An Introduction to Engineering Fluid Mechanics by J. A. Fox (auth.)

By J. A. Fox (auth.)

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Now 1 al/J Q --=u = - r (}e r al/J 27tr al/J and -= - ar ue = 0 Q al/J 3l/J=-ar+-1)e=ox1)r+rx-1)e ar ae ·27tr 1)l/J=Q1)e 27t ,I, 'I' = -Q e + constant = -Q tan -I Y- + const 27t 27t x The expression for the stream function of a source given by this equation agrees with the idea that the stream function is the flux. If Q units of fluid are being emitted in all directions every second, then (Q/27t)e units will be emitted in an angle e. p + -ae ae = -Q -1)r + 0 27t r = JL loge r + constant 27t An Introduction to Engineering Fluid Mechanics 48 jP= 3 -+------_+----~--~~----_+----r_------~jPo=O jP=-2 jP=-4 jP=-3 Fig.

This can be justified as follows Uc = Usource Vc + Usink = Vsource + Vsink These two statements are equivalent to the following two statements (~~) (~~) = c - (~:)c source + (~~)~k (~:Lurce (~:tnk =- - And 81/1 = (01/1) c ax c 8x + (01/1) oy c 8y 8x t (01/1) 8y + (01/1) 8y (01/1) 8x +(01/1) = - ax source I/Ic = l/lsource ax sink oy source oy sink + I/Isink The foregoing argument applies to the process of combining any set of stream functions and an exactly similar argument can be used to show that velocity potentials can be added and also result in vector addition.

The flow is irrotational and because the velocity potential I{) exists such a flow is often called potential flow. 9) 40 An Introduction to Engineering Fluid Mechanics The operator V denotes and V2 denotes It is used in vector algebra which permits the generalisation of results obtained in one system of coordinates to any other. e. there is no flow velocity in, say, the z direction the process of obtaining velocity potentials is greatly simplified. 7 The stream function If a function l/I (psi) is postulated such that u = (al/l/ay) and v = -(al/l/ax) the continuity equation will demonstrate its nature au av -+-=0 ax ay so ax ay ayax A streamline is a line in the flow field to which the adjacent flow is parallel.

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