Q15Mathematics I

Question

Whether the fluid motion given by

V = (y + z) i + (z + x) j + (x + y) k

is incompressible or not?

Answer 1

In the advanced mathematical study of fluid dynamics and vector calculus, the physical compressibility of a fluid flow is directly and exclusively determined by the divergence of its velocity vector field, denoted as $V$ . The divergence physically represents the volumetric rate at which fluid is expanding outward from (or compressing inward to) any given microscopic point in space.

If $div V > 0$ , the fluid is actively expanding (acting as a physical source).
If $div V < 0$ , the fluid is actively compressing (acting as a physical sink).
If $div V = 0$ , the volume of any fluid element remains absolutely constant during its motion. The flow is strictly classified as mathematically incompressible (or solenoidal).

Mathematical Evaluation

We are given the velocity vector field of the fluid as:

V = (y + z) i + (z + x) j + (x + y) k

To prove incompressibility, we must calculate the divergence using the standard differential operator $\nabla$ (del):

div V = \nabla \cdot V = \frac{\partial V _{x}}{\partial x} + \frac{\partial V _{y}}{\partial y} + \frac{\partial V _{z}}{\partial z}

We carefully substitute the respective vector components into this formula:

div V = \frac{\partial}{\partial x} (y + z) + \frac{\partial}{\partial y} (z + x) + \frac{\partial}{\partial z} (x + y)

Now, we execute the partial derivatives strictly. Because we are differentiating $(y + z)$ with respect to $x$ , both $y$ and $z$ are treated mathematically as constants, making the derivative identically zero. This logic holds symmetrically for all three terms:

div V = 0 + 0 + 0 = 0

Because the total calculated divergence is absolutely zero at every coordinate point in space, we have rigorously proven that the given fluid motion is completely and unconditionally incompressible.

Q15Mathematics I

Question

Answer

Mathematical Evaluation