(b) $d\psi$ between two streamlines is proportional to the Volumetric Flow

Figure 4.4: Flow between two Stream Lines.

 

Consider the volumetric flow through a small element of thickness ds placed on a streamline as shown in Fig. 4.4. The volumetric flow through the element is given by

$$dQ = u dy - vdx$$

$$={{\partial \psi} \over {\partial y}} dx - {(-{{\partial \psi} \over {\partial x}}}) dy$$

$$={{\partial \psi} \over {\partial y}} dx + {{{\partial \psi} \over {\partial x}}} dy$$

$$= d\psi$$ (4.22)

which indicates that the volumetric flow rate is proportional to the difference between stream functions. If we now integrate the Eqn.4.22 between two stream lines $\psi=C$ and $\psi=D$, we have,

$$Q = \int_1^2 dQ = \int_{\psi_1} ^{\psi_2}  d\psi = \psi_2 - \psi_1$$ (4.23)

(c) Aerospace, Mechanical & Mechatronic Engg. 2005
University of Sydney