GATE-CH-2013-11-fm-1mark

The mass balance for a fluid with density ($\rho$) and velocity vector ($\overrightarrow{V}$) is

• ${\displaystyle \frac{\partial \rho }{\partial t}+\mathrm{\nabla }\cdot \left(\rho \overrightarrow{V}\right)=0}$

• ${\displaystyle \frac{\partial \rho }{\partial t}+\overrightarrow{V}\cdot \left(\mathrm{\nabla }\rho \right)=0}$

• ${\displaystyle \frac{\partial \rho }{\partial t}+\rho (\mathrm{\nabla }\cdot \overrightarrow{V})=0}$

• ${\displaystyle \frac{\partial \rho }{\partial t}-\overrightarrow{V}\cdot \left(\mathrm{\nabla }\rho \right)=0}$

GATE-ME-2015-S2-58-fm-2mark

Match the following pairs:

Equation	Physical Interpretation
P. $\mathrm{\nabla }\times \overrightarrow{V}=0$	I. Incompressible continuity equation
Q. $\mathrm{\nabla }\cdot \overrightarrow{V}=0$	II. Steady flow
R. ${\displaystyle \frac{D\overrightarrow{V}}{Dt}}=0$	III. Irrotational flow
S. ${\displaystyle \frac{\partial \overrightarrow{V}}{\partial t}}=0$	IV. Zero acceleration of fluid particle

 

      

• P-IV, Q-I, R-II, S-III

• P-IV, Q-III, R-I, S-II

• P-III, Q-I, R-IV, S-II

• P-III, Q-I, R-II, S-IV

GATE-CH-2003-34-fm-2mark

A fluid element has a velocity $\overrightarrow{V}=-y^{2}x\overrightarrow{i}+2y{x}^2\overrightarrow{j}$. The motion at $(x,y)=(1/\sqrt{2},1)$ is ____________

• rotational and incompressible

• rotational and compressible

• irrotational and compressible

• irrotational and incompressible

GATE-CH-2008-40-fm-2mark

A steady flow field of an incompressible fluid is given by $\overrightarrow{V}=(Ax+By)\hat{i}-Ay\hat{j}$, where $A=1$ s${}^{-1}$, $B=1$ s${}^{-1}$, and $x,y$ are in meters. The magnitude of the acceleration (in m/s²) of a fluid particle at $(1,2)$ is

• 1

• $\sqrt{2}$

• $\sqrt{5}$

• $\sqrt{10}$

GATE-CH-2009-31-fm-2mark

For an incompressible flow, the $x$- and $y$- components of the velocity vector are $$v_x=2(x+y);v_y=3(y+z)$$ where $x,y,z$ are in meters and velocities are in m/s. Then the $z$-component of the velocity vector $(v_z)$ of the flow for the boundary condition $v_z=0$ at $z=0$ is

• $5z$

• $-5z$

• $2x+3z$

• $-2x-3z$

GATE-CH-2010-19-fm-1mark

The stream function in a $xy$-plane is given below: $$\psi =\frac{1}{2}{x}^2{y}^3$$ The velocity vector for this stream function is

• ${\displaystyle x{y}^3\overrightarrow{ı}-\frac{3}{2}{x}^2{y}^2\overrightarrow{ȷ}}$

• ${\displaystyle \frac{3}{2}{x}^2{y}^2\overrightarrow{ı}-x{y}^3\overrightarrow{ȷ}}$

• ${\displaystyle \frac{3}{2}{x}^2{y}^2\overrightarrow{ı}+x{y}^3\overrightarrow{ȷ}}$

• ${\displaystyle x{y}^3\overrightarrow{ı}+\frac{3}{2}{x}^2{y}^2\overrightarrow{ȷ}}$

GATE-ME-2009-31-fm-2mark

You are asked to evaluate assorted fluid flows for their suitability in a given laboratory application. The following three flow choices, expressed in terms of the two-dimensional velocity fields in the $xy$-plane, are made available. 

$$\begin{array}{rlrl}\text{P.}& u=2y,& v& =-3x\\ \text{Q.}& u=3xy,& v& =0\\ \text{R.}& u=-2x,& v& =2y\end{array}$$

Which flow(s) should be recommended when the application requires the flow to be incompressible and irrotational?

• P and R

• Q

• Q and R

• R

GATE-XE-2011-B-19-fm-2mark

A flow has a velocity field given by $$\overrightarrow{v}=2x\hat{ı}-2y\hat{ȷ}$$ The velocity potential $\varphi (x,y)$ for the flow is

• $2x-2y+\text{ const.}$

• $2xy+\text{ const.}$

• $x^2+y^2+\text{ const.}$

• $x^2-y^2+\text{ const.}$

GATE-CH-2014-39-fm-2mark

An incompressible fluid is flowing through a contraction section of length $L$ and has a 1-D ($x$-direction) steady state velocity distribution, ${\displaystyle u=u_0(1+\frac{2x}{L})}$. If $u_0=2$ m/s and $L=3$ m, the convective acceleration (in m/s²) of the fluid at $L$ is ____________

• Answer

GATE-CH-2014-48-fm-2mark

In a steady incompressible flow, the velocity distribution is given by $\overrightarrow{v}=3x\hat{ı}-Py\hat{ȷ}+5z\hat{k}$, where, $v$ is in m/s and $x,y$, and $z$ are in m. In order to satisfy the mass conservation, the value of the constant $P$ (in s${}^{-1}$) is ____________

• Answer

GATE-CH-1992-13-b-fm-6mark

For flow over a flat plate where in a laminar boundary layer is present for the case of a zero pressure gradient, the parabolic velocity profile for velocity $u$ is given by $$\begin{array}{rl}u& =a_{1}y+a_2{y}^2\text{ for }y\le \delta \\ u& =v_o\text{ for }y\ge \delta \end{array}$$ Find $a_1$ and $a_2$.

• Answer

GATE-CH-2002-6-fm-5mark

Consider the flow in a liquid film of constant thickness ($\delta$) along a vertical wall as shown in figure below.

Assuming laminar, one-dimensional, fully developed flow, the $y$-direction Navier Stokes equation reduces to $$\mu \frac{d^2{v}_y}{d{x}^2}+\rho g=0$$ where $v_y$ is the velocity in $y$ direction, $\mu$ is the viscosity and $\rho$ is the density of the liquid.

1. State the boundary conditions to be used for the solution of velocity profile.

2. Solve for the velocity profile.

3. If $Q$ is the volumetric flow rate per unit width of the wall, how is it related to the film thickness, $\delta$.

• Answer