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Divergence of a Vector Field

The divergence of a vector field at a point is a scalar and is defined as the amount of flux diverging from a unit volume element per unit time around that point.

 

The divergence of a vector A (= iAx +jAY+ kAz) differentiable at each point (x, y, z) in a region of space is defined as

 

 

Div A = ∆ .A(∂a x/∂x +∂AY/∂Y+∂AZ)

Physical Significance of Divergence or Divergence in Cartesian Coordinates (or Expression for the Divergence of a Vector Field).

 

Let there be a small rectangular parallelopiped ABCDEFGH with its centre at 0 (x, y , z ) and the sides having lengths dx, dy and dz are parallel to x , y and z-axes as shown in Figure 7.9.

Let a vector v = i v x + j v y + k v z represent the velocity of the fluid at the centre O of parallelopiped. The rate of change of flow of the liquid in a direction of x-axis may be given

 

Thus the magnitude of component of v along x-axis at the centre of face BFGC, i.e., v1 is given by

V1=VX -1/2 ∂V/∂X DX

 

 

Similarly the magnitude of the component of v along x-axis at the centre of the face AEHD, i.e., v1 is given by

V1=VX -1/2 ∂V/∂X DX

Here negative sign is taken because the face AEHD is in negative direction of the centre O and the magnitude of v along x-axis is v x.

 

But the volume of the fluid passing through a face per unit time is defined as,

Flow of volume per second= (Normal component of velocity to the face) x (Area of face)

Therefore, the net fluid entering the face AEHD per unit time

 

= ( v x -1/2 v1=vX -1/2 ∂V/∂X DX)

 

Similarly the fluid leaving the face BFGC per unit time

 

=(vx+1/2 1=VX -1/2 ∂V/∂X DX ) dy dz

 

Thus the excess of fluid leaving the parallelopiped along x-axis is

 

= (v x ½ ∂v /∂x dx) dydz-(vx -1/2 ∂v /∂x dx ) dydx

=∂v x /∂x ddydz

In a similar manner, one can find the values of the fluid leaving parallelopiped along y and z-axes as

(∂vy=/∂xdydz and∂vz ∂z )dxdydz

 

Therefore, the net volume of fluid leaving or diverging or moving out of parallelepiped per unit time.

 

=(∂vy=/∂x+∂vy /∂y +∂vz/∂z 0 dx Dydz)

 

where dx dy dz be the volume of parallelopiped. Thus the amount of diverging per unit volume, which is defined as divergence of v may be given as

 

div v =∂v x /∂xy /∂y +∂vz/∂z

We know the differential operator v is defined as

∆  =I ∂/∂x +j ∂ ∂y +k∂ /∂z

 

Therefore, the divergence of a vector field v may also be defined as

 

div v = ‘V.v     =         (i ∂/∂x +j∂/∂y+ k∂/ ∂z).  (ivx =­j u y +kuz)

= ∂u x/∂y+∂u y /∂y+ ∂uz/)

 

The divergence being the scalar product of two vectors is scalar. The following conclusions may be drawn from the above expression:

 

(i) When the div v or ‘∆.v is positive then either the fluid is undergoing expansion or point itself is the source of fluid.

 

(ii) When the div v is negative then either the fluid is undergoing contraction or the point is the sink of the fluid.

 

(iii) When div v = 0, then the volume of the fluid entering and leaving are same and hence we can say that the fluid is incompressible.