Boundary Layer · Fluid Mechanics · GATE ME

Air (density = 1.2 kg/m³, kinematic viscosity = 1.5 × 10^-5 m²/s) flows over a flat plate with a free-stream velocity of 2 m/s. The wall shear stress at a location 15 mm from the leading edge is τ_w. What is the wall shear stress at a location 30 mm from the leading edge?

Within a boundary layer for a steady incompressible flow, the Bernoulli equation

Consider an incompressible laminar boundary layer flow over a flat plate of length $$L$$, aligned with the direction of an incoming uniform free stream. If $$F$$ is the ratio of the drag force on the front half of the plate to the drag force on the rear half, then

Flow separation in flow past a solid object is caused by

If $$'x'$$ is the distance measured from the leading edge of a flat plate, then laminar boundary layer thickness varies as

As the transition from laminar to turbulent flow is induced in a cross flow past a circular cylinder the value of the drag coefficient drops.

The necessary and sufficient condition which brings about separation of boundary layer is
$${{dP} \over {dx}} > 0$$

The predominant forces acting on an element of fluid in the boundary layer over a flat plate in a uniform parallel stream are:

A streamlined body is defined as a body about which

The velocity profile inside the boundary layer for flow over a flat plate is given as $${u \over {{U_\infty }}} = \sin \left( {{\pi \over 2}\,{y \over \delta }} \right),$$ where $${U_\infty }$$ is the free stream velocity and $$'\delta '$$ is the local boundary layer thickness. If $${\delta ^ * }$$ is the local displacement thickness, the value of $${{{\delta ^ * }} \over \delta }$$ is

A steady laminar boundary layer is formed over a flat plate as shown in the figure. The free stream velocity of the fluid is $${U_0}.$$ The velocity profile at the inlet $$a$$-$$b$$ is uniform, while that at a downstream location $$c$$-$$d$$ is
given by $$u = {U_0}\left[ {2\left( {{y \over \delta }} \right) - {{\left( {{y \over \delta }} \right)}^2}} \right]$$

The ratio of the mass flow rate, $$\mathop {m{}_{bd}}\limits^ \bullet ,$$ leaving through the horizontal section $$b$$-$$d$$ to that entering through the vertical section $$a$$-$$b$$ is

Air ( $${\rho = 1.2\,\,kg/{m^3}}$$ and kinematic viscosity, $${v = 2 \times {{10}^{ - 5}}{m^2}/s}$$ ) with a velocity of $$2m/s$$ flows over the top surface of a flat plate of length $$2.5m.$$ If the average value of friction coefficient is $${C_f} = {{1.328} \over {\sqrt {{{{\mathop{\rm Re}\nolimits} }_x}} }},\,\,$$ the total drag force (in $$N$$) per unit width of the plate is ____________

An incompressible fluid flows over a flat plate with zero pressure gradient. The boundary layer thickness is $$1mm$$ at a location where the Reynolds number is $$1000$$. If the velocity of the fluid alone is increased by a factor of $$4,$$ then the boundary layer thickness sat the same location, in $$mm$$ will be

Consider a steady incompressible flow through a channel as shown below.

The velocity profile is uniform with a value of $${u_0}$$ at the inlet section $$A$$. The velocity profile at section B down stream is

$$$u\left\{ {\matrix{ {{V_m}{y \over \delta },} & {0 \le y \le \delta } \cr {{V_m},} & {\delta \le y \le H - \delta } \cr {{V_m}{{H - y} \over \delta },} & {H - \delta \le y \le H} \cr } } \right.$$$

The ratio $${{{V_m}} \over {{u_0}}}$$ is

Consider a steady incompressible flow through a channel as shown below.

The velocity profile is uniform with a value of $${u_0}$$ at the inlet section $$A$$. The velocity profile at section B down stream is

$$$u\left\{ {\matrix{ {{V_m}{y \over \delta },} & {0 \le y \le \delta } \cr {{V_m},} & {\delta \le y \le H - \delta } \cr {{V_m}{{H - y} \over \delta },} & {H - \delta \le y \le H} \cr } } \right.$$$

The ratio $${{{P_A} - {P_B}} \over {{1 \over 2}\rho {u_0}^2}}$$ (where $${{P_A}}$$ and $${{P_B}}$$ are the pressure at section $$A$$ and $$B$$ respectively and $$\rho $$ is the density of the fluid ) is

A smooth flat plate with a sharp leading edge is placed along a gas stream flowing at $$U = 10\,m/s.$$ The thickness of the boundary layer at section $$r$$- $$s$$ is $$10$$ $$mm,$$ the breadth of the plate is $$1$$ $$m$$ (into the paper) and the density of the gas, $$\rho = 1.0\,kg/{m^3}.$$ Assume that the boundary layer is thin, two-dimensional, and follows a linear velocity distribution, $$u = U\left( {y/\delta } \right),$$ at the section $$r$$-$$s$$, where $$y$$ is the height from plate.

The mass flow rate (in kg/s) across the section $$q$$-$$r$$ is

A smooth flat plate with a sharp leading edge is placed along a gas stream flowing at $$U = 10\,m/s.$$ The thickness of the boundary layer at section $$r$$- $$s$$ is $$10$$ $$mm,$$ the breadth of the plate is $$1$$ $$m$$ (into the paper) and the density of the gas, $$\rho = 1.0\,kg/{m^3}.$$ Assume that the boundary layer is thin, two-dimensional, and follows a linear velocity distribution, $$u = U\left( {y/\delta } \right),$$ at the section $$r$$-$$s$$, where $$y$$ is the height from plate.

The integrated drag force (in $$N$$) on the plate, between $$p$$-$$s$$, is

For air flow over a flat plate, velocity $$(U)$$ and boundary layer thickness $$\left( \delta \right)$$ can be expressed respectively, as $$${U \over {{U_\infty }}} = {3 \over 2}{y \over \delta } - {1 \over 2}{\left( {{y \over \delta }} \right)^3}\,\,\,\,;\,\,\,\,\delta = {{4.64x} \over {\sqrt {{{{\mathop{\rm Re}\nolimits} }_x}} }}$$$
If the free stream velocity is $$2$$ $$m/s$$, and air has Kinematic viscosity of $$1.5 \times {10^{ - 5}}{m^2}/s$$ and density of $$1.23$$ $$kg/{m^3}$$, then wall shear stress at $$x=1$$ $$m$$, is

The velocity profile across a boundary layer on a flat plate may be approximated as linear $$${V_x}\left( {x,y} \right) = {{{V_0}y} \over {\delta \left( x \right)}}$$$

Where $${{V_0}}$$ is the velocity far away and $$\delta \left( x \right)$$ is the boundary layer thickness at a distance $$x$$ from the leading edge, as shown below.

(a)$$\,\,\,\,\,\,$$ Use an appropriate control volume to determine the rate of mass influx into the
$$\,\,\,\,\,\,$$$$\,\,$$$$\,\,\,\,\,\,$$boundary layer up to $$x.$$

(b)$$\,\,\,\,\,\,$$ Obtain the momentum thickness into the boundary layer up to $$x.$$

(c)$$\,\,\,\,\,\,$$ In which direction (up or down) does the shear stress act on the face $$AB$$ of
$$\,\,\,\,\,\,$$$$\,\,\,\,\,\,\,$$the fluid element shown near the plate?