Question

A two-dimensional flow is described by velocity components
u = 2x and v= -2y. The discharge between points (1,1) and (2,2) is equal to

• A. 9 units
• B. 8 units
• C. 7 units
• D. 6 units

Answer 1

The correct option is D 6 units

(d)

$v=\frac{\partial \psi }{\partial x}=-2y...........\left(i\right)$

$u=\frac{\partial \psi }{\partial y}=-2x.............\left(ii\right)$

Integrating (i), we get

$\psi =-2xy+f\left(y\right)........\left(iii\right)$
Differentiating (iii) w.r.t. y, we get

$\frac{\partial \psi }{\partial y}=-2x+f^{\prime }\left(y\right)...........\left(iv\right)$

Equating (ii) and (iv), we get

f'(y) = 0 .........(v)

Integrating (v), we get

f(y) =C
where C is a numerical constant which can be treated as zero.
From (iii), we get

$\psi =-2xy$

At (1,1);${\psi }_1=-2units$

At (2,2);${\psi }_2=-8units$

$dQ=|d\psi |=|{\psi }_2-{\psi }_1|$

= 8 -2 = 6 units