The cross-sectional area of a horizontal tube | vedclass

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(A) According to the continuity equation, $A(x)v(x) = 	ext{constant}$, where $A(x)$ is the cross-sectional area and $v(x)$ is the velocity of the flu

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(A) According to the continuity equation, $A(x)v(x) = 	ext{constant}$, where $A(x)$ is the cross-sectional area and $v(x)$ is the velocity of the fluid.Since the area $A(x)$ increases linearly with $x$, we can write $A(x) = A_0 + kx$ (where $k > 0$).Thus, the velocity $v(x) = \frac{	ext{constant}}{A_0 + kx}$.From Bernoulli's equation for a horizontal tube, $P(x) + \frac{1}{2}ho v(x)^2 = 	ext{constant}$.Therefore, $P(x) = 	ext{constant} - \frac{1}{2}ho \left(\frac{C}{A_0 + kx}ight)^2$, where $C$ is a constant.As $x$ increases, the denominator $(A_0 + kx)^2$ increases, which means the term $\frac{1}{2}ho v(x)^2$ decreases.Consequently, the pressure $P(x)$ must increase as $x$ increases.Since $P(x)$ is inversely proportional to the square of a linear function of $x$, the curve of $P$ versus $x$ will be increasing and concave down, which matches the shape shown in Graph $A$.

Column-$I$	Column-$II$
$(a)$ Rain drops move downwards with constant velocity.	$(i)$ Viscous liquids
$(b)$ Floating clouds at a height in air.	$(ii)$ Viscosity
	$(iii)$ Less density

The cross-sectional area of a horizontal tube increases along its length linearly as we move in the direction of flow. The variation of pressure, as we move along its length in the direction of flow ($x$-direction), is best depicted by which of the following graphs?

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