Finally,in Figure 1.4.4,we illustrate a constant downward flow interacting with a diverging flow (source).The diverging flow is able to make some headway "upwards" against the downward constant flow,but eventually turns and flows downward, overwhelmed by the strength of the "downward"flow. Figure 1.4.4 A constant downward flow interacting with a diverging flow (source). In the language of vector calculus,we represent the flow field of a fluid by V=vi+v,j+v.k (1.4.2) A point (x,y,z)is a source if the divergence of v(x,y,z)is greater than zero.That is, 7.xy)=++a>0 (1.4.3) ax dy Oz where =0i+k+ (1.4.4) is the del operator.On the other hand,(x,y,z)is a sink if the divergence of v(x,y,z)is less than zero.When V.v(x,y,z)=0,then the point (x,y,z)is neither a source nor a sink.A fluid whose flow field has zero divergence is said to be incompressible Animation 1.2:Circulations A flow field which is neither a source nor a sink may exhibit another class of behavior- circulation.In Figure 1.4.5(a)we show a physical example of a circulating flow field where particles are not created or destroyed (except at the beginning of the animation), but merely move in circles.The purely circulating flow can also be represented by textures,as shown in Figure 1.4.5(b). 1-101-10 Finally, in Figure 1.4.4, we illustrate a constant downward flow interacting with a diverging flow (source). The diverging flow is able to make some headway “upwards” against the downward constant flow, but eventually turns and flows downward, overwhelmed by the strength of the “downward” flow. Figure 1.4.4 A constant downward flow interacting with a diverging flow (source). In the language of vector calculus, we represent the flow field of a fluid by ˆ ˆ ˆ xyz v i =++ vvv j k G (1.4.2) A point ( , , ) x y z is a source if the divergence of ( , , ) v x y z G is greater than zero. That is, (, ,) 0 x y z v v v xyz xyz ∂ ∂ ∂ ∇⋅ = + + > ∂∂∂ v G (1.4.3) where ˆ ˆ ˆ x y z ∂ ∂ ∂ ∇= + + ∂ ∂ ∂ ikk (1.4.4) is the del operator. On the other hand, ( , , ) x y z is a sink if the divergence of ( , , ) v x y z G is less than zero. When ( , , ) 0 ∇⋅ = v xyz G , then the point ( , , ) x y z is neither a source nor a sink. A fluid whose flow field has zero divergence is said to be incompressible. Animation 1.2: Circulations A flow field which is neither a source nor a sink may exhibit another class of behavior - circulation. In Figure 1.4.5(a) we show a physical example of a circulating flow field where particles are not created or destroyed (except at the beginning of the animation), but merely move in circles. The purely circulating flow can also be represented by textures, as shown in Figure 1.4.5(b)