**ABSTRACT**

This current research presents a mathematical model and analysis of flow and heat transfer in a Johnson-Segalman fluid in a circular pipe. A numerical solution to the governing equations is found by applying a finite difference technique. The effects of relevant parameters, such as the relaxation time parameter, viscosity parameter, and Brinkman number, on the fluid velocity and temperature distributions of the pipe flow are examined using the numerical solutions that were produced. The results demonstrate that as the relaxation time parameter and Brinkman number increase, so do the fluid velocity and temperature. Additionally, it was shown that the relaxation time parameter rises as fluid velocity increases but falls as fluid temperature rises. The impact of relaxation factors on the velocity distribution has insignificant effect on the flow process when the viscosity parameter gets closer to and larger than unity. The findings of this investigation are generally consistent with those of the earlier study as documented in the literature.

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