Abstarct: Upper Convected Maxwell Fluids are an important and widely used model of viscoelastic fluids that are classified in the group of non-Newtonian fluids. In this study, the behavior of These fluids is modeled in conditions such that the fluid hammer phenomenon occurs in a pipe. Here, this phenomenon is called “Upper Convected Maxwell fluid hammer”. The main purpose of this study is to establish the governing equations for viscoelastic transient flow in the pipe. The system studied in this thesis consists of the valve, pipe and reservoir. Maxwell equations are used as constitutive equations for the relationship of stresses, and the two-step variant of the Lax-Friedrichs (LxF) finite difference numerical method is used for discretization of the equations. Subsequently, the governing non-dimensional groups are computed. These groups are the Deborah, Reynolds and Mach numbers. The results of viscoelastic fluid hammer modeling showed that the transient flow damping in a viscoelastic fluid is lower than in a Newtonian fluid, and so attenuation time is longer compared to a Newtonian fluid. It was argued that this is mainly related to the elastic properties of the viscoelastic fluid, such as the relaxation time constant. The elastic properties play an important role in storing the potential energy imposed on the fluid, whereas the viscous properties result in wasting of the imposed energy. These opposing actions in a viscoelastic fluid cause the damping time of the transient flow to become longer compared to a Newtonian fluid. As for the effects of viscoelasticity on the fluid-structure interaction during the fluid hammer phenomenon, results showed that the compound coupling and junction coupling have the greatest effect on the pressure head, and Poisson coupling has the least effect. In the definition of the boundary conditions for junction coupling, the valve is allowed to vibrate; this causes the effect of junction coupling on the pressure head to be more prominent, But in Poisson's case, the boundary conditions imposed on the system are largely dependent on the structure; this means the coupling effect on the system is minimal. On the other hand, coupling effects in a viscoelastic fluid do not significantly impact the shear stresses in the fluid hammer phenomenon. In a detailed comparison, the Poisson coupling and junction coupling exhibit the lowest and maximum shear stresses, respectively.