Name: Fluid Mechanics
Uploaded: 2019-07-19
Duration: 6 min 9 s
Description: This video enables students to understand that fluid mechanics comprises fluid statics and fluid dynamics. Students also learn about steady and turbulent flo...

Question

What do you mean by an ideal liquid? Prove that when an ideal liquid flows through a tube in streamlined motion, the product of the cross-sectional area of the tube and the velocity of flow is constant.

Accepted Answer

An ideal liquid is one which has the following two properties:

$(1)$ Zero Compressibility and

$(2)$ Zero Viscosity.

An ideal liquid is incompressible, that is, on pressing the liquid there is no change in its volume (or density). Most of the liquids may be considered approximately incompressible, because on pressing them the change in their volume is negligible.

An ideal liquid is non-viscous, that is, when there is a relative motion between different layers of the liquid then there is no tangential frictional-force in between the layers. In actual practice, however, there is some viscosity in all liquids (and gases). It is less in gases, but larger in liquids.

Let us consider an incompressible and non-viscous liquid flowing in streamlined motion through a non- uniform tube $X Y$ .

Let $A_{1}$ and $A_{2}$ be the cross-sectional areas of the tube and $v_{1}$ and $v_{2}$ , the velocities of flow at the ends $X$ and $Y$ respectively. Let $ρ$ be the density of the liquid. The volume of the liquid entering at the end
$X$ in $1$ second is, $A_{1} \times v_{1}$
Therefore, mass of the liquid entering at the end $X$ in $1$ second is,
$ρ \times A_{1} \times v_{1}$
Similarly, mass of the liquid coming out from the end $Y$ in $1$ second is,
$ρ \times A_{2} \times v_{2}$
But the liquid which enters at one end must leave at the other. Hence, both these masses are equal, that is,
$ρ \times A_{1} \times v_{1} = ρ \times A_{2} \times v_{2}$ or

$A_{1} \times v_{1} = A_{2} \times v_{2}$ or

$A \times v = c o n s \tan t .$
Thus, at every place in the tube the product of the area of cross-section of the tube and the velocity of flow of the liquid is a constant. Therefore, the velocity of the liquid is smaller in the wider parts of the tube and larger in the narrower parts.

What do you mean by an ideal liquid? Prove that when an ideal liquid flows through a tube in streamlined motion, the product of the cross-sectional area of the tube and the velocity of flow is constant.

Important Questions on Motion in Fluids : Viscosity

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