91Ó°ÊÓ

The velocity of fluid:

$v=i\frac{dx}{dt}+j\frac{dy}{dt}+k\frac{dz}{dt}$

The equation of the continuity:

role="math" localid="1665141513332" $∇·v+\frac{∂\mathrm{ÒÏ}}{∂t}=0$

If the fluid is incompressible then,_{$\frac{∂\mathrm{ÒÏ}}{∂t}=0$} .

The fluid is called incompressible, $∇×v=0$.

In the case of fluid flow, Curl v at point is a measure of the angular velocity of the fluid in the neighbourhood of the point. When$∇×v=0$everywhere in the same region, the velocity field v is called an rotational in that region.

In the case of mathematical conditions, the force is aid to be conservative.

Making$v=∇\mathrm{ϕ}$which is the force of fluid,

Suppose$\mathrm{ÒÏ}$is the density of a fluid varies from point to point as well as with time such that$\mathrm{ÒÏ}=\mathrm{ÒÏ}(x,y,z,t)$, along the stream of fluid and x,y,z are the function of t and the velocity of fluids is as follows:

$v=i\frac{∂x}{∂t}+j\frac{∂y}{∂t}+k\frac{∂z}{∂t}$

By the equation of the continuity:

$∇·v+\frac{∂\mathrm{ÒÏ}}{∂t}=0$

Fluid flow is called irrational if$∇×v=0$where v is the velocity of the fluid.

Suppose the fluid is incompressible, then$\frac{∂\mathrm{ÒÏ}}{∂t}=0$.

So, the fluid is rotational, and from equation of continuity as follows:

$$\begin{gathered} ∇·v+0=0 \\ ∇·v=0 \end{gathered}$$

From the fluid force:

$v=∇\mathrm{ϕ}$

Putting this equation in above:

$∇·\left(∇\mathrm{ϕ}\right)=0$

So, the equation will be,_{${∇}^2\mathrm{ϕ}=0$} .

Hence, function $\mathrm{Ï•}$ satisfy the equation.