Laminar Flow Between Plates MCQ Quiz - Objective Question with Answer for Laminar Flow Between Plates - Download Free PDF

The velocity distribution through laminar flow in fixed parallel plates is given by:

 

\(u = \frac{1}{{2\mu }}\left( {\frac{{ - \partial P}}{{\partial x}}} \right)(dy - {y^2})\)

Now

\(\frac{{du}}{{dy}} = \frac{1}{{2\mu }}\left( {\frac{{ - \partial P}}{{\partial x}}} \right)\left( {d - 2y} \right)\)

Shear stress distribution, τ is given by

\(\tau = \frac{{\mu \;du}}{{dy}}\)

\(\tau = \frac{1}{2}\left( {\frac{{ - \partial P}}{{\partial x}}} \right)\left( {d - 2y} \right)\)

From above, following can be concluded:

1. Shear Stress distribution, ‘τ’ is linear.

2. At y = d/2 i.e. mid point; τ = 0 i.e. At center shear stress = 0.

3. At y = 0 i.e. at boundary; \(\tau = {\tau _{max}} = \frac{1}{2}\left( {\frac{{ - \partial P}}{{\partial x}}} \right)\left( d \right)\) i.e. shear stress is maximum at boundary.

∴ Shear stress is maximum at the plate boundaries and zero at plane \(\frac{d}{2}\)  away from each plate

Explanation:

Attraction between Two Closely Parallel Moving Boats

The phenomenon where two closely parallel moving boats experience an attraction towards each other can be explained using Bernoulli's equation. This principle is rooted in fluid dynamics and describes the behavior of fluid flow.

• Bernoulli's Principle: It states that an increase in the speed of a fluid occurs simultaneously with a decrease in pressure or a decrease in the fluid's potential energy.

• Application to Boats: When two boats move parallel to each other, the water flow between the boats speeds up, leading to a decrease in pressure between them.

• Resulting Attraction: The lower pressure between the boats compared to the higher pressure on their outer sides creates a net force that pulls the boats towards each other.

Analyzing the Given Options

1. Macaulay’s equation  (Incorrect)

   • Macaulay’s equation is used in structural analysis for determining deflections and moments in beams, which is unrelated to the fluid dynamics of moving boats.

2. Euler’s equation  (Incorrect)

   • Euler’s equation describes the motion of a fluid, but it does not specifically explain the pressure changes due to fluid speed in the context of two parallel moving boats.

3. Bernoulli’s equation  (Correct)

   • Bernoulli’s equation directly relates the speed of the fluid to the pressure, explaining the attraction due to the faster water flow between the boats creating lower pressure.

4. Momentum equation  (Incorrect)

   • The momentum equation deals with the conservation of momentum in fluid flow but does not specifically describe the pressure variations causing the boats to attract each other.

Explanation:

For the flow between two parallel spaced plates (as shown in the figure below): 

The velocity distribution is given by:

\(\rm{v = \frac{1}{{2\mu }}\left( {\frac{{ - \partial p}}{{\partial x}}} \right)\left( {By - {y^2}} \right)}\)

Shear stress is given by:

\(\rm{\begin{array}{l} \rm{\tau = \mu \frac{{du}}{{dy}}}\\ \rm{\tau= \frac{1}{2} \times \left( { - \frac{{\partial p}}{{\partial x}}} \right)\left( {B - 2y} \right)}\\ y = \frac{B}{2},\;\tau = 0 \end{array}}\)

At y = 0, τ = τmax

Thus, the stress distribution across a section is depicted below:

∴ The shear stress is maximum at the boundaries and zero at the center.

Velocity distribution profile equation is written as:

\(u = \frac{{ - 1}}{{2\mu }}\left( {\frac{{\partial p}}{{\partial x}}} \right)\left( {Hy - {y^2}} \right)\)

It is a parabolic equation, So the velocity distribution is parabolic.

Explanation:

For a fully developed laminar viscous flow through a circular pipe, the maximum velocity is equal to twice the average velocity.

i.e. \({U_{max}} = 2 \times {U_{avg}}\)

\({U_{max}} = \frac{1}{{4\mu }}\left( { - \frac{{\partial P}}{{\partial x}}} \right).{R^2}\)

\(U_{avg}=\bar U = \frac{1}{{8\mu }}\left( { - \frac{{\partial P}}{{\partial x}}} \right).{R^2}\)

Hence \(\frac{{{U_{max}}}}{{\bar U}} = 2\)

\({U\over U_{max}} = {1\over 2}\)

Important Points 

For the fully developed laminar flow through the parallel plates, the maximum velocity is equal to the (3/2) times of the average velocity.

i.e. \({U_{max}} = \frac{3}{2} \times {U_{avg}}\)

For a laminar flow between parallel plates:

