This thesis is devoted to the creation of a rheometer based on the boundary value problem of the two eccentric cylinders. This device is the first one after the cone-and-plate rheometer which is able to determine more material constants other than the viscosity. However, the eccentric cylinder rheometer shows to be one of the future instruments which is capable to further developments without the difficulties due to the effect of inertia encountered in the cone-and-plate rheometer. The preliminary results obtained by this apparatus proved to be sufficient for the determination of the viscosity coefficient and the second order coefficient over a range of shearing which overcomes the range covered by the cone-and-plate. Further investigation are required which is expected to allow the determination of other material constants. The theoretical basis of the present problem is discussed in details [1, 2], where the fluid performs rotational and axial motions. In the present thesis the rotational part of the motion is considered and the stream function for this motion is determined.