11.2 Fluid Kinematics, Continuity & Bernoulli's Equation

Fluid Kinematics, Continuity & Bernoulli's Equation

Fluid kinematics describes fluid motion in space and time without consideration of the forces causing the motion. In mining and process engineering—such as calculating flow rates in coal washing circuits, slurry distribution networks, and mine ventilation airways—kinematics and energy conservation equations form the theoretical core of fluid dynamic calculations.

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1. Lagrangian vs. Eulerian Descriptions

Fluid motion can be tracked via two distinct classical mechanics frameworks:

  1. Lagrangian Framework:             2. Eulerian Framework:
     Track specific fluid particle        Fix observation point (x, y, z);
     P(t) over its trajectory:            measure velocity field V(x, y, z, t):

        P(t_1) -> P(t_2) -> P(t_3)               (x, y, z)
          *--------->*--------->*                   [ + ] ---> V(x, y, z, t)

1. Lagrangian Description: Focuses on tracking individual, identifiable fluid particles as they move through space and time. Position vector $\vec{r} = \vec{r}(x_0, y_0, z_0, t)$ and velocity $\vec{V} = \frac{d\vec{r}}{dt}$. (Commonly used in solid mechanics and discrete multiphase slurry particle tracking).
2. Eulerian Description: Focuses on fixed spatial control points $(x,y,z)$ in the flow field, measuring fluid properties (velocity $\vec{V}$, pressure $P$, density $\rho$) passing through those fixed points as a function of time: $\vec{V} = u\hat{i} + v\hat{j} + w\hat{k} = f(x,y,z,t)$. (Standard approach in fluid mechanics and turbomachinery).

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1.1 The Substantial (Material) Derivative and Acceleration

The total acceleration of a fluid particle in an Eulerian field is given by the Substantial / Material Derivative $\frac{D\vec{V}}{Dt}$:

$\vec{a} = \frac{D\vec{V}}{Dt} = \underbrace{\frac{\partial\vec{V}}{\partial t}}_{\text{Local / Temporal Acceleration}} + \underbrace{\left(u\frac{\partial\vec{V}}{\partial x} + v\frac{\partial\vec{V}}{\partial y} + w\frac{\partial\vec{V}}{\partial z}\right)}_{\text{Convective / Spatial Acceleration}} = \frac{\partial\vec{V}}{\partial t} + (\vec{V} \cdot \nabla)\vec{V}$

Scalar Acceleration Components in Cartesian Coordinates:

$a_x = \frac{\partial u}{\partial t} + u\frac{\partial u}{\partial x} + v\frac{\partial u}{\partial y} + w\frac{\partial u}{\partial z}$ $a_y = \frac{\partial v}{\partial t} + u\frac{\partial v}{\partial x} + v\frac{\partial v}{\partial y} + w\frac{\partial v}{\partial z}$ $a_z = \frac{\partial w}{\partial t} + u\frac{\partial w}{\partial x} + v\frac{\partial w}{\partial y} + w\frac{\partial w}{\partial z}$

Summary of Acceleration Combinations:

• Steady Uniform Flow: Local Accel = $0$, Convective Accel = $0$ $\implies \vec{a} = 0$ (e.g., constant discharge through constant diameter pipe).
• Steady Non-Uniform Flow: Local Accel = $0$, Convective Accel $\ne 0$ $\implies \vec{a} \ne 0$ (e.g., constant discharge through a tapering pipe or nozzle).
• Unsteady Uniform Flow: Local Accel $\ne 0$, Convective Accel = $0$ $\implies \vec{a} \ne 0$ (e.g., pulsating flow through a constant diameter pipe).
• Unsteady Non-Uniform Flow: Local Accel $\ne 0$, Convective Accel $\ne 0$ $\implies \vec{a} \ne 0$ (e.g., changing discharge through a tapering nozzle).

