fluid. The left panel (conformal
gravity BH) shows that the accretion rate is decreasing for larger value of
$r$, that is $\dot{M}=7000$ for $r=0.8$ and Killing horizon is at
$r_{KH}\approx 0.8$ whereas the universal horizon is at $r_{UH}\approx 0.1$.
The accretion rate is increasing for smaller values of the radius, that is
$\dot{M}>8000$ for $r\simeq 0.65$ and the Killing horizon is at $r_{KH}\approx
0.65$ whereas the universal horizon is at $r_{UH}\approx 0.2$. We can say that
the mass of the conformal gravity BH decreases whereas the radius increases,
on the other hand the accretion rate is an increasing function of radius. In
this case, the critical points are overlapping at the universal horizon. This
implies that the mass of Schwarzschild de-Sitter BH decreases whereas the
radius increases, on the other hand the accretion rate is an increasing
function of radius in the presence of cosmological constant $k$.

Now, the mass of Schwarzschild BH occurs in the bottom panel where we observe
the maximum accretion rate acquire for smaller radius. The critical points are
overlapping at the universal horizon but these points are away from the
killing horizon in the presence of mass function $\beta$. Four key points are
observed for the Schwarzschild BH:

  * •

$\beta=0.7$, corresponding to $r_{UH}\approx 0.135$, $r_{KH}\approx 0.65$.

  * •

$\beta=0.8$, corresponding to $r_{UH}\approx 0.133$, $r_{KH}\approx 0.70$.

  * •

$\beta=0.9$, corresponding to $r_{UH}\approx 0.131$, $r_{KH}\approx 0.75$.

  * •

$\beta=1.0$, corresponding to $r_{UH}\approx 0.129$, $r_{KH}\approx 0.80$.

The accretion rate of Schwarzschild BH is increasing with the decreasing
values of radius. So, it is a decreasing function of $r$. Hence, we conclude
that the mass of Schwarzschild BH is larger as compare to conformal gravity BH
and Schwarzschild de-Sitter BH.

Figure 9: In the mass accretion rate, left panel (conformal gravity BH)
displays the behavior of Eq. (8) with conformal parameters $\gamma=0.1$,
$k=-1$, $\beta=1$. Right panel (Schwarzschild de-Sitter BH) displays the
behavior of Eq. (8) with Schwarzschild de-Sitter BH parameters $\gamma=0$,
$k=-1$, $\beta=1$. Bottom panel (Schwarzschild BH) displays the behavior of
Eq. (8) with Schwarzschild BH parameters $\gamma=0$, $k=0$, $\beta=1$. Other
constants are taken as $L_{0}=0.90$, $L_{1}=0.5$.

Exact solution for radiation fluid $\omega=1/3$:
By assuming $\omega=1$ in Eqs. (63) and (67), we calculate the radial velocity
and the energy-density of the ultra-stiff fluids, that is

$\displaystyle u=$
$\displaystyle\Big{[}\frac{\Big{(}-32A(r)L_{1}^{4}+\sqrt{1024A(r)^{2}L_{1}^{8}-27r^{4}L_{0}^{6}}\Big{)}^{1/3}}{4L_{1}^{2}}$
$\displaystyle+\frac{3r^{4/3}L_{0}^{2}}{4L_{1}^{2/3}\Big{(}-32A(r)L_{1}^{4}+\sqrt{1024A(r)^{2}L_{1}^{8}-27r^{4}L_{0}^{6}}\Big{)}^{1/3}}\Big{]}^{2/3}.$

The energy density of the fluid is given by

$\displaystyle\varrho=$
$\displaystyle\Big{[}\frac{L_{1}}{r^{2}}\Big{]}^{\frac{4}{3}}\Big{[}\frac{\Big{(}-32A(r)L_{1}^{4}+\sqrt{1024A(r)^{2}L_{1}^{8}-27r^{4}L_{0}^{6}}\Big{)}^{1/3}}{4L_{1}^{2}}$
$\displaystyle+\frac{3r^{4/3}L_{0}^{2}}{4L_{1}^{2/3}\Big{(}-32A(r)L_{1}^{4}+\sqrt{1024A(r)^{2}L_{1}^{8}-27r^{4}L_{0}^{6}}\Big{)}^{1/3}}\Big{]}^{\frac{-8}{9}}.$

