Fluid Mechanics for MAP/Fluid Dynamics

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Introduction
Differential Approach: We seek solution at every point $$ \displaystyle (x_{1},x_{2},x_{3})$$, i.e describe the detailed flow pattern at all points.

Integral Approach: We focus on a control volume (CV), which is a finite region. It determines gross flow effects such as force or torque on a body or the total energy exchange. For this purpose, balances of incoming and outgoing flux of mass, momentum and energy are made through this finite region. It gives very fast engineering answers, sometimes crude but useful.



Lagrangian versus Eulerian Approach: Substantial Derivative
Let $$ \displaystyle \alpha$$ be any flow variable (pressure, velocity, etc.). Eulerian approach deals with the description of $$ \displaystyle \alpha$$ at each location $$ \displaystyle (x_{i})$$ and time (t). For example, measurement of pressure at all $$ \displaystyle x_{i}$$ defines the pressure field: $$ \displaystyle P(x_{1},x_{2},x_{3},t)$$. Other field variables of the flow are:

$$ \displaystyle U_{j}(x_{1},t), P(x_{1},t), \rho(x_{1},t), T(x_{1}), \tau_{jk}(x_{1},t) \Rightarrow \alpha(x_{i},t)$$

Lagrangian approach tracks a fluid particle and determines its properties as it moves.

$$ \displaystyle x_{i}|_{p}(t+\Delta t) = x_{i}|_{p}(t) + \int^{t+\Delta t}_{t}U_{i}|_{p}(t')dt'$$

Oceanographic measurements made with floating sensors delivering location, pressure and temperature data, is one example of this approach. X-ray opaque dyes, which are used to trace blood flow in arteries, is another example.

Let $$ \displaystyle \alpha_{p}$$ be the variable of the particle (substance) P, this $$ \displaystyle \alpha_{p}$$ is called "substantial variable".

For this variable:

$$ \displaystyle \alpha_{p}(\vec{x_{p}},t)$$ and $$ \displaystyle \vec{x_{p}} = \vec{x_{p}}(t) \rightarrow \alpha_{p}(\vec{x_{p}},t) = \alpha_{p}(t)$$

In other words, one observes the change of variable $$ \displaystyle \alpha$$ for a selected amount of mass of fixed identity, such that for the fluid particle, every change is a function of time only.

In a fluid flow, due to excessive number of fluid particles, Lagrangian approach is not widely used.

Thus, for a particle P finding itself at point $$ \displaystyle x_{i}$$ for a given time, we can write the equality with the field variable:

$$ \displaystyle \alpha_{p}(t) = \alpha \left[(x_{i})_{p},t\right]$$

Along the path of the particle:

$$ \displaystyle \alpha_{p}(t + \Delta t) = \alpha \left[(x_{i} + \Delta x_{i})_{p}, t + \Delta t\right]$$

Hence,

$$ \displaystyle d\alpha_{p} = \alpha \left[((x_{i} + \Delta x_{i})_{p}, t + \Delta t) - \alpha ((x_{i})_{p},t)\right]$$

$$ \displaystyle d\alpha_{p} = \frac{\partial \alpha}{\partial t}dt + \frac{\partial \alpha}{\partial x_{i}}dx_{ip}$$

$$ \displaystyle \frac{d\alpha_{p}}{dt} = \frac{\partial \alpha}{\partial t} + \left(\frac{\partial \alpha}{\partial x_{i}}\right)\left(\frac{dx_{i}}{dt}\right)_{p} = \underbrace{\frac{\partial \alpha}{\partial t}}_{Local\ change\ in\ time} + \underbrace{\frac{\partial \alpha}{\partial x_{i}}U_{i}}_{Change\ in\ space}$$

The local change in time is the local time derivative (unsteadiness of the flow) and the change in space is the change along the path of the particle by means of the convective derivative.

$$ \displaystyle \frac{d\alpha_{p}}{dt} = \frac{D\alpha}{Dt} = \left(\frac{\partial }{\partial t} + U_{i}\frac{\partial }{\partial x_{i}}\right)\alpha$$

The substantial derivative connects the Lagrangian and Eulerian variables.

