$1.($a$)$

$1^{\text{T}\text{h}\text{e}\text{v}\text{e}\text{l}\text{o}\text{c}\text{i}\text{t}\text{y}\text{c}\text{o}\text{m}\text{p}\text{o}\text{n}\text{e}\text{n}\text{t}\text{s}in\text{a}\text{t}\text{w}\text{o}-\text{d}\text{i}\text{m}\text{e}\text{n}\text{s}\text{i}\text{o}\text{n}\text{a}\text{l}\text{f}\text{l}\text{o}\text{w}\text{a}\text{r}\text{e}u}=8x^{2}y-\frac{8}{3}{y}^3$

$5$

and $v=-8x{y}^2+\frac{8}{3}{x}^3.$ Show that these velocity components represent a possible case of an irrotational flow.

$($b$)$ Show that the equipotential lines and stream lines are orthogonal to each other.

$2.($a$)$ Write the Euler's equation of motion. Hence obtain Bernoulli's $/$ equation from it and name the different terms.

$($b$)$ State all the assumptions made in the derivation of Bernoulli's equation. Show that each term in Bernoulli's equation represent energy per unit weight of fluid.

$3.$ A $30$ cm diameter pipe carries water under a head of $25$ m with a $$ velocity of $3\mathrm{.}0\frac{m}{sec}.$ If the avis of the pipe tuins through $45,$ find the magnitude and direction of the resultant force at the bend.

$4.$ Define all the hydraulic coefficients for flow through a small ofirice. Describe experimental methods to determine them.

$5.($a$)$ Water flows over a rectangular weir at $15$ cm head over its crest. If the length of weir is $1$ m and $C_d=0\mathrm{.}62$ find the discharge. Afterwards if this water flows over a right$-$angled triangular weir of $C_d=0\mathrm{.}6$ find the head over the crest of triangular weir.

$($b$)$ Draw neat sketches to show different types of weirs classified on the basis of their shape of crest.

$6.($a$)$ For viscous flow through a pipe of radius $R$ the average velocity of flow $\frac{?}{\text{b}\text{a}\text{r}}(u)=\frac{1}{8}(\frac{-p}{x})R^2.$ Find the loss of head for flowing through a pipe length L

$($b$)$ The average velocity of flow of crude oil of specific gravity $0\mathrm{.}9$ and viscosity $0\mathrm{.}97$ poise through a pipe of diameter $0\mathrm{.}1$ m is $0\mathrm{.}471\frac{m}{sec}.$ Find the head loss if the pipe length is $10$ m $.$