Some new exact analytical solutions for helical flows of second grade fluids

The helical flow of a second grade fluid, between two infinite coaxial circular cylinders, is studied using Laplace and finite Hankel transforms. The motion of the fluid is due to the inner cylinder that, at time t = 0+ begins to rotate around its axis, and to slide along the same axis due to hyperbolic sine or cosine shear stresses. The components of the velocity field and the resulting shear stresses are presented in series form in terms of Bessel functions J0(
• ), Y0(
• ), J1(
• ), Y1(
• ), J2(
• ) and Y2(
• ). The solutions that have been obtained satisfy all imposed initial and boundary conditions and are presented as a sum of large-time and transient solutions. Furthermore, the solutions for Newtonian fluids performing the same motion are also obtained as special cases of general solutions. Finally, the solutions that have been obtained are compared and the influence of pertinent parameters on the fluid motion is discussed. A comparison between second grade and Newtonian fluids is analyzed by graphical illustrations.

► The exact analytical solutions are obtained, when the hyperbolic sine or cosine shear stresses are given on the boundary of inner cylinder.
► The required time to reach the large-time state increases with respect to α, a or b and decreases with regard to the kinematic viscosity ν.
► Effect of a or b on the fluid motion is the scaled version of the effect of time t on fluid motion.
► Qualitatively, the effects of the material parameters a and ν are the same. Both components of the velocity are decreasing functions with respect to α and ν.
► The Newtonian fluid flows faster in comparison with second grade fluid.