The general form of the mass of BH is given by

$\displaystyle\dot{M}=$ $\displaystyle\Big{[}\frac{8\pi
L_{1}^{\frac{7}{3}}}{r^{\frac{8}{3}}}\Big{]}\Big{[}\Big{(}-32A(r)L_{1}^{4}+\sqrt{1024A(r)^{2}L_{1}^{8}-27r^{4}L_{0}^{6}}\Big{)}^{1/3}\Big{(}{4L_{1}^{2}}\Big{)}^{-1}+\frac{3r^{4/3}L_{0}^{2}}{4L_{1}^{2/3}}$
$\displaystyle\times\Big{(}-32A(r)L_{1}^{4}+\sqrt{1024A(r)^{2}L_{1}^{8}-27r^{4}L_{0}^{6}}\Big{)}^{-1/3}\Big{]}^{\frac{-8}{9}}.$

1\. The mass accretion rate of conformal gravity BH is

$\displaystyle\dot{M}=$ $\displaystyle\Big{[}\frac{8\pi
L_{1}^{\frac{7}{3}}}{r^{\frac{8}{3}}}\Big{]}\Big{[}\Big{(}-32(1-\frac{\beta(2-3\beta\gamma)}{r}-3\beta\gamma+\gamma
r-kr^{2})L_{1}^{4}$
$\displaystyle+\sqrt{1024\Big{(}1-\frac{\beta(2-3\beta\gamma)}{r}-3\beta\gamma+\gamma
r-kr^{2}\Big{)}^{2}L_{1}^{8}-27r^{4}L_{0}^{6}}\Big{)}^{1/3}\Big{(}{4L_{1}^{2}}\Big{)}^{-1}$
$\displaystyle+\frac{3r^{4/3}L_{0}^{2}}{4L_{1}^{2/3}}\times\Big{(}-32(1-\frac{\beta(2-3\beta\gamma)}{r}-3\beta\gamma+\gamma
r-kr^{2})L_{1}^{4}$
$\displaystyle+\sqrt{1024\Big{(}1-\frac{\beta(2-3\beta\gamma)}{r}-3\beta\gamma+\gamma
r-kr^{2}\Big{)}^{2}L_{1}^{8}-27r^{4}L_{0}^{6}}\Big{)}^{-1/3}\Big{]}^{\frac{-8}{9}}.$

2\. The mass accretion rate of Schwarzschild de-Sitter BH is

$\displaystyle\dot{M}$ $\displaystyle=\Big{[}\frac{8\pi
L_{1}^{\frac{7}{3}}}{r^{\frac{8}{3}}}\Big{]}\Big{[}\Big{(}-32(1-\frac{2\beta}{r}-kr^{2})L_{1}^{4}$
$\displaystyle+\sqrt{1024\Big{(}1-\frac{2\beta}{r}-kr^{2}\Big{)}^{2}L_{1}^{8}-27r^{4}L_{0}^{6}}\Big{)}^{1/3}\Big{(}{4L_{1}^{2}}\Big{)}^{-1}$
$\displaystyle+\frac{3r^{4/3}L_{0}^{2}}{4L_{1}^{2/3}}\times\Big{(}-32(1-\frac{2\beta}{r}-kr^{2}))L_{1}^{4}$
$\displaystyle+\sqrt{1024\Big{(}1-\frac{2\beta}{r}-kr^{2}\Big{)}^{2}L_{1}^{8}-27r^{4}L_{0}^{6}}\Big{)}^{-1/3}\Big{]}^{\frac{-8}{9}}.$

3\. The mass accretion rate of Schwarzschild BH is

$\displaystyle\dot{M}$ $\displaystyle=\Big{[}\frac{8\pi
L_{1}^{\frac{7}{3}}}{r^{\frac{8}{3}}}\Big{]}\Big{[}\Big{(}-32(1-\frac{2\beta}{r})L_{1}^{4}$
$\displaystyle+\sqrt{1024\Big{(}1-\frac{2\beta}{r}\Big{)}^{2}L_{1}^{8}-27r^{4}L_{0}^{6}}\Big{)}^{1/3}\Big{(}{4L_{1}^{2}}\Big{)}^{-1}$
$\displaystyle+\frac{3r^{4/3}L_{0}^{2}}{4L_{1}^{2/3}}\times\Big{(}-32(1-\frac{2\beta}{r})L_{1}^{4}$
$\displaystyle+\sqrt{1024\Big{(}1-\frac{2\beta}{r}\Big{)}^{2}L_{1}^{8}-27r^{4}L_{0}^{6}}\Big{)}^{-1/3}\Big{]}^{\frac{-8}{9}}.$

In Fig. 10, we plot mass accretion-rate $\dot{M}$ versus the radius $r$ for
aforemention BHs for radiation fluid with the parameters $\gamma$, $k$ and
$\beta$. The left panel (conformal gravity BH) shows that the accretion rate
is decreasing for smaller values of $r$, that is the red curve shows the
minimum accretion rate between the universal and Killing horizon. It is noted
that the maximum accretion rate occurs at the universal, but far from the
Killing horizon. It is the increasing function of the radius in the presence
of $\gamma$. The red curve in the right panel depicts the minimum accretion
rate at the universal and the Killing horizon for Schwarzschild de-Sitter BH
in the presence of cosmological constant $k$. It is noted here that the
accretion rate increases for smaller values of $k$ and we can see the maximum
accretion rate near $r\approx 1.08,0.82,0.75,0.69$ for
$k=-1.0,-1.5,-2.0,-2.5$, respectively. It is the decreasing function of radius
that is mass increases when radius decreases.

