11.3 Viscosity, Drag Force, Equation of Continuity & Bernoulli's Principle

Viscosity, Drag Force, Equation of Continuity & Bernoulli's Principle

Fluid dynamics investigates the mechanical behavior of liquids and gases in motion, forming a prominent component of of the Intermediate Physics Part I syllabus and UET ECAT evaluations. In contrast to solid mechanics where rigid bodies move as cohesive units, fluid particles slide past one another with internal shear stresses, vary in flow velocity across conduit cross-sections, and exchange pressure energy with kinetic and gravitational potential energy. Mastering viscous drag regimes, Stokes' law, terminal velocity scaling, mass conservation through the equation of continuity, and energy conservation through Bernoulli's equation is vital for solving both theoretical and numerical ECAT problems.

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1. Fluid Viscosity and Internal Shear

All real fluids exhibit internal resistance to relative motion between adjacent fluid layers, a property termed viscosity (often described as internal fluid friction).

Mechanism of Viscous Action

Consider a fluid layer confined between two parallel plates separated by a distance $\Delta y$. The bottom plate is fixed while the top plate of surface area $A$ moves horizontally with steady velocity $v$. Due to molecular adhesion, the fluid layer immediately contacting the upper plate moves at velocity $v$, while the layer contacting the stationary plate remains at rest. The intermediate fluid layers establish a steady velocity gradient $\frac{\Delta v}{\Delta y}$.

The tangential dragging force $F$ required to maintain this steady motion is proportional to the contact area $A$ and the transverse velocity gradient:

$F = \eta A \frac{\Delta v}{\Delta y}$

Where $\eta$ (eta) is the coefficient of viscosity of the fluid:

$\eta = \frac{F / A}{\Delta v / \Delta y} = \frac{\text{Shear Stress}}{\text{Velocity Gradient}}$

SI Units and Dimensional Analysis

• SI Unit: Newton-second per meter squared ($\text{N}\cdot\text{s}\cdot\text{m}^{-2}$) or kilogram per meter-second ($\text{kg}\cdot\text{m}^{-1}\cdot\text{s}^{-1}$), also termed the Pascal-second ($\text{Pa}\cdot\text{s}$).
• CGS Unit: Poise ($1\text{ Poise} = 0.1\text{ Pa}\cdot\text{s} \implies 1\text{ Pa}\cdot\text{s} = 10\text{ Poise}$).
• Dimensional Formula:

$[\eta] = \frac{[F]}{[A][\Delta v / \Delta y]} = \frac{[M L T^{-2}]}{[L^2][(L T^{-1})/L]} = \frac{[M L T^{-2}]}{[L^2 T^{-1}]} = [M L^{-1} T^{-1}]$

Temperature Dependence of Viscosity

A favorite ECAT conceptual contrast relates to how temperature affects fluid viscosity:

1. Liquids: Cohesive intermolecular forces dominate liquid viscosity. As temperature increases, thermal kinetic energy weakens these cohesive bonds, causing liquid viscosity to decrease (roughly exponentially; water falls from about $1.79\text{ mPa}\cdot\text{s}$ at $0^\circ\text{C}$ to about $0.28\text{ mPa}\cdot\text{s}$ at $100^\circ\text{C}$).
2. Gases: Molecular diffusion and momentum exchange dominate gas viscosity. As temperature increases, root-mean-square molecular velocity increases, accelerating momentum transfer across streamlines and causing gas viscosity to increase ($\eta_{\text{gas}} \propto \sqrt{T}$).

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2. Stokes' Law and Terminal Velocity

Stokes' Law for Spherical Droplets

When a smooth spherical body of radius $r$ moves at relatively low speed $v$ through a homogeneous, quiescent, viscous fluid of viscosity $\eta$, the fluid exerts a retarding drag force $F_d$ opposing the motion. Sir George Gabriel Stokes derived this relationship theoretically:

$F_d = 6\pi \eta r v$

Validity Conditions:

• The moving body must be strictly spherical with a smooth surface.
• The fluid medium must be infinite in extent compared to the sphere's dimensions.
• The flow around the sphere must be laminar (streamline) without turbulence (low Reynolds number, typically $Re < 1$).

