**1. Introduction**

Cattaneo-Christov model is an improved version of Fourier law as Fourier law does not detect the initial temperature disturbance; to overcome this ambiguity, Cattaneo added a thermal relaxation parameter. This parameter covers the ambiguity of Fourier law. The classical Fourier law is obtained while vanishing the relaxation parameter [1]. Cattaneo-Christov heat flux model gives us heat transfer rate in stretching cylinders as well as sheets. Heat transfer is a wonderful natural phenomenon that occurs when two bodies have a thermal difference until both bodies are at thermal equilibrium. The Cattaneo-Christov model is in the form of a heat equation. The thermal convection effect is studied using the Christov heat model in conjunction with the Cattaneo heat model [2]. It has been realized that the development of stretchy surfaces and the flow field that surrounds them speaks to a variety of technological and industrial applications, such as paper making, glass blowing, crystal growth, and aerodynamic plastic sheet extrusion [3]. Heat transfer is a common natural occurrence as long as there is a temperature differential between things or between various regions of the same object, heat transfer will occur. As a result, a lot of effort has gone into predicting the heat transport behavior. In several starting and boundary problems, the uniqueness and structural stability of the solutions for the temperature governing equations using the Cattaneo-Christov heat flow model have been demonstrated. The chapter released uses the Cattaneo-Christov heat flux model to analyze the flow and heat transfer of upper-convective Maxwell fluid across a stretching sheet [4]. Efforts have been undertaken to increase the thermal efficiency of processes during the last many decades. On the one hand, there has been an attempt to lower the size of the equipment by increasing the thermal exchange surface, such as with fins, and on the other hand, novel fluid exchangers with higher thermal conductivity have been developed. Different NPS types (metallic, nonmetallic, and carbon based) have been synthesized and dispersed in conventional fluids such as water, oil, or ethylene glycol referred to as nanofluids since the advent of nanotechnology and the possibility of synthesizing materials on a nanometric scale [5]. The boundary layer flow and heat transfer caused by stretching flat plates or cylinders are both practical and theoretically interesting in fiber technology and extrusion operations. This method is used to produce polymer sheets and plastic films. The cooling of an infinite metallic plate in a cooling bath, the boundary layer along material handling conveyors, the aerodynamic extrusion of plastic sheets, the boundary layer along a liquid film in condensation processes, paper production, glass blowing, metal spinning and drawing plastic films, and polymer extrusion are all examples of boundary layers [6]. The aim of this chapter is to manipulate the heat transfer rate of titanium oxide nanofluid with the Cattaneo-Christov heat flux model over a vertical stretching cylinder.