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2. Classification of Fluid Flows

1. Steady vs. Unsteady Flow:
   • Steady Flow: Flow properties at any point do not change with time: $\frac{\partial \vec{V}}{\partial t} = 0, \frac{\partial P}{\partial t} = 0, \frac{\partial \rho}{\partial t} = 0$.
   • Unsteady Flow: Fluid properties at a point change with respect to time: $\frac{\partial \vec{V}}{\partial t} \ne 0$.
2. Uniform vs. Non-Uniform Flow:
   • Uniform Flow: Velocity vector does not change with spatial position along the direction of flow at any instant: $\left(\frac{\partial \vec{V}}{\partial s}\right)_{t=\text{const}} = 0$.
   • Non-Uniform Flow: Velocity changes along the path of flow: $\left(\frac{\partial \vec{V}}{\partial s}\right)_{t=\text{const}} \ne 0$.
3. Laminar vs. Turbulent Flow:
   • Laminar Flow: Fluid particles move in smooth, parallel layers or laminas without macroscopic cross-mixing. Governed by viscous forces (low Reynolds number).
   • Turbulent Flow: Fluid particles move in highly erratic, chaotic trajectories with intense momentum cross-mixing and eddy formation (high Reynolds number).
4. Rotational vs. Irrotational Flow:
   • Rotational Flow: Fluid particles rotate about their own mass centers while translating (Vorticity $\vec{\zeta} = \nabla \times \vec{V} \ne 0$).
   • Irrotational Flow: Fluid particles do not rotate about their own mass centers ($\nabla \times \vec{V} = 0$).
5. Compressible vs. Incompressible Flow:
   • Incompressible: Density is constant throughout ($\rho = \text{constant} \implies \nabla \cdot \vec{V} = 0$). (Valid for liquids and gas flows with Mach number $M = V/c < 0.3$).
   • Compressible: Density varies significantly ($M > 0.3$).

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3. Flow Lines: Streamlines, Pathlines, Streaklines, and Streamtubes

                     ds = dx i + dy j + dz k
                   ----->----------> Streamline
                  /      V = u i + v j + w k  (V is tangent to ds everywhere)
                 /
                + (V x ds = 0)

1. Streamline: An imaginary continuous line drawn through a flowing fluid such that the tangent at every point gives the direction of the velocity vector at that instant. There can be no fluid flow across a streamline.
   • Differential Equation of a Streamline: Since $\vec{V} \times d\vec{s} = 0$: $\frac{dx}{u} = \frac{dy}{v} = \frac{dz}{w}$
   • In 2D plane flow ($xy$-plane): $\frac{dx}{u} = \frac{dy}{v} \implies \left(\frac{dy}{dx}\right)_{\text{streamline}} = \frac{v}{u}$
2. Pathline: The actual trajectory traced by an individual fluid particle over a period of time (Lagrangian concept).
3. Streakline: The instantaneous locus of all fluid particles that have successively passed through a fixed point in space (e.g., continuous dye injection or smoke filament).
4. Streamtube: A tubular surface formed by a bundle of streamlines passing through a closed curve. Since velocity is everywhere tangent to the tube boundary, no fluid can cross the walls of a streamtube.

Exam Axiom: In steady flow, streamlines, pathlines, and streaklines are completely identical and coincide. In unsteady flow, they diverge into distinct curves.

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4. The 3D Continuity Equation (Conservation of Mass)

Applying mass conservation to a differential control volume $dx \times dy \times dz$:

$\left[\text{Net Mass Inflow Rate}\right] = \left[\text{Rate of Mass Accumulation inside CV}\right]$

4.1 General 3D Continuity Equation in Cartesian Coordinates:

$\frac{\partial \rho}{\partial t} + \frac{\partial(\rho u)}{\partial x} + \frac{\partial(\rho v)}{\partial y} + \frac{\partial(\rho w)}{\partial z} = 0$

In vector notation: $\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \vec{V}) = 0$

