LAUSR

The exact solutions of unsteady MHD viscoelastic fluid (Maxwell) of three-dimensional flow in a porous medium is calculated by the traveling wave method. The governing equations can be reduced into ordinary differential equations by wave parameter $$\xi = a_{1} x + a_{2} y + a_{3} z + \Omega t$$ ξ = a 1 x + a 2 y + a 3 z + Ω t . The varieties of new accurate traveling wave solutions are attained for three varying cases. In special cases, the solution of Newtonian fluid can be achieved by putting relaxation time $$\lambda \rightarrow 0$$ λ → 0 in general solutions and further the solutions of Maxwell and Newtonian fluids with and without MHD and porous effects can be obtained by putting $$B, \Phi \rightarrow 0$$ B , Φ → 0 in general solutions. In the end, the effect of pertinent parameters on the fluid motion is discussed (with respect to the x-variable) as well as the difference among various Maxwell fluids is explored through 2D and 3D graphical illustration.