Still now it is an interesting area for the cosmologists to study and discover the unknown phenomenon of the universe that have yet to observe to study and explore its hidden knowledge. So cosmologists have taken considerable interest to understand the past evolution, present state and evolution in the future of the universe. Before the formation of particles, the strings took major role in the creation and evolution of the universe in early era. Authors in ^{1, 2 }initiated the general relativistic study of the strings, and they developed the classical theory of the geometric strings. Because of the major role of strings in describing the evolution of our universe in the early epoch, in recent times, many famous researchers are interested in cosmic strings in general relativity ^{7, 8, 9}.

The bulk viscosity assumes an extraordinary part in the development of the early universe. There are numerous events inside the development of the universe wherein the bulk viscosity could emerge. The size of the viscous stress comparative with the expansion is controlled by the coefficients of bulk viscosity. Spatially homogeneous and anisotropic Bianchi type-I models are attempted to comprehend the universe in its beginning phase of the evolution of the universe. The various pictures of the universe may show up at the beginning phase of the cosmological development of the universe because of the dissipative process brought about by viscosity which counteracts the cosmological breakdown (collapse). A few authors endeavored to find the specific solutions of field equations by considering viscous effects in general relativity in isotropic as well as anisotropic cosmological model universes. Authors in^{12}. In ^{13} the authors have endeavored to introduce another solution for the field equations obtained for Bianchi type-III cosmological model in Lyra manifold by utilizing the law of variation of Hubble's Parameter (H), which yields constant DP.

Nowadays it is very interesting to study string cosmology in five-dimensional space-time in general relativity. The possibility of space-time having more than four dimensions (Extra dimensions) has fascinated many researchers. In the recent years, to study cosmological models, the higher dimensional space-time has been given more importance. Generally, the higher dimensional model was introduced by ^{14} and ^{15} in an effort to unify gravity with electromagnetism. Higher dimensional model can be regarded as a tool to illustrate the late time expedited expanding paradigm ^{16}. Investigation of higher dimensional space-time can be regarded as a task of paramount importance as the universe might have come across a higher dimensional era during the initial epoch ^{17}. Marciano ^{18} asserts that the detection of a time varying fundamental constants can possibly show us the proof for extra dimensions. According to ^{19} and ^{20}, extra dimensions generate huge amount of entropy which gives possible solution to atness and horizon problem. Since we are living in a 4D space-time, the hidden extra dimension in 5D is highly likely to be associated with the invisible DM and DE ^{21}. Several authors have investigated various Bianchi type problems in the field of higher dimensional space-time. An in homogeneous higher dimensional cosmological model with massive string in general relativity was constructed by ^{22}. An LRS Bianchi type-I inflationary string Cosmological model with massive scalar field in general relativity in the field of five-dimensional space-time was constructed by^{ }

Above discussion and investigations motivated us to study the five-dimensional LRS Bianchi type-I string model in cosmology with particles attached to them in general relativity. The paper is presented as: In Section 1, a brief presentation of strings, bulk viscosity and their importance are discussed. In Section 2, the five-dimensional LRS Bianchi type-I metric is introduced and the field equations in the framework of general relativity are determined; In Section 3, using some simplifying assumptions we find the determinate solutions of the survival field equations. In Section 4, we examined the geometrical and physical properties of our model universe with the help of graphs; In Section 5 conclusions are presented.

We consider the 5-dimensional LRS Bianchi type-I metric as

Here a, b and c are the metric functions of cosmic time t alone and the extra coordinate "m" is taken to be space-like.

For the above metric lets

In general relativity the Einstein's field equation is written as

For a cloud string the energy-momentum tensor is

Where, ρ is the energy density and λ the tension density of the string and they are related as

Here without loss of generality we can take

The spatial volume is given by

Where R(t) is the average scale factor of the universe.

The Scalar Expansion is given by

Hubble Parameter is given by

Deceleration parameter is

The shear scalar is given by

And the mean anisotropy parameter is given by

Where, H_{i} (i=1,2,3,4) represents the directional Hubble Parameters in the directions of x, y, z and m axes and are defined as

Using the equations (3)-(6), the field equation (2) takes the form

Here, the overhead dots mean differentiation with time `t'.

In this part, we intend to derive the solutions of the four highly non-linear independent equations (13)-(16) with 6 unknown variables a, b, c, ξ, λ and ρ. For deterministic solution we considered the following physical plausible conditions:

We consider that the shear scalar and expansion scalar are proportional, which leads to

Here,

The reason of assuming the above condition depends on observations of the velocity red-shift relation for extragalactic sources recommended that the Hubble expansion of the universe is isotropic today to within 30% ^{38, 39, 40}. If H is Hubble constant and σ is the shear then the red-shift studies limit

From (14),(15) by using (17), we get

Using (17) in (14) and comparing with (15), we get n=1.

From (17) we get

Equation (18) yields the following cases:

We intend to determine the cosmological models for the above two cases separately.

Solving we get,

Clearly, the solution are not unique because b(t) can be obtained for any given a(t). So for further studies here we consider

Now solving (22) and using (19) we get,

Here,

The following metric describes the geometry of the model

The tension density of the string is obtained as

The energy density of the string is

The particle density is given by

From (15) we get

The spatial volume is obtained as

The scalar expansion is obtained as

Using (32) in (30) we get

The Hubble parameter is obtained as

The deceleration parameter is

The shear scalar of the model is

Solving it we get,

From the equation (19) together with (37) we have,

Now using equations (37)-(38) in the field equations (13)-(16) we get,

Here ξ, λ, ρ and a are four unknowns involved in three equation (39)-(41). To obtain a determinate solution we have assumed following different plausible conditions of equations of state as Geometric String

Where,

This yields,

Where,

This case leads to the five-dimensional LRS Bianchi type-I vacuum model universe in Einstein's theory of relativity.

The following metric described the geometry of our model,

Using (41) in (43) we get,

Which yields either

Since,

In case I we have obtained a five-dimensional LRS Bianchi type-I cosmological model universe with string in general relativity given by (26). The variation of parameters with time for this model are shown below by taking,

At initial epoch t=0, the metric (model) (26) becomes flat.

It is seen that at time t=0 the evolution of energy density ρ is infinite and it decreases gradually as the time t increases and become constant after some finite time (

It is also seen from

Initially at

The average Hubble parameter H is a decreasing function of time. It is large constant when

Since,

From the

The DP “q” in the model is decelerating

The bulk viscosity for this model is decreasing function of time. The function of the bulk viscosity is to retard the expansion of the universe and since bulk viscosity deceases with the time so retardness also decreases which supports in the expansion in faster rate in the late time universe. From the above discussion it can be seen that the bulk viscosity plays a great function in the evolution of the universe.

The case II, leads to LRS Bianchi type-I vacuum model in Einstein's theory of relativity in five-dimensional space-time represented by which is not realistic because strings don't survive for this model.

In this study, we have investigated a five-dimensional LRS Bianchi type-I String cosmological model in general theory of relativity in presence of bulk viscous fluid given by (26) which is an inflationary model. The model universe obtained here is anisotropic, accelerating and expanding. The DP “q” obtained here is decelerating at initial stage and accelerates after some finite time, indicating inflation in the model after an epoch of deceleration which is in accordance with the present day observational scenario of the accelerated expansion of our universe as type la supernovae