4.2 Special Cases:

• Steady Compressible Flow ($\frac{\partial \rho}{\partial t} = 0$): $\frac{\partial(\rho u)}{\partial x} + \frac{\partial(\rho v)}{\partial y} + \frac{\partial(\rho w)}{\partial z} = 0$
• Incompressible Flow ($\rho = \text{constant}$): $\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} = 0 \iff \nabla \cdot \vec{V} = 0$
• 1D Steady Flow along a Conduit: $\rho_1 A_1 V_1 = \rho_2 A_2 V_2 = \dot{m} = \text{constant}$ For incompressible fluid ($\rho_1 = \rho_2$): $A_1 V_1 = A_2 V_2 = Q = \text{constant}$

4.3 Cylindrical Polar Coordinates ($r, \theta, z$):

$\frac{\partial \rho}{\partial t} + \frac{1}{r}\frac{\partial(r \rho v_r)}{\partial r} + \frac{1}{r}\frac{\partial(\rho v_{\theta})}{\partial \theta} + \frac{\partial(\rho v_z)}{\partial z} = 0$ For 2D steady incompressible axisymmetric flow ($v_{\theta} = 0, \partial/\partial z = 0$): $\frac{1}{r}\frac{\partial(r v_r)}{\partial r} = 0 \implies r v_r = \text{constant}$

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5. Potential Flow Theory: Velocity Potential & Stream Function

               psi = c_2 (Streamline)     psi = c_1
                     |                        |
         phi = k_1 --+------------------------+----
                     |                        |
                     |     Orthogonal (90 deg)|
                     |        Intersection    |
         phi = k_2 --+------------------------+----
          (Equipotential Lines)

5.1 Velocity Potential Function ($\phi$)

A scalar spatial function $\phi(x,y,z,t)$ defined such that its negative spatial gradient yields the velocity components:

$u = -\frac{\partial \phi}{\partial x}, \quad v = -\frac{\partial \phi}{\partial y}, \quad w = -\frac{\partial \phi}{\partial z} \iff \vec{V} = -\nabla \phi$

(Note: Some textbooks omit the negative sign; standard PSU/GATE convention includes the negative sign indicating flow occurs in direction of decreasing potential).

Key Properties of $\phi$:

1. Existence Condition: $\phi$ exists if and only if the flow is irrotational: $\nabla \times \vec{V} = -\nabla \times (\nabla \phi) \equiv 0$ If $\phi$ exists for a flow field, the flow is guaranteed to be irrotational.
2. Laplace Equation for Incompressible Flow: Substituting $\phi$ into the continuity equation $\nabla \cdot \vec{V} = 0$ yields: $\frac{\partial^2 \phi}{\partial x^2} + \frac{\partial^2 \phi}{\partial y^2} + \frac{\partial^2 \phi}{\partial z^2} = 0 \iff \nabla^2 \phi = 0$ Any function satisfying Laplace's equation represents a possible steady, incompressible, irrotational fluid flow.
3. Equipotential Lines: Lines along which $\phi = \text{constant}$ ($d\phi = 0$): $d\phi = \frac{\partial \phi}{\partial x}dx + \frac{\partial \phi}{\partial y}dy = -u \, dx - v \, dy = 0$ $\left(\frac{dy}{dx}\right)_{\phi = \text{const}} = -\frac{u}{v}$

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5.2 Stream Function ($\psi$)

A scalar function $\psi(x,y,t)$ defined for 2D incompressible flow such that it automatically satisfies the continuity equation:

$u = -\frac{\partial \psi}{\partial y}, \quad v = \frac{\partial \psi}{\partial x}$

*(Or alternatively $u = \frac{\partial \psi}{\partial y}, v = -\frac{\partial \psi}{\partial x}$). Checking continuity: $\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = \frac{\partial}{\partial x}\left(-\frac{\partial \psi}{\partial y}\right) + \frac{\partial}{\partial y}\left(\frac{\partial \psi}{\partial x}\right) = -\frac{\partial^2 \psi}{\partial x \partial y} + \frac{\partial^2 \psi}{\partial y \partial x} = 0$.