System versus Control volume
In mechanics, system is a collection of matter of fixed identity (always the same atoms or fluid particles) which may move, flow and interact with its surroundings.

Hence, the mass is constant for a system, although it may continually change size and shape. This approach is very useful in statics and dynamics, in which the system can be isolated from its surrounding and its interaction with the surrounding can be analysed by using a free-body diagram.

In fluid dynamics, it is very hard to identify and follow a specific quantity of the fluid. Imagine a river and you have to follow a specific mass of water along the river.

Mostly, we are rather interested in determining forces on surfaces, for example on the surfaces of airplanes and cars. Hence, instead of system approach, we identify a specific volume in space (associated with our geometry of interest) and analyse the flow within, through or around this volume. This specific volume is called "Control Volume". This control volume can be fixed, moving or even deforming.

The control volume is a specific geometric entity independent of the flowing fluid. The matter within a control volume may change with time, and the mass may not remain constant.



Conservation of mass
The mass of a system do not change:

$$ \displaystyle \displaystyle \frac{dM}{dt}_{system}\bigg| = 0$$

where, $$ \displaystyle \displaystyle M = \int_{mass(system)}{dm} = \int_{V(system)}{\rho dV}$$

Newton's second law
For a system moving relative to a inertial reference frame, the sum of all external forces acting on the system is equal to the time rate of change of linear momentum ($$ \displaystyle \vec{P}$$) of the system:

$$ \displaystyle F_{i} = \frac{dP_{i}}{dt}\bigg|_{system}$$

$$ \displaystyle \displaystyle P_{i(system)} = \int_{mass(system)}{U_{i}dm} = \int_{V(system)}{U_{i}\rho dV}$$

The first law of Thermodynamics
$$ \displaystyle \underbrace{dQ}_{heat\ added\ on\ system} + \underbrace{dW}_{work\ done\ on\ system} = dE$$

in the rate form:

$$ \displaystyle \dot{Q} + \dot{W} = \frac{dE}{dt}\bigg|_{system}$$

where $$ \displaystyle \displaystyle E_{system} = \int_{mass(system)}{e dm} = \int_{V(system)}{e\rho dV}$$

and  $$ \displaystyle \displaystyle e = \underbrace{u}_{Intenral \ energy} + \underbrace{\frac{U_{i}U_{i}}{2}}_{Kinetic\ energy} + \underbrace{gz}_{Potential\ energy}$$

There are also other laws like the conservation of moment of momentum (angular momentum) and second law of thermodynamics, but they are not the subject of this course and will not be treated here.

Note that all basic laws are written for a system, i.e defined mass with fixed identity. We should rephrase these laws for a control volume.

One dimensional Reynolds Transport Theorem
Consider a flow through a nozzle.

If $$ \displaystyle B $$ is an extensive variable of the system.

$$ \displaystyle B_{sys}(t) = B_{cv}(t)$$

$$ \displaystyle \displaystyle B_{sys}(t + \Delta t) = B_{cv}(t + \Delta t) - B_{I}(t + \Delta t) +

B_{II}(t + \Delta t)$$

$$ \displaystyle \frac{\Delta B_{sys}}{\Delta t} = \frac{B_{sys}(t + \Delta t) - B_{sys}(t)}{\Delta t} = \frac{B_{cv}(t + \Delta t) - B_{cv}(t)}{\Delta t}\ -\ \frac{B_{I}(t + \Delta t)}{\Delta t}\ +\ \frac{B_{II}(t + \Delta t)}{\Delta t}$$

