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Sources And Sinks

PYQs

Sources And Sinks

1) Two sources, each of strength $m$ , are placed at the point at the points $(- a, 0)$ , $(a, 0)$ and a sink of strength $2 m$ at origin. Show that the stream lines are the curves $(x^{2} + y^{2})^{2}$ = $a^{2} (x^{2} - y^{2} + λ x y)$ , where $λ$ is variable parameter.

Show also that the fluid speed at any point is $(2 m a^{2}) / (r_{1} r_{2} r_{3})$ , where $r_{1}$ , $r_{2}$ and $r_{3}$ are the distances of the points from the sources and the sink, respectively.

[2019, 20M]

First Part: The complex potential $w$ at any point $P (z)$ is given by $w = - m \log (z - a) - m \log (z + a) + 2 m \log z \dots (1)$

⟹ w = m [\log z^{2} - \log (z^{2} - a^{2})]

$⟹ ϕ + i ψ = m [\log (x^{2} - y^{2} + 2 b y) - \log (x^{2} - y^{2} - a^{2} + 2 b y)],$ as $z = x + i y$

2019-8(c)

Equating the imaginary parts, we have $\begin{matrix} ψ = m [\tan^{- 1} {2 x y / (x^{2} - y^{2})} - \tan^{- 1} {2 x y / (x^{2} - y^{2} - a^{2})}] \\ ψ = m \tan^{- 1} [\frac{- 2 a^{2} x y}{{(x^{2} + y^{2})}^{2} - a^{2} (x^{2} - y^{2})}], on simplification \end{matrix}$

The desired streamlines are given by $ψ =$ constant $= m \tan^{- 1} (- 2 / λ) .$ Then we obtain

(- 2 / λ) = (- 2 a^{2} x y) / [{(x^{2} + y^{2})}^{2} - a^{2} (x^{2} - y^{2})] or {(x^{2} + y^{2})}^{2} = a^{2} (x^{2} - y^{2} + λ x y)

Second Part: We have,

\begin{aligned} \begin{aligned} \frac{d w}{d z} & = - \frac{m}{z - a} - \frac{m}{z + a} + \frac{2 m}{z} = - \frac{2 a^{2} m}{z (z - a) (z + a)} \\ ∴ & q = | \frac{d w}{d z} | = \frac{2 a^{2} m}{| z v e r t z - a | | z + a |} = \frac{2 a^{2} m}{r_{1} r_{2} r_{3}} \end{aligned} \\ \begin{array}{llll} where & r_{1} = | z - a |, & r_{2} = | z + a | & and & r_{3} = | z | \end{array} \end{aligned}

2) If fluid fills the region of space on the positive side of the $x - a x i s$ , which is a right boundary and if there be a sources $m$ at the point $(0, a)$ and an equal sink at $(0, b)$ and if the pressure on the negative side be the same as the pressure at infinity, show that the resultant pressure on the boundary is $\frac{π ρ m^{2} (a - b)^{2}}{{2 a b (a + b)}}$ where $ρ$ is the density of the fluid.

[2013, 15M]

The object system consists of source $+ m$ at $A$ $(o, a)$ , i-e. at $z = i a$ and sink $- m$ at $z = i b$ . The image system consists of source $+ m$ at $A^{'} (z = - i a)$ and sink $- m$ at $B^{'} (z = - i b)$ w.r.t the positive line $O X$ which is rigid boundary.

The complex potential due to object system with rigid boundary is equivalent to the object system and its image system with no rigid boundary.

\begin{aligned} ∴ w & = - m \log (z - i a) + m \log (z - i b) \\ - m \log (z + i a) + m \log (z + i b) \\ ⟹ & = - m \log (z^{2} + a^{2}) + m \log (z^{2} + b^{2}) \end{aligned}

Now, $\begin{aligned} \frac{d w}{d z} & = - 2 m \\ = [\frac{1}{z^{2} + a^{2}} - \frac{1}{z^{2} + b^{2}}] \\ = \frac{2 m z (a^{2} - b^{2})}{(z^{2} + a^{2}) (2^{2} + b^{2})} \\ ⟹ q & = | \frac{d ω}{d z} | = \frac{2 m (a^{2} - b^{2}) | z |}{\vertz^{2} + a^{2} | \vertz^{2} + b |} \end{aligned}$

For any point on $x$ -axis, we have $z = x$ so that $q = \frac{2 m x (a^{2} - b^{2})}{(x^{2} + a^{2}) (x^{2} + b^{2})}$

This is the expression for velocity at any point on $x$ -axis. Let $P_{0}$ be the pressure at $x = \infty$ . By Bernoulli equation for steady motion.

∴ \frac{P}{ρ} + \frac{1}{2} q^{2} = c

In view of $p = ρ_{0}, q = 0$ when $x = \infty$ we get $c = p_{0} / ρ$ . $\frac{P_{0} - P}{ρ} = \frac{1}{2} q^{2}$