Key Properties of $\psi$:

1. Existence Condition: $\psi$ exists for any 2D flow (rotational or irrotational) provided it satisfies the continuity equation.
2. Streamlines: Lines of constant $\psi$ ($d\psi = 0$): $d\psi = \frac{\partial \psi}{\partial x}dx + \frac{\partial \psi}{\partial y}dy = v \, dx - u \, dy = 0$ $\left(\frac{dy}{dx}\right)_{\psi = \text{const}} = \frac{v}{u}$
3. Volumetric Discharge between Streamlines: The volume flow rate per unit depth $q$ between two streamlines $\psi_1$ and $\psi_2$ is: $q = \psi_2 - \psi_1$
4. Irrotationality Condition for $\psi$: If the flow is also irrotational ($\omega_z = 0$): $\omega_z = \frac{1}{2}\left(\frac{\partial v}{\partial x} - \frac{\partial u}{\partial y}\right) = \frac{1}{2}\left(\frac{\partial^2 \psi}{\partial x^2} + \frac{\partial^2 \psi}{\partial y^2}\right) = 0 \iff \nabla^2 \psi = 0$

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5.3 Cauchy-Riemann Equations & Orthogonality of Flow Nets

Comparing definitions of $u$ and $v$:

$\frac{\partial \phi}{\partial x} = \frac{\partial \psi}{\partial y}, \quad \frac{\partial \phi}{\partial y} = -\frac{\partial \psi}{\partial x}$

These are the celebrated Cauchy-Riemann Equations from complex variable theory, proving that the complex potential $W(z) = \phi(x,y) + i\psi(x,y)$ is an analytic function of $z = x + iy$.

Proof of Orthogonality:

Multiplying the slopes of the equipotential line and the streamline at their point of intersection:

$\left(\frac{dy}{dx}\right)_{\phi = \text{const}} \times \left(\frac{dy}{dx}\right)_{\psi = \text{const}} = \left(-\frac{u}{v}\right) \times \left(\frac{v}{u}\right) = -1$

Flow Net Theorem: Equipotential lines ($\phi = \text{const}$) and streamlines ($\psi = \text{const}$) intersect orthogonally (at $90^{\circ}$) at every point in the flow field, except at stagnation points where $u = v = 0$ and slopes become indeterminate.

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5.4 Circulation ($\Gamma$) and Vorticity ($\vec{\zeta}$)

• Vorticity ($\vec{\zeta}$): Defined as the curl of velocity vector, representing twice the angular velocity ($\vec{\omega}$): $\vec{\zeta} = \nabla \times \vec{V} = 2\vec{\omega}$ In 2D plane flow ($xy$): $\zeta_z = 2\omega_z = \frac{\partial v}{\partial x} - \frac{\partial u}{\partial y}$
• Circulation ($\Gamma$): Line integral of the tangential velocity component around a closed contour $C$: $\Gamma = \oint_C \vec{V} \cdot d\vec{r} = \iint_A (\nabla \times \vec{V}) \cdot d\vec{A} = \iint_A \zeta_z \, dA = 2\omega_z A$ For an irrotational vortex, $\Gamma = \text{constant}$ and velocity $v_{\theta} = \frac{\Gamma}{2\pi r}$.

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6. Euler's Equation of Motion & Bernoulli's Principle

6.1 Euler's Equation along a Streamline

Applying Newton's Second Law ($\sum dF_s = dm \cdot a_s$) to a fluid element moving along a streamline $s$ without viscous friction:

$- \frac{\partial P}{\partial s} ds \, dA - \rho g \, ds \, dA \cos\theta = \rho \, ds \, dA \left(V\frac{\partial V}{\partial s}\right)$

Since $\cos\theta = \frac{dz}{ds}$, dividing by $\rho , ds , dA$ yields Euler's Equation of Motion:

$\frac{dP}{\rho} + V dV + g dz = 0$

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6.2 Bernoulli's Equation

Integrating Euler's equation along a streamline for an incompressible fluid ($\rho = \text{constant}$):

$\int \frac{dP}{\rho} + \int V dV + \int g dz = \text{constant}$

$\frac{P}{\rho} + \frac{V^2}{2} + g z = \text{constant}$

Dividing throughout by acceleration due to gravity $g$ yields the standard head form:

$\frac{P}{\rho g} + \frac{V^2}{2g} + z = \text{constant} = H$

$\underbrace{\frac{P}{\gamma}}_{\text{Pressure Head}} + \underbrace{\frac{V^2}{2g}}_{\text{Velocity Head / Kinetic Head}} + \underbrace{z}_{\text{Datum / Potential Head}} = \underbrace{H}_{\text{Total Energy Head}}$

• Piezometric Head: Sum of pressure head and datum head ($h_{\text{piezo}} = \frac{P}{\gamma} + z$).
• Total Head ($H$): Sum of piezometric head and dynamic velocity head.

