Tham khảo tài liệu 'kundu fluid mechanics 2 episode 5', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 154 Irrotational Haul which implies dw u iv. dz It is easy to show that taking Sz parallel to the y-axis leads to an identical result. The derivative dw dz is therefore a complex quantity whose real and imaginary parts give Cartesian components of the local velocity dw dz is therefore called the complex velocity. If the local velocity vector has a magnitude q and an angle a with the x-axis then qe ia. dz It may be considered remarkable that any twice differentiable function w z z X iy is an identical solution to Laplace s equation in the plane x y . A general function of the two variables x y may be written as z z where z X iy is the complex conjugate of z. It is the very special case when z z w z alone that we consider here. As Laplace s equation is linear solutions may be superposed. That is the sums of elemental solutions are also solutions. Thus as we shall see flows over specific shapes may be solved in this way. 4. Flow al a Wall Angle Consider the complex potential w Az n where A is a real constant. If r and 0 represent the polar coordinates in the z-plane then w A re 9 Ar cosnớ i sinnớ giving ộ Arn cosnO Ỷ Ar sinnớ. For a given n lines of constant yịr can be plotted. Equation shows that i f 0 for all values of r on lines 0 0 and 6 lĩ In. As any streamline including the 0 line can be regarded as a rigid boundary in the z-plane it is apparent that Eq. is the complex potential for flow between two plane boundaries of included angle a Tt n. Figure shows the flow patterns for various values of n. Flow within a certain sector of the z-plane only is shown that within other sectors can be found by symmetry. It is clear that the walls form an angle larger than 180 for n 1 and an angle smaller than 180 for n 1. The complex velocity in terms of a n n is Az -1 z jr- dz a which shows that at the origin dw dz 0 for a n and dw dz oo for a n. Thus the comer is a stagnation point for flow in a wall angle smaller than 180 4. Ĩ ĨOUĨ al a Hall