Derivation of Terminal Velocity ($v_t$)

Consider a spherical droplet (such as a fog droplet, rain drop, or oil droplet in Millikan's apparatus) of radius $r$ and density $\rho$ falling vertically under gravity through a medium of density $\sigma$ and viscosity $\eta$. As the sphere falls, three vertical forces act on it:

1. Downward Gravitational Weight ($F_g$):

$F_g = m g = \left(\frac{4}{3}\pi r^3 \rho\right) g$

1. Upward Archimedean Buoyancy Force ($F_b$): Equal to the weight of fluid displaced:

$F_b = m_{\text{fluid}} g = \left(\frac{4}{3}\pi r^3 \sigma\right) g$

1. Upward Viscous Drag Force ($F_d$): Given by Stokes' law:

$F_d = 6\pi \eta r v$

By Newton's second law, the net downward force and instantaneous downward acceleration $a$ are:

$F_{\text{net}} = F_g - F_b - F_d = m a$

$\frac{4}{3}\pi r^3 (\rho - \sigma) g - 6\pi \eta r v = m a$

Initially, at $v = 0$, drag force $F_d = 0$, so the droplet experiences maximum downward acceleration $a = \frac{\rho - \sigma}{\rho} g$. As velocity increases, drag force increases linearly with $v$. Consequently, the net downward force decreases until it becomes zero ($F_{\text{net}} = 0 \implies a = 0$).

At this dynamic equilibrium, the droplet stops accelerating and continues downward at a constant maximum velocity termed the terminal velocity ($v_t$):

$F_g = F_b + F_d \implies \frac{4}{3}\pi r^3 (\rho - \sigma) g = 6\pi \eta r v_t$

Solving for $v_t$:

$v_t = \frac{4\pi r^3 (\rho - \sigma) g}{3 \times 6\pi \eta r} = \frac{2 r^2 (\rho - \sigma) g}{9 \eta}$

When a dense droplet falls in air where the medium density is negligible compared to the droplet density ($\sigma \ll \rho$):

$v_t \approx \frac{2 r^2 \rho g}{9 \eta}$

Essential ECAT Proportionalities for Terminal Velocity

1. Radius Dependency: Terminal velocity is directly proportional to the square of droplet radius:

$v_t \propto r^2$

ECAT Numerical Shortcut: If two spherical droplets of radii $r_1$ and $r_2$ fall in the same medium, their terminal velocities scale as $\frac{v_{t1}}{v_{t2}} = \left(\frac{r_1}{r_2}\right)^2$. If the radius is doubled ($2r$), terminal velocity increases fourfold ($4\times$).

1. Droplet Mass Dependency: Because mass $m = \frac{4}{3}\pi r^3 \rho \propto r^3$, the radius scales with mass as $r \propto m^{1/3}$. Substituting into $v_t \propto r^2$ gives:

$v_t \propto (m^{1/3})^2 = m^{2/3}$

1. Coalescence of Droplets: If $N$ identical droplets each of radius $r$ and terminal speed $v_t$ coalesce into a single large drop of radius $R$:

$V_{\text{new}} = N \times V_{\text{old}} \implies \frac{4}{3}\pi R^3 = N \left(\frac{4}{3}\pi r^3\right) \implies R = N^{1/3} r$

The terminal velocity $V_t$ of the coalesced drop is:

$V_t = v_t \left(\frac{R}{r}\right)^2 = v_t (N^{1/3})^2 = N^{2/3} v_t$

For example, if 8 identical water droplets coalesce ($N = 8$), $V_t = 8^{2/3} v_t = (2)^2 v_t = 4 v_t$.

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3. Ideal Fluid Flow and the Equation of Continuity

Criteria for an Ideal Fluid

To apply analytical fluid laws, real fluids are approximated as ideal fluids exhibiting four specific characteristics:

1. Incompressible Flow: The fluid density $\rho$ remains uniform and constant throughout ($\frac{d\rho}{dt} = 0$).
2. Non-viscous Flow: Internal friction is negligible ($\eta = 0$), so no mechanical energy is dissipated as heat.
3. Steady (Streamline) Flow: The velocity, pressure, and density of fluid particles passing any fixed spatial coordinate are invariant over time.
4. Irrotational Flow: Fluid elements experience no net angular momentum or vortex rotation about their centers of mass.

The Equation of Continuity

The Equation of Continuity is the mathematical expression of the Law of Conservation of Mass for flowing fluids.

Consider an ideal fluid flowing steadily through a conduit of variable cross-section. In time interval $\Delta t$:

• At cross-section 1 (area $A_1$, flow speed $v_1$), fluid traverses distance $\Delta x_1 = v_1 \Delta t$. The volume passing section 1 is $\Delta V_1 = A_1 v_1 \Delta t$, and the mass entering is $\Delta m_1 = \rho_1 A_1 v_1 \Delta t$.
• At cross-section 2 (area $A_2$, flow speed $v_2$), fluid traverses distance $\Delta x_2 = v_2 \Delta t$. The mass leaving is $\Delta m_2 = \rho_2 A_2 v_2 \Delta t$.

