Kaluza–Klein Bulk Viscous Fluid Cosmological Models and the Validity of the Second Law of Thermodynamics in <i>f</i>(R, T) Gravity

Abstract: The authors considered the bulk viscous fluid in f ( R , T ) gravity within the framework of Kaluza–Klein space time. The bulk viscous coefficient ( ξ ) expressed as <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>ξ</m:mi> <m:mo>=</m:mo> <m:msub> <m:mi>ξ</m:mi> <m:mn>0</m:mn> </m:msub> <m:mo>+</m:mo> <m:msub> <m:mi>ξ</m:mi> <m:mn>1</m:mn> </m:msub> <m:mfrac> <m:mover accent="true"> <m:mi>a</m:mi> <m:mo>˙</m:mo> </m:mover> <m:mi>a</m:mi> </m:mfrac> <m:mo>+</m:mo> <m:msub> <m:mi>ξ</m:mi> <m:mn>2</m:mn> </m:msub> <m:mfrac> <m:mover accent="true"> <m:mi>a</m:mi> <m:mo>¨</m:mo> </m:mover> <m:mover accent="true"> <m:mi>a</m:mi> <m:mo>˙</m:mo> </m:mover> </m:mfrac> <m:mo>,</m:mo> </m:mrow> </m:math> $\xi = {\xi _0} + {\xi _1}{{\dot a} \over a} + {\xi _2}{{\ddot a} \over {\dot a}},$ where ξ 0 , ξ 1 , and ξ 2 are positive constants. We take p =( γ −1) ρ , where 0≤ γ ≤2 as an equation of state for perfect fluid. The exact solutions to the corresponding field equations are given by assuming a particular model of the form of f ( R , T )= R +2 f ( T ), where f ( T )= λT , λ is constant. We studied the cosmological model in two stages, in first stage: we studied the model with no viscosity, and in second stage: we studied the model involve with viscosity. The cosmological model involve with viscosity is studied by five possible scenarios for bulk viscous fluid coefficient ( ξ ). The total bulk viscous coefficient seems to be negative, when the bulk viscous coefficient is proportional to <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>ξ</m:mi> <m:mn>2</m:mn> </m:msub> <m:mfrac> <m:mover accent="true"> <m:mi>a</m:mi> <m:mo>¨</m:mo> </m:mover> <m:mover accent="true"> <m:mi>a</m:mi> <m:mo>˙</m:mo> </m:mover> </m:mfrac> <m:mo>,</m:mo> </m:mrow> </m:math> ${\xi _2}{{\ddot a} \over {\dot a}},$ hence, the second law of thermodynamics is not valid; however, it is valid with the generalised second law of thermodynamics. The total bulk viscous coefficient seems to be positive, when the bulk viscous coefficient is proportional to <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>ξ</m:mi> <m:mo>=</m:mo> <m:msub> <m:mi>ξ</m:mi> <m:mn>1</m:mn> </m:msub> <m:mfrac> <m:mover accent="true"> <m:mi>a</m:mi> <m:mo>˙</m:mo> </m:mover> <m:mi>a</m:mi> </m:mfrac> <m:mo>,</m:mo> </m:mrow> </m:math> $\xi = {\xi _1}{{\dot a} \over a},$ <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>ξ</m:mi> <m:mo>=</m:mo> <m:msub> <m:mi>ξ</m:mi> <m:mn>1</m:mn> </m:msub> <m:mfrac> <m:mover accent="true"> <m:mi>a</m:mi> <m:mo>˙</m:mo> </m:mover> <m:mi>a</m:mi> </m:mfrac> <m:mo>+</m:mo> <m:msub> <m:mi>ξ</m:mi> <m:mn>2</m:mn> </m:msub> <m:mfrac> <m:mover accent="true"> <m:mi>a</m:mi> <m:mo>¨</m:mo> </m:mover> <m:mover accent="true"> <m:mi>a</m:mi> <m:mo>˙</m:mo> </m:mover> </m:mfrac> </m:mrow> </m:math> $\xi = {\xi _1}{{\dot a} \over a} + {\xi _2}{{\ddot a} \over {\dot a}}$ and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>ξ</m:mi> <m:mo>=</m:mo> <m:msub> <m:mi>ξ</m:mi> <m:mn>0</m:mn> </m:msub> <m:mo>+</m:mo> <m:msub> <m:mi>ξ</m:mi> <m:mn>1</m:mn> </m:msub> <m:mfrac> <m:mover accent="true"> <m:mi>a</m:mi> <m:mo>˙</m:mo> </m:mover> <m:mi>a</m:mi> </m:mfrac> <m:mo>+</m:mo> <m:msub> <m:mi>ξ<

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