We note that for Schwarzschild BH, the range of the maximum accretion rate is
between the radius $0.5$ to $0.6$, for larger value of $\beta=4$. It is also
the decreasing function of the radius that is accretion rate increases whereas
the radius decreases. So, four key points are observed for the Schwarzschild
BH: __

  * •

$\beta=1.0$_, corresponding to_ $r_{UH}\approx 0.4$, $r_{KH}\approx 1.0$.

  * •

$\beta=2.0$_, corresponding to_ $r_{UH}\approx 0.4$, $r_{KH}\approx 0.90$.

  * •

$\beta=3.0$_, corresponding to_ $r_{UH}\approx 0.4$, $r_{KH}\approx 0.70$.

  * •

$\beta=4.0$_, corresponding to_ $r_{UH}\approx 0.4$, $r_{KH}\approx 0.60$.

The mass accretion rate of Schwarzschild BH is increasing and the radius
decreases for increasing values of the mass function. Hence, we conclude that
the mass of Schwarzschild BH is larger as compared to conformal gravity BH and
Schwarzschild de-Sitter BH.

Figure 10: For the mass accretion rate, left panel (conformal gravity BH)
displays the behavior of Eq. (8) with conformal parameters $\gamma=0.1$,
$k=-1$, $\beta=1$. Right panel (Schwarzschild de-Sitter BH) displays the
behavior of Eq. (8) with parameters $\gamma=0$, $k=-1$, $\beta=1$. Bottom
panel (Schwarzschild BH) displays the behavior of (8) with the parameters
$\gamma=0$, $k=0$, $\beta=1$. Other constants are taken as $L_{0}=0.90$,
$L_{1}=0.5$ .

Figure 11: For the mass accretion rate, left panel displays the behavior of
conformal gravity BH with conformal parameters $\gamma=0.1$, $k=-1$,
$\beta=1$. Right panel displays the behavior of Schwarzschild de-Sitter BH
with parameters $\gamma=0$, $k=-1$, $\beta=1$. Bottom panel displays the
behavior of Schwarzschild BH with parameters $\gamma=0$, $k=0$, $\beta=1$.
Other constants are taken as $L_{0}=0.90$, $L_{1}=0.5$.

We plot mass accretion-rate ($\dot{M}$) versus the radius ($r$) for
aforemention BHs in the case of sub-relativistic fluid with particular values
of conformal parameters $\gamma$, $k$ and $\beta$, as shown in Fig. 11. The
left panel (conformal gravity BH) shows that the accretion rate is decreasing
for larger values of $\gamma$. One can see the maximum accretion rate at
$r\approx 0.87$ for $\gamma=0.4$, $r\approx 0.82$ for $\gamma=0.3$, $r\approx
0.76$ for $\gamma=0.2$, $r\approx 0.69$ for $\gamma=0.1$. While, the mass
decreases but the radius increases in this case between the universal and
Killing horizon. The red curve in the right panel depicts the maximum
accretion rate between the universal and Killing horizon for Schwarzschild de-
Sitter BH in the presence of cosmological constant $k$. It is noted here that
the accretion rate decreases for smaller values of $k$ and we can see the
maximum accretion rate near $r\approx 0.69,0.82,0.92,1.0$ for
$k=-1.0,-1.5,-2.0,-2.5$, respectively. Now, the mass of Schwarzschild BH is
given in the bottom panel where we observe the maximum accretion rate for
smaller radius between the universal and Killing horizons (see red curve). The
maximum accretion rate occurs between $r\approx 0.2$ to $r\approx 0.4$. While,
the minimum accretion rate occurs at the Killing horizon in the bottom panel.
We have noted that the minimum accretion rate is between $r\approx 0.2$ to
$r\approx 1.0$ for the larger value of $\beta=4.0$. Four key points are
observed for the Schwarzschild BH:

  * •

$\beta=1.0$, corresponding to $r_{UH}\approx 0.22$, $r_{KH}\approx 0.40$.

  * •

$\beta=2.0$, corresponding to $r_{UH}\approx 0.21$, $r_{KH}\approx 0.60$.

  * •

$\beta=3.0$, corresponding to $r_{UH}\approx 0.19$, $r_{KH}\approx 0.80$.

  * •

$\beta=4.0$, corresponding to $r_{UH}\approx 0.18$, $r_{KH}\approx 1.0$.

