# Physical significance of divergence

In my textbook They considered a parallelopiped $$ABCDEFGH$$ with sides $$dx,dy,dz$$ parallel to $$x,y,z$$ axis respectively

• $$\vec V$$ represents the vector velocity of the fluid at the centre $$P$$ of f the volume (parallelopiped) with components $$V_x,V_y,V_z$$ along three axis
• the fluid flows in or out through the all six faces.
• let us consider the fluid flow through the two opposite faces 1 and 2 of the volume element each being normal to x-axis and has area=dydz
• The value of the x component of $$\vec V$$ at the centre of the face 1 and 2 will be different from $$V_x$$ at the at the centre $$P$$.

• $$\frac {\partial V_x}{\partial x}$$ is the rate of change of $$V_x$$ along x-axis, then the change in value of $$V_x$$ in going from $$P$$ to the centre of vertical faces 1 or 2 = $$\frac{\partial V_xdx}{\partial x2}$$

• the value of x-component of the velocity $$\vec V$$ at the centre of face 1 = $$V_x-\frac{1}{2}\frac {\partial V_x}{x}dx$$

Now my problem is

• "The value of the $$x$$ component of $$\vec V$$ at the centre of the face 1 and 2 will be different from $$V_x$$ at the at the centre $$P$$" Why?

• $$\frac {\partial V_x}{\partial x}$$ is the rate of change of $$V_x$$ along $$x$$-axis, then the change in value of $$V_x$$ in going from $$P$$ to the centre of vertical faces 1 or 2 = $$\frac{\partial V_xdx}{\partial x2}$$

• $$\frac {\partial V_x}{\partial x}$$ is the rate of change of $$V$$ along x-axis because here water is flowing in all directions and we are concerned with the $$x$$ axis only, okay. But, change in the value of $$V_x$$ in going from $$p$$ to the centre of vertical faces of 1 or 2 = $$\frac{\partial V_xdx}{\partial x2}$$ but how and why. Please explain

• the value of $$x$$-component of the velocity $$\vec V$$ at the centre of face 1 = $$V_x-\frac{1}{2}\frac {\partial V_x}{x}dx$$

• Possible duplicates: physics.stackexchange.com/q/264509/2451 and links therein. – Qmechanic Oct 28 '18 at 6:40
• 1. Because $v_x$ changes with $x$. – Rob Jeffries Oct 28 '18 at 8:01
• @RobJeffries what?? – Akash Oct 28 '18 at 8:02
• @Qmechanici checked out that...i didn't get my answer ..that is not a duplicate – Akash Oct 28 '18 at 8:10

The velocity $$V$$ is actually a vector field i.e it has different values of velocity at different points in space. That is why you get different components at different points.
Rate of change of $$x-component$$ of velocity in the x- direction is $$\frac{\partial V}{\partial x}$$. Now since the centre of parallelopiped is $$\frac{dx}{2}$$ units away from the centres of faces of parallelopiped, we multiply by $$\frac{dx}{2}$$ to obtain change in velocity.
Using the same logic, since the centre of face 1 is $$\frac{dx}{2}$$ away from the centre of parallelopiped, we subtract the change in velocity from the velocity at the centre.
• So,according to your 3rd paragraph, the velocity $V_x$ is changing at a constant rate as ths velocity at face 1 is find out by $V_x-\frac{1}{2}\frac {\partial V_x}{x}dx$. – Akash Oct 28 '18 at 23:52