Because fluid cannot accumulate or be destroyed within the pipe:

$\Delta m_1 = \Delta m_2 \implies \rho_1 A_1 v_1 \Delta t = \rho_2 A_2 v_2 \Delta t \implies \rho_1 A_1 v_1 = \rho_2 A_2 v_2$

For an incompressible fluid ($\rho_1 = \rho_2 = \rho$):

$A_1 v_1 = A_2 v_2 = \text{constant}$

Volume Flow Rate ($Q$)

The product of cross-sectional area and flow speed represents the volume flow rate (or discharge rate) $Q$:

$Q = A v = \frac{\Delta V}{\Delta t}$

• SI Unit: cubic meters per second ($\text{m}^3\cdot\text{s}^{-1}$), dimensions $[L^3 T^{-1}]$.
• Conduit Diameter Relationship: For a circular pipe of diameter $d$ and radius $r$, $A = \pi r^2 = \frac{\pi d^2}{4}$:

$v \propto \frac{1}{A} \propto \frac{1}{r^2} \propto \frac{1}{d^2}$

ECAT Speed Rule: If the diameter of a garden hose or artery is constricted to half its initial size ($d_2 = d_1 / 2$), the fluid flow speed increases by a factor of $2^2 = 4$.

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4. Bernoulli's Principle and Bernoulli's Equation

Derivation from the Work-Energy Theorem

Bernoulli's Equation represents the Law of Conservation of Mechanical Energy applied to ideal fluid flow.

Consider a fluid segment moving through an elevated pipe whose cross-sectional area varies from $A_1$ at elevation $h_1$ to $A_2$ at elevation $h_2$:

1. Work Done by Pressure Forces:
   • At the inlet: Force $F_1 = P_1 A_1$ moves fluid through displacement $\Delta x_1$, performing positive work $W_1 = P_1 A_1 \Delta x_1 = P_1 \Delta V$.
   • At the outlet: Downstream pressure $P_2$ opposes forward motion, performing negative work $W_2 = -P_2 A_2 \Delta x_2 = -P_2 \Delta V$.
   • Net mechanical work done on the fluid element: $W_{\text{net}} = (P_1 - P_2) \Delta V$.
2. Change in Kinetic Energy:

$\Delta K = \frac{1}{2} \Delta m (v_2^2 - v_1^2) = \frac{1}{2} (\rho \Delta V) (v_2^2 - v_1^2)$

1. Change in Gravitational Potential Energy:

$\Delta U = \Delta m g (h_2 - h_1) = (\rho \Delta V) g (h_2 - h_1)$

By the Work-Energy Theorem ($W_{\text{net}} = \Delta K + \Delta U$):

$(P_1 - P_2) \Delta V = \frac{1}{2} \rho \Delta V (v_2^2 - v_1^2) + \rho g \Delta V (h_2 - h_1)$

Canceling volume $\Delta V$ across all terms and rearranging:

$P_1 + \frac{1}{2}\rho v_1^2 + \rho g h_1 = P_2 + \frac{1}{2}\rho v_2^2 + \rho g h_2$

In generalized form along any streamline:

$P + \frac{1}{2}\rho v^2 + \rho g h = \text{constant}$

Terminology and Energy Densities

Every term in Bernoulli's equation represents energy per unit volume (energy density), with units of $\text{J}\cdot\text{m}^{-3} = \text{N}\cdot\text{m}^{-2} = \text{Pa}$ and dimensions $[M L^{-1} T^{-2}]$:

• $P$: Static Pressure (internal pressure energy per unit volume).
• $\frac{1}{2}\rho v^2$: Dynamic Pressure (kinetic energy per unit volume).
• $\rho g h$: Hydrostatic Pressure (gravitational potential energy per unit volume).

Dividing each term by $\rho g$ yields the hydraulic head representation in meters of fluid column:

$\frac{P}{\rho g} + \frac{v^2}{2g} + h = \text{constant}$

Where $\frac{P}{\rho g}$ is pressure head, $\frac{v^2}{2g}$ is velocity head, and $h$ is elevation head.

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5. Applications of Bernoulli's Principle

1. Horizontal Flow and Pressure-Velocity Tradeoff

For a horizontal conduit ($h_1 = h_2$):

$P + \frac{1}{2}\rho v^2 = \text{constant} \implies P_1 - P_2 = \frac{1}{2}\rho(v_2^2 - v_1^2)$

Bernoulli's Principle: Along a horizontal streamline, fluid pressure is lowest where fluid velocity is highest, and fluid pressure is highest where fluid velocity is lowest.

2. Torricelli's Theorem (Speed of Efflux)

Consider an open storage tank filled with liquid of density $\rho$ to total depth $H$. A small orifice is punctured in the tank wall at depth $h$ below the liquid surface.