The mass accretion rate of Schwarzschild BH decreases with the increases of
radius for the various values of the mass function.

##  9 Conclusion

In this paper, we have investigated the spherically symmetric accretion around
the conformal gravity BH, with four kinds of fluid as ultra-stiff fluid,
ultra-relativistic fluid, radiation-fluid and sub-relativistic fluid by using
the Hamiltonian approach. It is demonstrated that the energy density is always
equal to pressure in ultra-stiff fluids. In this case, it has been observed
that supersonic as well as subsonic accretion flow would exist for the
particular values of the parameters. The critical radius in Schwarzschild BH
is larger than the conformal gravity BH and Schwarzschild de-Sitter BH. The
energy density is double of the pressure for ultra-relativistic fluid and
there exists a supersonic flow, which is followed by subsonic flow. The fluid
flow around Schwarzschild BH for ultra-relativistic fluid is entirely
different as compared to conformal gravity BH and Schwarzschild de-Sitter BH.
The $3D$-speed $v$ is very small but the radial distance is larger for
Schwarzschild BH as compare to conformal gravity BH and Schwarzschild de-
Sitter BH. It is also noted that the critical radius is very close to the
horizon for Schwarzschild BH as compare to conformal gravity BH and
Schwarzschild de-Sitter BH for ultra-relativistic fluid (see Fig.(4)). The
nature of radiation-fluid and sub-relativistic fluid in which the energy
density is greater than the pressure is similar for $v>v_{c}$. A very simple
behavior has been observed for the radiation-fluid that is only supersonic
flow exists for Schwarzschild BH while subsonic accretion exists for conformal
gravity BH and Schwarzschild de-Sitter BH. Further, for sub-relativistic
fluid, the flow around Schwarzschild BH is absolutely closer to ultra-
relativistic fluid that is the critical radius is closer to the horizon as
compared to conformal gravity BH and Schwarzschild de-Sitter BH. The $3D$
speed for radial motion is very small in Schwarzschild BH as compared to
conformal gravity BH and Schwarzschild de-Sitter BH.

In addition, we have explored the results of mass accretion rate $\dot{M}$,
radial velocity $u$ and the energy density $\varrho$ corresponding
Schwarzschild BH, conformal gravity BH and Schwarzschild de-Sitter BH. We have
investigated the mass accretion rate with four types of fluid around conformal
gravity BH, Schwarzschild de-Sitter BH and Schwarzschild BH which is shown in
Figs.(8-11). The Schwarzschild BH acquires the higher accretion rate as
compare to conformal gravity BH and Schwarzschild de-Sitter BH for ultra-stiff
fluids. The mass accretion rate is smaller in Schwarzschild de-Sitter BH than
the conformal gravity BH and Schwarzschild BH for the ultra-relativistic
fluid, radiation-fluid and sub-relativistic fluid. It is concluded that the
maximum mass accretion rate occur for conformal gravity BH at $\dot{M}>8000$
versus $(r\leq 0.8)$ for ultra-relativistic fluid. The maximum mass accretion
rate for Schwarzschild de-Sitter BH mass occurs at $\dot{M}>2500$ versus
$(r\leq 1.0)$ for ultra-relativistic fluid. Similarly, the maximum mass
accretion rate for Schwarzschild BH occurs at $\dot{M}>8000$ versus $(r\leq
0.8)$ for ultra-relativistic fluid.

Further, we have discussed the ultra-stiff, ultra-relativistic, radiation and
the sub-relativistic fluids with the equation of state which helps to identify
that what kind of fluids is accreting onto the BH. Moreover, critical points
and conserved quantities have been found for these fluids. The behavior of
accreting fluid has been discussed as subsonic and supersonic according to
equation of state. We have compared the fluid flow for all models and observed
that the fluid flow and CPs are closer to Case $3$ instead of Case $2$ and
$1$, respectively.

It has been analyzed that the subsonic accretion is followed by the supersonic
accretion inside the BH horizon and it does not support to the claim that ”the
flow must be supersonic at the horizon” [91]. So, for conformal gravity BH the
fluid flow is neither supersonic nor transonic near the horizon [93, 94] .
This outflow is unsteady because it follows a subsonic path after passing
through the saddle point ($r_{c},v_{c}$) and becomes supersonic with speed
approaches to the speed of the light. In the cosmological point of view, the
point ($v=0,r=r_{h}$) can be observed as repeller as well as attractor where
the solution curves diverge and converge, respectively, [92, 98]. These
results open a new window correspond to fine-tuning and variability problems
in dynamical systems.

## Acknowledgments

We are very grateful to the honorable referee, who put his/her efforts and
give valuable suggestions for improving the manuscript.

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