\(\begin{array}{l} {V_{max}}\; = \;\frac{{{B^2}}}{{8u}}\left( {\frac{{ - \partial P}}{{\partial x}}} \right)\\ {V_{avg}}\; = \;\frac{{{B^2}}}{{12u}}\left( {\frac{{ - \partial P}}{{\partial x}}} \right)\\ \Rightarrow \frac{{{V_{avg}}}}{{{V_{max}}}}\; = \;\frac{2}{3} \end{array}\)

 

Explanation:

For the flow between two parallel spaced plates (as shown in the figure below): 

The velocity distribution is given by:

\(\rm{v = \frac{1}{{2\mu }}\left( {\frac{{ - \partial p}}{{\partial x}}} \right)\left( {By - {y^2}} \right)}\)

Shear stress is given by:

\(\rm{\begin{array}{l} \rm{\tau = \mu \frac{{du}}{{dy}}}\\ \rm{\tau= \frac{1}{2} \times \left( { - \frac{{\partial p}}{{\partial x}}} \right)\left( {B - 2y} \right)}\\ y = \frac{B}{2},\;\tau = 0 \end{array}}\)

At y = 0, τ = τmax

Hence, shear stress varies directly as the distance from the midplane.

Thus, the stress distribution across a section is depicted below:

∴ The shear stress is maximum at the boundaries and zero at the centre.

Concept:

Viscous flow in Plates:

Velocity equation:

\(u=-\frac{1}{2\mu}\left( \frac{\partial p}{\partial x}\right)(ty\;-\;y^2)\)

where t = distance between two plates and y = distance measured from lower plate towards the upper plate.

Shear stress equation:

\(\tau=-\frac{1}{2}\left(\frac{\partial p}{\partial x}\right)(t-2y)\)

Maximum velocity happens at, \(y =\frac{t}{2}\), which is given by \(u_{max}=-\frac{1}{8\mu}\left( \frac{\partial p}{\partial x}\right)t^2\)

Ratio of Maximum velocity to the Average velocity in case of plates:

\(\frac{u_{max}}{u_{avg}}=\frac{3}{2}\)

Calculation:

Given:

Parallel plates:

U_max = 6 ms^-1, Uavg = ??

\(\frac{U_{max}}{U_{avg}}=\frac{3}{2}\)

\({U_{avg}}=\frac{2}{3}\times6\Rightarrow 4\) ms^-1

Additional Information

Viscous flow in Pipes:

Velocity equation:

\(u=-\frac{1}{4\mu}\left( \frac{\partial p}{\partial x}\right)(R^2\;-\;r^2)\)

where r = distance measured from the centre of the pipe.

Shear stress equation:

\(\tau=-\left(\frac{\partial p}{\partial x}\right)\frac{r}{2}\)

Maximum velocity happens at r = 0, which is given by \(u_{max}=-\frac{1}{4\mu}\left( \frac{\partial p}{\partial x}\right)R^2\)

Ratio of Maximum velocity to the Average velocity in case of pipe:

\(\frac{U_{max}}{U_{avg}}=2\)

For the flow between two parallel spaced plates (as shown in the figure below): 

The velocity distribution is given by:

\(\rm{v = \frac{1}{{2\mu }}\left( {\frac{{ - \partial p}}{{\partial x}}} \right)\left( {By - {y^2}} \right)}\)

Shear stress is given by:

\(\rm{\begin{array}{l} \rm{\tau = \mu \frac{{du}}{{dy}}}\\ \rm{\tau= \frac{1}{2} \times \left( { - \frac{{\partial p}}{{\partial x}}} \right)\left( {B - 2y} \right)}\\ y = \frac{B}{2},\;\tau = 0 \end{array}}\)

At y = 0, τ = τmax

Thus, the stress distribution across a section is depicted below:

∴ The shear stress is maximum at the boundaries and zero at the center.

Concept:

Pressure difference b/w two points in Laminar flow b/w two fixed parallel plate

\({P_2} - {P_1} = - \frac{{12\mu vl}}{{{t^2}}}\)

Maximum shear stress of Laminar flow b/w two fixed parallel plate

\({τ _{max}} = - \frac{1}{2}\left( {\frac{{dp}}{{dx}}} \right)t\)

Calculation:

Given:

Specific Gravity = 0.9 ⇒ Density (ρ) = 900 kg/m³

μ = 1 poise ⇒ 0.1 Ns/m²

v = 1 m/s, t = 1 cm ⇒ 0.01 m

The pressure difference is:

\({P_2} - {P_1} = - \frac{{12\mu vl}}{{{t^2}}}\)

\(\frac{{{P_2} - {P_1}}}{l} = - \frac{{12\mu v}}{{{t^2}}}\)

\(\left( {\frac{{dp}}{{dx}}} \right) = \; - \frac{{12\mu v}}{{{t^2}}}\)

\(\left( {\frac{{dp}}{{dx}}} \right) = - \frac{{12 \times .1 \times 1}}{{{{\left( {.01} \right)}^2}}} \Rightarrow - 12000\frac{N}{{{m^2}}}\)

Maximum shear stress is:

\({τ _{max}} = - \frac{1}{2}\left( {\frac{{dp}}{{dx}}} \right)t\)

\({τ _{max}} = - \frac{1}{2} \times \left( { - 12000} \right) \times 0.01\)

τ_max = 60 N/m²

Concept:

Laminar flow between the parallel fixed plate

Velocity distribution equation is given by:

\(u = \frac{{ - 1}}{{2\mu }}\left( {\frac{{\partial p}}{{\partial x}}} \right)\left( {Hy - {y^2}} \right)\)

Where,

u = velocity of the fluid at any distance y from the boundary

H = Distance between two parallel fixed plates.