Applying Bernoulli's equation between surface 1 (pressure $P_1 = P_{\text{atm}}$, elevation $h_1 = H$, area $A_1$) and orifice 2 (pressure $P_2 = P_{\text{atm}}$, elevation $h_2 = H - h$, area $A_2$):

$P_{\text{atm}} + \frac{1}{2}\rho v_1^2 + \rho g H = P_{\text{atm}} + \frac{1}{2}\rho v_2^2 + \rho g (H - h)$

Because the tank surface area is vastly larger than the orifice ($A_1 \gg A_2$), the equation of continuity dictates $v_1 = \frac{A_2}{A_1} v_2 \approx 0$. Canceling $P_{\text{atm}}$ and simplifying:

$\rho g H = \frac{1}{2}\rho v_2^2 + \rho g H - \rho g h \implies \frac{1}{2}\rho v_2^2 = \rho g h$

$v_2 = \sqrt{2 g h}$

Torricelli's Law: The speed of efflux of liquid through an orifice under gravity is identical to the speed acquired by a body falling freely from rest through the same vertical height $h$.

Horizontal Range of Efflux Jet: The jet exits horizontally at height $y = H - h$ above the base. Time of free flight to ground is $t = \sqrt{\frac{2y}{g}} = \sqrt{\frac{2(H-h)}{g}}$. The horizontal landing range $R$ is:

$R = v_2 t = \sqrt{2gh} \times \sqrt{\frac{2(H-h)}{g}} = 2\sqrt{h(H - h)}$

• Maximum horizontal range occurs when the hole is drilled exactly at mid-depth: $h = \frac{H}{2}$.
• Maximum range value: $R_{\text{max}} = 2\sqrt{\frac{H}{2} \left(H - \frac{H}{2}\right)} = H$.
• Orifices equidistant from top and bottom (at depths $h$ and $H - h$) yield identical horizontal ranges.

3. Venturi Meter and Venturi Effect

A Venturi meter measures fluid flow speed in a pipeline. It consists of a converging section leading to a constricted throat of cross-sectional area $A_2 < A_1$.

• By Continuity: $v_2 = \frac{A_1}{A_2} v_1 > v_1$.
• By Bernoulli: $P_1 - P_2 = \frac{1}{2}\rho(v_2^2 - v_1^2) > 0$. Pressure drops significantly at the constricted throat.
• Measuring pressure drop with a U-tube manometer of liquid density $\rho_m$ yielding column height difference $\Delta h$:

$P_1 - P_2 = \rho_m g \Delta h$

Equating and solving for inlet speed $v_1$:

$v_1 = \sqrt{\frac{2(P_1 - P_2)}{\rho \left[\left(\frac{A_1}{A_2}\right)^2 - 1\right]}} = A_2 \sqrt{\frac{2 \rho_m g \Delta h}{\rho (A_1^2 - A_2^2)}}$

4. Dynamic Lift on Aerofoils and Spinning Balls

• Airplane Wing (Aerofoil): An aerofoil is shaped with a curved upper camber and flat lower surface. Incoming streamlines compress and deflect upwards, forcing air to traverse the top surface faster than the bottom surface ($v_{\text{top}} > v_{\text{bottom}}$). By Bernoulli's principle, pressure above the wing drops below pressure underneath ($P_{\text{top}} < P_{\text{bottom}}$). The resulting net upward force provides aerodynamic lift:

$F_{\text{lift}} = (P_{\text{bottom}} - P_{\text{top}}) A = \frac{1}{2}\rho (v_{\text{top}}^2 - v_{\text{bottom}}^2) A$

• Magnus Effect on Spinning Cricket/Tennis Ball: A ball spinning about an axis perpendicular to its translational flight drags a boundary air layer. On one side, spin velocity adds to relative wind velocity, creating higher speed and lower pressure; on the opposite side, spin opposes oncoming air, creating lower speed and higher pressure. The lateral pressure gradient curves the ball's trajectory (swing / swerve).

5. Atomizers, Paint Sprayers, and Carburetors

When a rubber bulb pushes high-speed air across the top of a narrow vertical dip tube immersed in liquid, velocity rises and local pressure drops below atmospheric pressure. Ambient air pressure pressing on the open container surface forces liquid up the tube. The high-speed airstream shears the ascending liquid into a fine atomized spray.

6. Vascular Blood Dynamics and Occlusion

In human circulatory mechanics, blood flows through arteries under pumping pressure. If an artery develops atherosclerotic plaque, the lumen narrows ($A \downarrow$). Continuity forces blood speed to surge ($v \uparrow$). By Bernoulli's equation, internal static pressure inside the narrowed artery drops sharply ($P \downarrow$). If external tissue pressure exceeds internal arterial pressure, the narrowed vessel can momentarily collapse and then reopen, producing the vascular flutter that textbooks use to illustrate Bernoulli's principle in the body.

Fluid Mechanics Laws Summary Table