The Four Strict Assumptions of Bernoulli's Equation:

1. Steady Flow: Flow parameters at any point do not vary with time ($\partial/\partial t = 0$).
2. Incompressible Flow: Fluid density $\rho$ is constant throughout.
3. Frictionless / Inviscid Fluid: Dynamic viscosity $\mu = 0$ (viscous shear stresses are negligible).
4. Flow along a Streamline: The constant $H$ applies strictly along an individual streamline. (Exception: If the entire flow field is irrotational, $\nabla \times \vec{V} = 0$, then $H$ has the identical constant value throughout the entire flow domain).

Real Fluid Flow with Friction Head Loss ($h_L$):

Between section 1 (upstream) and section 2 (downstream):

$\frac{P_1}{\rho g} + \frac{V_1^2}{2g} + z_1 = \frac{P_2}{\rho g} + \frac{V_2^2}{2g} + z_2 + h_L$

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7. Flow Measurement Devices

                 Venturimeter:                     Orifice Meter:
     Converging    Throat   Diverging               Pipe   Orifice Plate
     ================\====/===============         =========[ | ]=========
                     |    |                                  \|/
                     | h  |                             Vena  |  Recirculation
                     +----+                          Contracta|  Eddies

7.1 The Venturimeter

Consists of:

1. Convergent Cone: Short cone with convergence angle $\approx 20^{\circ} - 22^{\circ}$ (accelerates fluid smoothly).
2. Throat: Cylindrical throat section of diameter $d_2 \approx \left(\frac{1}{3} \text{ to } \frac{1}{2}\right) d_1$ (minimum area, maximum velocity, minimum pressure).
3. Divergent Cone: Long, gradual cone with divergence angle $\approx 5^{\circ} - 7^{\circ}$. The small angle is strictly engineered to prevent boundary layer separation, turbulence, and eddy formation.

Theoretical and Actual Discharge:

Applying Bernoulli's and Continuity ($a_1 V_1 = a_2 V_2$) between inlet (1) and throat (2):

$Q_{\text{th}} = \frac{a_1 a_2}{\sqrt{a_1^2 - a_2^2}} \sqrt{2gh}$

$Q_{\text{act}} = C_d Q_{\text{th}} = C_d \frac{a_1 a_2}{\sqrt{a_1^2 - a_2^2}} \sqrt{2gh}$

Where:

• $C_d$ is the Coefficient of Discharge (for Venturimeters, $C_d \approx 0.95 - 0.98$ due to streamlined flow and minimal friction losses).
• $h$ is the differential piezometric head: $h = \left(\frac{P_1}{\gamma} + z_1\right) - \left(\frac{P_2}{\gamma} + z_2\right) = x\left(\frac{S_m}{S} - 1\right)$ for a heavy manometric fluid.

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7.2 The Orifice Meter

A flat circular plate with a sharp-edged concentric hole (diameter $d_0$) clamped between pipe flanges.

• Vena Contracta: The section downstream of the orifice where fluid streamlines converge to a minimum cross-sectional area $a_c$, yielding maximum velocity. Occurs at distance $\approx 0.5 d_0$ downstream.
• Hydraulic Coefficients:
  1. Coefficient of Contraction ($C_c$): $C_c = \frac{a_c}{a_0} \approx 0.61 - 0.64$
  2. Coefficient of Velocity ($C_v$): $C_v = \frac{V_{\text{act}}}{V_{\text{th}}} \approx 0.97 - 0.98$
  3. Coefficient of Discharge ($C_d$): $C_d = C_c \times C_v \approx 0.60 - 0.65$
• Actual Discharge Formula: $Q_{\text{act}} = C_d \frac{a_1 a_0}{\sqrt{a_1^2 - a_0^2}} \sqrt{2gh}$
• Venturi vs. Orifice Comparison: Orifice meters are inexpensive and require short installation lengths, but cause substantial permanent head loss ($30% - 40%$ of differential head) due to severe eddy dissipation downstream of the vena contracta.

