Viscothermal longitudinal wave propagation in incompressible viscous liquid with non-uniform shear flow confined by rigid pipeline wall of constant temperature is mathematically formulated, which is a set of coupled fourth-order nonlinear differential equations. Based on Bessel-Fourier theory, a general solution in the form of Bessel-Fourier sequences, which are orthogonal and complete in Lebesgue Space, is proposed to transform the fourth-order nonlinear differential equations to a set of homogeneous linear algebraic equations. Then acoustic wave number can be solved using the existence condition of non-trivial solution to the homogeneous linear algebraic equations, leading to the vanishment of the corresponding determinant which can be numerically calculated by various iteration methods. Based on the proposed scheme, phase velocity and attenuation of viscothermal longitudinal wave is numerically analysed in shear flow. In the end, measurement performance of ultrasonic flow meter based on wave propagation is investigated.