The first term for $$ \displaystyle \Delta t \rightarrow 0$$

$$ \displaystyle \lim_{\Delta t \rightarrow 0} \frac{B_{cv}(t + \Delta t) - B_{cv}(t)}{\Delta t} = \frac{\partial B_{cv}}{\partial t}$$

$$ \displaystyle B_{II}(t + \Delta t)$$ for $$ \displaystyle \Delta t \rightarrow 0$$

$$ \displaystyle B_{II}(t + \Delta t) = \rho_{2}b_{2}\Delta V_{2}$$

$$ \displaystyle \displaystyle B_{II}(t + \Delta t) = \rho_{2}b_{2}A_{2}l_{2} = \rho_{2}b_{2}A_{2}U_{2}\Delta t$$

$$ \displaystyle \displaystyle B_{out} = \rho_{2}b_{2}A_{2}U_{2}\Delta t$$

Similarly

$$ \displaystyle \displaystyle B_{I}(t + \Delta t) = \rho_{1}b_{1}\Delta V_{1} = \rho_{1}b_{1}A_{1}U_{1}\Delta t = B_{in}$$

Thus, for $$ \displaystyle \Delta t \rightarrow 0$$, the terms in the equality for the time derivative of the system are

$$ \displaystyle \frac{\Delta B_{sys}}{\Delta t} \rightarrow \frac{dB_{sys}}{dt}$$

$$ \displaystyle \frac{B_{in}}{\Delta t}=\dot{B}_{in}$$

$$ \displaystyle \frac{B_{out}}{\Delta t}=\dot{B}_{out}$$

so that, $$ \displaystyle \frac{dB_{sys}}{dt} = \frac{\partial B_{cv}}{\partial t} + \dot{B}_{out} - \dot{B}_{in}$$

This is the equation of 1 dimensional Reynolds transport theorem (RTT).

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The three terms on the RHS of RTT are:

1. The rate of change of B within CV indicates the local unsteady effect.

2. The flux of B passing out of the CS.

3. The flux of B passing into the CS.

There can be more than one inlet and outlet.

Three Dimensional Reynolds Transport Theorem
Hence, for a quite complex, unsteady, three dimensional situation, we need a more general form of RTT. Consider an arbitrary 3-D CV and the outward unit normal vector $$ (\vec{n}) $$ defined at each point on the CS. The outflow and inflow flux of $$ B$$ across CS can be written as:

$$ \displaystyle \dot{B}_{out} = \int_{CSout}{dB_{out}} = \int_{CSout}{\rho b \vec{U}\cdot\vec{n}dA}$$

$$ \displaystyle \dot{B}_{in} = \int_{CSin}{dB_{in}} = -\int_{CSin}{\rho b \vec{U}\cdot\vec{n}dA}$$

$$ \displaystyle \dot{B}_{in}$$ and $$ \displaystyle \dot{B}_{out}$$ are positive quantities. Therefore, the negative sign is introduced into $$ \displaystyle \dot{B}_{in}$$, to compensate the negative value of $$ \displaystyle\vec{U}\cdot\vec{n}$$.

$$ \displaystyle \frac{dB_{sys}}{dt} = \frac{\partial B_{cv}}{\partial t} + \int_{CSout}{\rho b \vec{U}\cdot\vec{n}dA} + \int_{CSin}{\rho b \vec{U}\cdot\vec{n}dA}$$

$$ \displaystyle \frac{dB_{sys}}{dt} = \frac{\partial}{\partial t}\int_{cv}{\rho b dV} + \int_{cs}{\rho b \vec{U}\cdot\vec{n}dA}$$

Since $$ \displaystyle \rho \vec{u}\cdot \vec{n} dA = dm$$, RTT can be written as:

$$ \displaystyle \frac{dB_{sys}}{dt} = \frac{\partial B_{cv}}{\partial t} + \underbrace{\int_{CS}{bdm}}_{net\ flux\ of\ B\ across\ CS}$$

It is possible that CV can move with constant velocity or arbitrary acceleration.