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7.3 The Pitot Tube and Pitot-Static Tube

Used for measuring local point velocities in pipes, mine ventilation shafts, and open channels.

• Stagnation Point: The point at the tip of the tube opening directly into the flow where fluid velocity is brought completely to rest ($V = 0$).
• Stagnation Pressure ($P_{\text{stag}}$): By Bernoulli's equation between free stream (0) and stagnation point ($s$): $P_{\text{stag}} = P_{\text{static}} + P_{\text{dynamic}} = P_0 + \frac{1}{2}\rho V^2$ $H_{\text{stag}} = h_{\text{static}} + \frac{V^2}{2g}$
• Local Flow Velocity ($V$): $V = C_v \sqrt{2g (H_{\text{stag}} - h_{\text{static}})} = C_v \sqrt{2g \Delta h} = C_v \sqrt{2g x \left(\frac{S_m}{S} - 1\right)}$ For standard calibrated pitot tubes, $C_v \approx 0.98 - 1.00$.

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8. Worked Engineering Example

Problem:

A horizontal Venturimeter with inlet diameter $d_1 = 300\text{ mm}$ and throat diameter $d_2 = 150\text{ mm}$ is installed in a pipeline carrying coal slurry of specific gravity $S = 1.25$. A differential U-tube mercury manometer ($S_m = 13.6$) connected between the inlet and throat registers a deflection of $x = 200\text{ mm} = 0.20\text{ m}$. Assuming $C_d = 0.97$, calculate:

1. The differential head in meters of the flowing slurry.
2. The actual volumetric discharge ($Q_{\text{act}}$) through the pipeline in $\text{m}^3/\text{s}$ and $\text{litres/s}$.

Solution:

1. Differential Piezometric Head ($h$): $h = x \left(\frac{S_m}{S} - 1\right) = 0.20 \left(\frac{13.6}{1.25} - 1\right) = 0.20 \left(10.88 - 1\right) = 0.20 \times 9.88 = 1.976\text{ m of slurry}$
2. Cross-Sectional Areas: $a_1 = \frac{\pi d_1^2}{4} = \frac{\pi \times 0.30^2}{4} = 0.070686\text{ m}^2$ $a_2 = \frac{\pi d_2^2}{4} = \frac{\pi \times 0.15^2}{4} = 0.017671\text{ m}^2$ $\frac{a_1}{a_2} = \left(\frac{d_1}{d_2}\right)^2 = \left(\frac{300}{150}\right)^2 = 4.0 \implies a_1 = 4 a_2$
3. Theoretical Velocity and Discharge: $\sqrt{a_1^2 - a_2^2} = \sqrt{(4 a_2)^2 - a_2^2} = \sqrt{15 a_2^2} = a_2 \sqrt{15} = 0.017671 \times 3.87298 = 0.06844\text{ m}^2$ $\sqrt{2gh} = \sqrt{2 \times 9.81 \times 1.976} = \sqrt{38.769} = 6.2265\text{ m/s}$ $Q_{\text{th}} = \frac{a_1 a_2 \sqrt{2gh}}{\sqrt{a_1^2 - a_2^2}} = \frac{0.070686 \times 0.017671 \times 6.2265}{0.06844} = 0.11364\text{ m}^3/\text{s}$
4. Actual Discharge ($Q_{\text{act}}$): $Q_{\text{act}} = C_d \times Q_{\text{th}} = 0.97 \times 0.11364 = 0.11023\text{ m}^3/\text{s}$ $Q_{\text{act}} = 0.11023 \times 1000 = 110.23\text{ litres/second}$ The pipeline flow rate is $110.23\text{ L/s}$.