Required pressure $P$ on boundary is given by:

\begin{aligned} P & = \int_{- \infty}^{\infty} (P_{0} - P) d x \\ = \int_{- \infty}^{\infty} \frac{1}{2} ρ q^{2} d x \\ = \frac{1}{2} ρ \int_{- \infty}^{\infty} \frac{4 m^{2} x^{2} {(a^{2} - b^{2})}^{2}}{{(x^{2} + a^{2})}^{2} {(x^{2} + b^{2})}^{2}} d x \\ = 4 ρ m^{2} {(a^{2} - b^{2})}^{2} \int_{0}^{\infty} \frac{x^{2} d x}{{(x^{2} + a^{2})}^{2} {(x^{2} + b^{2})}^{2}} \\ = 4 m^{2} ρ \int_{0}^{\infty} [\frac{a^{2} + b^{2}}{a^{2} - b^{2}} {\frac{1}{x^{2} + b^{2}} - \frac{1}{x^{2} + a^{2}}} - \frac{a^{2}}{{(x^{2} + a^{2})}^{2}} - \frac{b^{2}}{{(x^{2} + b^{2})}^{2}}] d x . \\ = 4 m^{2} f [\frac{a^{2} + b^{2}}{a^{2} - b^{2}} {\frac{π}{2 b} - \frac{π}{2 a}} - \frac{π}{4 a} - \frac{π}{4 b}] \\ = \frac{π ρ m^{2} (a - b)^{2}}{a b (a + b)} \\ = 4 m^{2} f [\frac{a^{2} + b^{2}}{a^{2} - b^{2}} {\frac{π}{2 b} - \frac{π}{2 a}} - \frac{π}{4 a} - \frac{π}{4 b}] \\ = \frac{π ρ m^{2} (a - b)^{2}}{a b (a + b)} \end{aligned}

(We have used below results) $\begin{aligned} \int_{0}^{\infty} \frac{d x}{x^{2} + a^{2}} & = {[\frac{1}{a} \tan - \frac{x}{a}]}_{0}^{\infty} \\ = \frac{π}{2 a} \\ Also, \int_{0}^{\infty} \frac{d x}{{(x^{2} + a^{2})}^{2}} & = \frac{1}{a^{3}} \int_{0}^{\infty} \cos^{2} θ d θ; x = a \tan θ \\ = \frac{x^{4}}{2 a^{3}} \int_{0}^{π / 2} (1 + \cos 2 θ) d θ = \frac{π}{2} \cdot \frac{1}{2 a^{3}} \\ = \frac{π}{4 a^{3}} \end{aligned}$

3) Two sources, each of strength $m$ are placed at the point $(- a, 0)$ , $(a, 0)$ and a sink of strength $2 m$ is at the origin. Show that the stream lines are the curves: ${(x^{2} + y^{2})}^{2} = a^{2} (x^{2} - y^{2} + λ x y)$ where $λ$ is a variable parameter. Show also that the fluid speed at any point is $(2 m a^{2}) / (r_{1} r_{2} r_{3})$ , where $r_{1} r_{2}$ and $r_{3}$ are the distance of the points from the source and the sink.

[2009, 12M]

First Part:

The complex potential $w$ at any point $P (z)$ is given by:

$w = - m \log (z - a) - m \log (z + a) + 2 m \log z \dots (1)$ or, $w = m [\log z^{2} - \log (z^{2} - a^{2})]$ or $ϕ + i ψ = m [\log (x^{2} - y^{2} + 2 i x y) - \log (x^{2} - y^{2} - a^{2} + 2 i x y)],$ as $z = x + i y$

The desired streamlines are given by $ψ =$ constant $= m \tan^{- 1} (- 2 / λ)$ . Then we obtain $(- 2 / λ) = (- 2 a^{2} x y) / [{(x^{2} + y^{2})}^{2} - a^{2} (x^{2} - y^{2})] or {(x^{2} + y^{2})}^{2} = a^{2} (x^{2} - y^{2} + λ x y)$

Second Part: From (1), we have

\frac{d w}{d z} = - \frac{m}{z - a} - \frac{m}{z + a} + \frac{2 m}{z} = - \frac{2 a^{2} m}{z (z - a) (z + a)}

$q = | \frac{d w}{d z} | = \frac{2 a^{2} m}{| z | | z - a | | z + a |} = \frac{2 a^{2} m}{r_{1} r_{2} r_{3}}$ , where

$r_{1} = | z - a |$ , $r_{2} = | z + a |$ and $r_{3} = | z |$

4) Let the fluid fills the region $x \geq 0$ (right half of $d$ plane). Let a source $α$ be $(0, y_{1})$ and equal sink at $(0, y_{2}), y_{1} > y_{2}$ . Let the pressure be same as pressure infinity i.e., $p_{0}$ . Show that the resultant pressure on the boundary $y - a x i s$ is $π ρ α^{2} {(y_{1} - y_{2})}^{2} / 2 y_{1} y_{2} (y_{1} + y_{2})$ , $ρ$ being the density of the fluid.

[2008, 30M]

5) Two sources, each of strength $m$ are placed at the points $(- a, 0)$ and $(a, 0)$ and a sink of strength $2 m$ is placed at the origin. Show that the stream lines are the curves ${(x^{2} + y^{2})}^{2} = a^{2} (x^{2} - y^{2} + λ x y)$ , where $λ$ is a variable parameter. Also show that fluid speed at any point is $\frac{2 m a^{2}}{r_{1} r_{2} r_{3}}$ where $r_{1} r_{2}$ and $r_{1}$ are respectively the distance of the point from the sources and sink.

[2003, 15M]

First Part:

The complex potential $w$ at any point $P (z)$ is given by:

Second Part: From (1), we have

\frac{d w}{d z} = - \frac{m}{z - a} - \frac{m}{z + a} + \frac{2 m}{z} = - \frac{2 a^{2} m}{z (z - a) (z + a)}

$q = | \frac{d w}{d z} | = \frac{2 a^{2} m}{| z | | z - a | | z + a |} = \frac{2 a^{2} m}{r_{1} r_{2} r_{3}}$ , where

$r_{1} = | z - a |$ , $r_{2} = | z + a |$ and $r_{3} = | z |$

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