This form of RTT is valid if the CV has no acceleration with respect to a fixed (inertial) reference frame. RTT is then valid for a moving CV with constant velocity when:

1. All velocities are measured relative to the CV.

2. All time derivative measure relative to the CV.

Thus for a CV moving with $$ \displaystyle \vec{U}_{s}$$

$$ \displaystyle \vec{U}_{r} = \vec{U} - \vec{U}_{s}$$

$$ \displaystyle \frac{dB_{sys}}{dt} = \frac{d}{dt}\left(\int_{cv}{b \rho dV}\right) + \int_{cs}{b \rho \vec{U_{r}}\cdot \vec{n}dA}$$

These issues will be covered again.


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Conservation of mass
$$ \displaystyle B = M, b = 1$$

$$ \displaystyle \frac{dM_{sys}}{dt} = \frac{\partial}{\partial t}\int_{cv}{b\rho dV} + \int_{cs}{b\rho \vec{U}\cdot\vec{n} dA} = 0$$

i.e

$$ \displaystyle \underbrace{\frac{\partial}{\partial t}\int_{cv}{\rho dV}}_{rate\ of \ change\ of\ mass\ in \ CV} + \underbrace{\int_{cs}{\rho \vec{U}\cdot \vec{n} dA}}_{net\ rate\ of\ mass\ flux\ through\ the\ CS} = 0$$

Assume $$ \displaystyle \rho$$ = constant (incompressible)

$$ \displaystyle 0 = \rho\frac{\partial}{\partial t} \int_{cv}{dV} + \rho\int_{cs}{\vec{U}\cdot\vec{n}dA}$$

As $$ \displaystyle V $$ of CV is also constant, the time derivative drops out:

$$ \displaystyle 0 = \rho \int_{cs}{\vec{U}\cdot\vec{n}dA}\rightarrow\underbrace{\int_{cs}{\vec{U}\cdot\vec{n}dA}}_{volume\ flow\ rate} = 0$$

The net volume flow rate should be zero through the control surfaces.

Note that we did not assume a steady flow. This equation is valid for both steady and unsteady flows.

If the flow is steady,

$$ \displaystyle 0 = \int_{cs}{\rho \vec{U}\cdot\vec{n}dA} \rightarrow \text{Net mass  flow rate is equal to zero} $$

there is no-mass accumulation or deficit in the control volume.

Linear Momentum equation for inertial control volume
$$ \displaystyle B = \vec{P}$$, $$ \displaystyle b  =  \vec{U}$$

$$ \displaystyle \frac{d\vec{P}}{dt}_{system} = \frac{\partial}{\partial t}\int_{cv}{\vec{U}\rho dV} + \underbrace{\int_{cs}{\vec{U}\rho \vec{U}\cdot\vec{n}dA}}_{momentum\ flux}$$

$$ \displaystyle \frac{d\vec{P}}{dt}_{system} = \vec{F}_{on\ system}$$

This equation states that the sum of all forces acting on a non-accelerating CV is equal to the sum of the net rate of change of momentum inside the CV and the net rate of momentum flux through the CS.

Force on the system is the sum of surface forces and body forces.

$$ \displaystyle \vec{F}_{on\ system} = \vec{F_{S}} + \vec{F_{B}}$$

The surface forces are mainly due to pressure, which is normal to the surface, and viscous stresses, which can be both normal or tangential to the surfaces.

$$ \displaystyle \vec{F}_{pressure} = -\int_{A}{P\vec{n}dA}$$

$$ \displaystyle \vec{F}_{v.stresses} = \int_{A}{\tau dA}$$

The body forces can be due to gravity or magnetic field.

at the initial moment $$ \displaystyle t$$

$$ \displaystyle \vec{F}_{on\ system} = \vec{F}_{on\ CV}$$

i.e. in component form:

$$ \displaystyle F_{Si} + F_{Bi} = \frac{\partial}{\partial t}\int_{cv}{U_{i}\rho dV} + \int_{cs}{U_{i}\rho U_{j}n_{j} dA}$$

