EC > GATE 2024 > Vector Calculus
Let $ρ(x,y,z,t)$ and $u(x,y,z,t)$ represent density and velocity, respectively, at a point $(x,y,z)$ and time t.
Assume $\frac{∂ρ}{∂t}$ is continuous. Let V be an arbitrary volume in space enclosed by the closed surface S and $n̂$ be the outward unit normal of S.
Which of the following equations is/are equivalent to $\frac{∂ρ}{∂t}+∇⋅(ρu)=0$?
A
$\int_{V}\frac{∂ρ}{∂t}dv=-\iint_{S}ρu⋅n̂ ds$
B
$\int_{V}\frac{∂ρ}{∂t}dv=[\oint_{S}ρu⋅n̂ ds$
C
$\int_{v}\frac{∂ρ}{∂t}dv=-\int_{V}∇⋅(ρu)dv$
D
$\int_{V}\frac{∂ρ}{∂t}dv=\int_{V}∇⋅(ρu)dv$

Correct : A;C

Similar Questions

î and ĵ be the unit vectors along x and y axes, respectively, and let A be a positive constant. Which one of the following statements is true for the vector f...
#37 MCQ
Let $F_1$, $F_2$ and $F_3$ be functions of (x, y, z).Suppose that for every given pair of points A and B in space, the line integral $∫_C (F_1 dx + F_2 dy + F_3...
#75 MSQ
Consider the two-dimensional vector field F⃗(x, y) = x î + y ĵ, where î and ĵ denote the unit vectors along the x-axis and the y-axis, respectively. A contour C...
#225 MCQ

Related Topics

continuity equation density velocity GATE EC 2024 fluid dynamics divergence theorem continuity equation EC fluid mechanics GATE divergence theorem EC fluid dynamics GATE

Unique Visitor Count

Total Unique Visitors

Loading......