Calculation:

H = 6 mm, u_centre = 1.8 m/sec

At \(y = \frac{H}{2} = \frac{6}{2} = 3\;mm\)

\(1.8 = {u_{centre}} = - \frac{1}{{2\mu }}\left( {\frac{{\partial p}}{{\partial x}}} \right)\left[ {6 \times 3 - {{\left( 3 \right)}^2}} \right]\)

\( - \frac{1}{{2\mu }}\left( {\frac{{\partial p}}{{\partial x}}} \right) = 0.2\)

Now, at y = 1 mm

\(u = - \frac{1}{{2\mu }}\left( {\frac{{\partial p}}{{\partial x}}} \right)\left[ {6 \times 1 - {{\left( 1 \right)}^2}} \right]\)

\(u = - \frac{1}{{2\mu }}\left( {\frac{{\partial p}}{{\partial x}}} \right)\left( {6 - 1} \right)\)

\(u = 0.2 \ \times \ 5\)

u = 1 m/s.

\(v = \frac{1}{{2\mu }}\left( {\frac{{ - \partial p}}{{\partial x}}} \right)\left( {By - {y^2}} \right)\)

Shear stress

\(\begin{array}{l} \tau = \mu \frac{{du}}{{dy}}\\ = \frac{1}{2} \times \left( { - \frac{{\partial p}}{{\partial x}}} \right)\left( {B - 2y} \right)\\ y = \frac{B}{2},\;\tau = 0 \end{array}\)

Y = 0, τ = τ_max

Explanation:

General velocity profile for Laminar flow

\(\frac{u}{{{u_\infty }}} = \frac{3}{2}\left( {\frac{y}{\delta }} \right) - \frac{1}{2}{\left( {\frac{y}{\delta }} \right)^3}\)

\({\tau _{wall}} = \mu {\left( {\frac{{\partial u}}{{dy}}} \right)_{y = 0}} = \frac{{3\mu }}{{2\delta }}\)

Now \(\delta = \frac{{4.65x}}{{\sqrt {R{e_X}} }}\)

\(\delta \propto {x^{\frac{1}{2}}}\)

\({\tau _{wall}} \propto {x^{\frac{{ - 1}}{2}}}\)

\( = \tau \propto \frac{1}{{\sqrt x }}\) as the distance from the leading edge increases, shear stress decreases.

Couette flow is the flow of a viscous fluid in the space between two surfaces, one of which is moving tangentially relative to the other. The configuration often takes the form of two parallel plates or the gap between two concentric cylinders.

In plug flow, the velocity of the fluid is assumed to be constant across any cross-section of the pipe perpendicular to the axis of the pipe. The plug flow model assumes there is no boundary layer adjacent to the inner wall of the pipe.

Stokes flow or creeping flow is a type of fluid flow where advective inertial forces are small compared with viscous forces. The Reynolds number is low i.e. Re≪1. This is a typical situation in flows where the fluid velocities are very slow, the viscosities are very large.

Explanation:

The laminar flow between two parallel fixed plate

Velocity \(V= \frac{-1}{2\mu}\)\(\left(\frac{-\partial P}{\partial x}\right)\) (By - y²)  

Discharge Q = \(\frac{1}{12\mu}\)\(\left(\frac{-\partial P}{\partial x}\right)\)B³

Shear stress on the boundary τ  = \(\frac{1}{2}\) \(\frac{-\partial P}{\partial x}\)(B-2y)

Shear Stress at Boundary:

at y = o, τ  = \(\frac{6 \mu Q}{B^3}\)× B

Q = 600× 10^-4 m³ /sec , B = 3×10^-2 m

μ = 1.2 pa -sec

τ  = \(\frac{6× 1.2×600×10^{-4}}{(3× 10^{-2})^3}\) × (3× 10^-2)

τ  = 480 pa

Explanation:

The velocity distribution across a section of two fixed parallel plates is parabolically given by

\(u = \frac{1}{{2\mu }}\left( { - \frac{{\partial P}}{{\partial x}}} \right)\left( {ty - {y^2}} \right)\)

where \(\frac{{\partial P}}{{\partial x}} = \) pressure gradient along the length of the plate

y = point of consideration from lower fixed plate

t = distance between the two fixed parallel plates

Note: The shear stress distribution across a section of two fixed parallel plates is Linear.