First law of Thermodynamics
$$ \displaystyle B = E, b = e$$

$$ \displaystyle \frac{dE}{dt}_{system} = \frac{\partial}{\partial t}\int_{cv}{e\rho dV} + \int_{cs}{e\rho \vec{U}\cdot\vec{n}dA}$$

at the initial moment $$ \displaystyle t $$, the following equality is valid:

$$ \displaystyle \frac{dE}{dt}_{system} = [\dot{Q} + \dot{W}]_{system} = [\dot{Q} + \dot{W}]_{cv}$$

thus, the integral form of energy equation is:

$$ \displaystyle [\dot{Q} + \dot{W}]_{cv}= \frac{\partial}{\partial t}\int_{cv}{e\rho dV} + \int_{cs}{e\rho \vec{U}\cdot\vec{n}dA}$$

Examples
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Example 1
Consider the mass balance in a stream tube by using the integral form of the conservatin of mass equation.

Let $$ \displaystyle A_{1}$$ and $$ \displaystyle A_{2}$$ be too small such that the velocities $$ \displaystyle \vec{U}$$ at position 1 and 2 are uniform across $$ \displaystyle A_{1}$$ and $$ \displaystyle A_{2}$$.

$$ \displaystyle \frac{\partial}{\partial t}\int_{cv}{\rho dV} + \int_{cs}{\rho \vec{U}\cdot\vec{n}dA} = 0$$

The first term is zero and the second term can be analyzed by decomposing the integration area.

$$ \displaystyle \int_{CS_I}{\rho \vec{U}\cdot\vec{n}dA} + \int_{CS_{III}}{\rho \vec{U}\cdot\vec{n}} dA + \int_{CS_{II}}{\rho\vec{U}\cdot\vec{n}dA} = 0$$

where the integration over $$ \displaystyle CS_{III}$$ is zero, because there is no flow across the streamtube. Thus,

$$ \displaystyle \displaystyle -\rho U_{1}A_{1} + 0 + \rho U_{2}A_{2} = 0$$

$$ \displaystyle \rho U_{1}A_{1} = \rho U_{2}A_{2} \rightarrow U_{2} = \frac{U_{1}A_{1}}{A_{2}}$$

$$ \displaystyle \dot{m}_{1} = \dot{m}_{2}$$


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Example 4


Water from a stationary nozzle strikes to a plate. Assume that the flow is normal to the plate and in the jet velocity is uniform. Determine the force on the plate in $$ \displaystyle x_{1}$$ direction.

Independent from the selected CV.

$$ \displaystyle 0 = \int_{cs}{\rho U_{i}n_{i}dA}\ (mass)$$

$$ \displaystyle F_{i} = F_{Si} + F_{Bi} = \int_{cs}{U_{i}\rho U_{j}n_{j}dA}\ (momentum)$$

No body force in $$ \displaystyle x_{1}$$ direction.

$$ \displaystyle \displaystyle F_{1} = F_{S1} = \int_{cs}{U_{1}\rho U_{j}n_{j}dA}$$

$$ \displaystyle \displaystyle F_{S1} = p_{a}A - p_{a}A + R_{1}$$

$$ \displaystyle R_{1} = \int_{cs}{U_{1}\rho U_{j}n_{j}dA} = \int_{CS1}{U_{1}^{1}\rho U_{j}^{1}n_{j}dA} + \underbrace{\int_{CS2}{U_{1}^{2}\rho U_{j}^{2}n_{j}dA}}_{0} + \underbrace{\int_{CS3}{U_{1}^{3}\rho U_{j}^{3}n_{j}dA}}_{0}$$

$$ \displaystyle U_{1} = 0$$ at 2 and 3.

$$ \displaystyle R_{1} = -U_{1}\rho U_{1}A_{Jet}$$

The force which acts on the plate (action-reaction) is $$ \displaystyle K_{1} = -R_{1} = U_{1}\rho U_{1}A_{Jet}$$

