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This paper is devoted to studying the generalized Chaplygin gas models in Bianchi type III space- time geometry with time varying bulk viscosity, cosmological and gravitational constants. We are considering the condition on metric potential
_{}. Also to obtain deterministic models we have considered physically reasonable relations like
_{} , and the equation of state for generalized Chaplygin gas given by
_{} . A new set of exact solutions of Einstein’s field equations has been obtained in Eckart theory, truncated theory and full causal theory. Physical behaviour of the models has been discussed.

The motivation behind the stimulated interest in anisotropic cosmological models is experimental study of isotropy of the cosmic microwave background radiation and speculation about the amount of the helium formed at the early stages of the evolution of the universe. The existence of anisotropic stage of the universe is supported by experimental data and numbers of scientific arguments in the literature which is supposed to be phased out during evolution. The present day universe is isotropic and homogeneous. In understanding the behavior of universe at early stages, anisotropic cosmological models have played a significant role. Singh and Singh [_{0}, III and Kantowski-Sachs space-times within framework of Lyra geometry. Bianchi type III cosmological model in f(R, T) theory of gravity has been discussed by Reddy et al. [

The astronomical observations of type Ia supernovae [

The idea of variability of G originated with the work of Dirac [

In the literature it has been discussed that during the early stages of evolution of the universe, bulk viscosity could arise in many circumstances and could lead to an effective mechanism of galaxy formation [

It has been observed that the universe has entered an acceleration phase and some exotic dark energy must presently dominate [

We consider the Bianchi type III metric in the form

For perfect fluid distribution Einstein’s field equations with gravitation and cosmological constant may be written as

where G is gravitational constant,

The energy momentum tensor

where

where p is equilibrium pressure,

Einstein’s filed Equation (2) for the metric (1) leads to

where the over head dot denote differentiation with respect to time t. An additional equation for the time changes of G and

Equation (10) splits into two equations as

For the full causal non-equilibrium thermodynamics the causal evolution equation for bulk viscosity is given by [

Since there are five basic Equations (5)-(9) and eight unknowns viz.

Case I: Non-Causal Cosmological Solution

For non causal solution

To find the complete solution of the system of equations, following relations are taken into consideration.

The power law relation for bulk viscosity is taken as

The equation of state is

We assume the solution of the system in the form

where n is constant. On integrating Equation (17), we get

where a and b are constants of integration.

Using Equations (16) and (17) in (11), we obtain

which on solving yields

where C is constant of integration.

From Equation (20) and

On differentiating Equation (20), one can get

Now with the help of Equations (17) and (18), Equation (8) becomes

which on differentiation leads to

Substituting Equations (12), (14) and (17) into Equation (23), we have

By use of Equations (15) and (21), Equation (24) yields

where

From Equation (25) and

Using Equations (20) and (25), Equation (22) gives

From Equation (26) and

Now from Equations (15) and (20), we have

From Equation (27) and

From Equations (14) and (17), the expression for bulk viscous stress is given by

Thus the metric (1) reduces to the form

The shear scalar [

For this model the Shear scalar is

From Equation (31) it is clear that as

The expansion scalar is defined by

For this model expansion scalar is given by

The deceleration parameter is related to the expansion scalar as

For this model

Foe accelerating expansion of the universe the deceleration parameter q < 0 for

Case II: Causal Cosmological Solution

In addition to physically plausible relations (16), (17), in this case we assume

where H is Hubble parameter, given by

From Equations (17) and (39), the Hubble parameter is given by

Using Equations (17)-(18), (38) and (40) in Equation (8), we get

where

From Equations (20) and (41),

From Equation (42) and

Substitute the values from Equations (17), (20), (38) and (42) in Equation (5), we get

where

By use of Equation (20), Equation (43) gives

(i) Evaluation of Bulk viscosity in Truncated Causal Theory

Now we study variation of bulk viscosity coefficient

In order to have exact solution of the system of equations one more physically plausible relation is required.

Thus, we consider the well known relation

Using Equations (17), (20), (44) and (46) in Equation (45) one can obtain

(47)

where

(ii) Evaluation of Bulk Viscosity in Full Causal Theory

It has already been mentioned that for full causal theory

On the basis of Gibb’s inerrability condition, Maartens [

which with the help of Equation (16) gives

using Equations (20), (40), (46) and (50) in Equation (48) one can obtain

which on simplification yields the expression for bulk viscosity

(51)

In this paper we have studied bulk viscous Bianchi type III space-time geometry with generalized Chaplygin gas and time-varying gravitational and cosmological constants. We have obtained a new set of Einstein’s equations

by considering

constant are decreasing as gravitational constant G(t) is increasing with time. Shear dies out with evolution of the universe for large value of t. For accelerating model of the universe, the deceleration parameter q < 0

for

which is considered to be fundamental and match with the observations. In order to have clear idea of variation in behavior of cosmological parameters, relevant graphs have been plotted. All graphs of cosmological parameters go with cosmological observations.

S. K. would like to thank U. G. C. New Delhi for providing financial support under the scheme of major research project F. No. 41-765/2012 (SR). S. K. and R. K. would like to thank Inter University Centre for Astronomy and Astrophysics for providing facilities.