It is also possible to solve the problem with $$ \displaystyle CV_{2}$$

$$ \displaystyle F_{S1} = p_{a}A + R_{1} = - U_{1}^{2}A_{jet}\rho$$

$$ \displaystyle R_{1} = -p_{a}A - U_{1}^{2}A_{Jet}\rho$$

Hence, the force exerted on the plate by the CV is

$$ \displaystyle K_{1} = - R_{1}$$

$$ \displaystyle \displaystyle F_{net} = -R_{1} - p_{a}A$$

$$ \displaystyle F_{net} = p_{a}A + U_{1}^{2}A\rho - p_{a}A$$

$$ \displaystyle F_{net} = U_{1}^{2}A\rho$$

Example 5
Consider the plate exposed to uniform velocity. The flow is steady and incompressible. A boundary layer builds up on the plate. Determine the Drag force on the plate. Note that $$ \displaystyle U_{1}$$ can be approximated at L.

$$ \displaystyle U_{1}(L, x_{2}) = U\left(2\frac{x_{2}}{\delta} - \left(\frac{x_{2}}{\delta}\right)^{2}\right)$$

Apply conservation of mass

$$ \displaystyle \displaystyle \rho\int_{cs}{U_{i}n_{i}dA} = 0$$

$$ \displaystyle \rho\int_{0}^{h}{-U_{0}\ w\ dx_{2}} + \rho\int_{0}^{\delta}{U\ w\ dx_{2}} = 0$$

$$ \displaystyle U_{0}h = \int_{0}^{\delta}{Udx_{2}}$$

$$ \displaystyle F_{i} = \underbrace{\frac{\partial}{\partial t}\int{U_{i}\rho dV}}_{=0\ steady\ state} + \int_{cs}{U_{i}\rho U_{j}n_{j}dA}$$

$$ \displaystyle F_{1} = -D = \int_{0}^{h}{U_{1}\rho U_{j}n_{j}dA} + \underbrace{\int_{CS2}{U_{1}\rho U_{j}n_{j}dA}}_{= 0\ streamline} + \int_{0}^{\delta}{U_{1}\rho U_{j}n_{j}dA} + \underbrace{\int{U_{1}\rho U_{j}n_{j}dA}}_{= 0\ wall}$$

$$ \displaystyle -D = -U_{0}\rho U_{0}\ h\ w + \int{U_{1}\rho U_{1}\ w\ dx_{2}}$$

$$ \displaystyle D = U_{0}^{2}\rho \ h\ w - \rho\ w\int_{0}^{\delta}{U_{1}^{2}dx_{2}}= \rho U_{0}\int_{0}^{\delta}{U_{1}dx_{2}}-\rho\ w\int_{0}^{\delta}{U_{0}U_{1}dx_{2}}$$

Insert the mass conservation result into the momentum equation.

$$ \displaystyle D = \rho w \int_{0}^{\delta}{U_{1}(U_{0} - U_{1})dx_{2}}|_{x_{1} = L}$$

$$ \displaystyle U_{1}$$ is known.Here,using $$ \displaystyle \frac{x_{2}}{\delta} = \eta$$

$$ \displaystyle D = \rho\ w\ U_{0}^{2}\delta \int_{0}^{1}{(2\eta - \eta^{2})(1 - 2\eta + \eta^{2})d\eta} = \frac{2}{15}\rho U_{0}^{2}w\ \delta$$

Momentum Equation for CV with rectilinear acceleration
For an inertial CV the following transport equation for momentum holds:

$$ \displaystyle \vec{F} = \frac{\partial}{\partial t}\int_{cv}{\vec{U}\rho\ dV} + \int_{cs}{\vec{U}\rho\vec{U}\cdot\vec{n}dA} $$

However, not all CV are inertial: for example a rocket must accelerate if it is to get off the ground.

Denote an inertial reference frame with $$ \displaystyle X_{1},X_{2},X_{3}$$ and another reference frame moving with the system $$ \displaystyle x_{1},x_{2},x_{3}$$. Hence, $$ \displaystyle x_{1},x_{2},x_{3}$$ becomes the non-inertial frame of reference. Let the system to move with a velocity and an acceleration $$ \displaystyle \vec{U}_{rf}$$ and $$ \displaystyle \vec{a}_{rf}$$ respectively. Here $$\displaystyle \overrightarrow U_{X}= \overrightarrow U_{x}+\overrightarrow U_{rf}$$ and accordingly with time derivative dt, $$\displaystyle \frac{d\overrightarrow U_{X}}{dt}= \frac{d\overrightarrow U_{x}}{dt}+\frac{d\overrightarrow U_{rf}}{dt} \Rightarrow \frac{d\overrightarrow U_{X}}{dt} \ne \frac{d\overrightarrow U_{x}}{dt}$$ or,$$ \displaystyle \vec{a}_{X} = \vec{a}_{x} + \vec{a}_{rf}\Rightarrow \frac{d\overrightarrow P_{X}}{dt} \ne \frac{d\overrightarrow P_{x}}{dt}$$ These above relationship implies that the velocity and the acceleration is not same when considered from inertial and moving reference frame. The Newton's second law states that:

$$ \displaystyle \left.\vec{F}\right|_{system} = \left.\frac{d\vec P_X}{dt}\right|_{system} $$



Thus the following relation holds for the fluid velocity in the system

$$ \displaystyle \vec{U}_{x} = \vec{U}_{X} - \vec{U}_{rf}$$

Where $$ \displaystyle \vec{U}_{x}$$ is the velocity of the fluid in the system with respect to a non-inertial reference frame, $$ \displaystyle \vec{U}_{X}$$ is the velocity of the fluid in the system with respect to the inertial reference frame. Accordingly, the acceleration reads:

$$ \displaystyle \frac{d\vec{U}_{x}}{dt} = \frac{d\vec{U}_{X}}{dt} - \frac{d\vec{U}_{rf}}{dt} (1) $$

$$ \displaystyle \vec{a}_{x} = \vec{a}_{X} - \vec{a}_{rf}$$

For a control volume moving with $$ \displaystyle \vec{U}_{rf}$$ and $$ \displaystyle \vec{a}_{rf}$$

$$ \displaystyle \frac{d\vec{U}_{CV}}{dt}=\frac{d\vec{U}_{rf}}{dt}$$

Thus for cases, where $$ \displaystyle \frac{dU_{cv}}{dt} \neq 0$$, the time derivative of $$ \displaystyle \vec{P}_{X}$$ and $$ \displaystyle \vec{P}_{x}$$ are not equal for a system accelerating relative to an inertial reference frame, i.e. RTT is not valid for an accelerating control volume.

To develop momentum equation for an accelerating CV, it is necessary to relate $$ \displaystyle \vec{P}_{X}$$ to $$ \displaystyle \vec{P}_{x}$$.

Previously we have seen that in a non-inertial reference frame having rectilinear acceleration, i.e. (translational acceleration).

$$ \displaystyle \vec{a}_{X} = \vec{a}_{x} + \vec{a}_{rf}$$ and also

$$ \displaystyle \vec{F} = \int_{mass\ (system)}{ \vec a_{X}dm}$$

$$ \displaystyle =\int_{mass\ (system)}{(\vec{a}_{x} + \vec{a}_{rf})dm}$$

$$ \displaystyle \vec{F} - \int_{mass\ (system)}{\vec{a}_{rf}dm} = \int_{mass\ (system)}{\vec{a}_{x}dm}$$

$$ \displaystyle \vec{F} - \int_{mass\ (system)}{\vec{a}_{rf}dm} = \int_{mass\ (system)}{\frac{d\vec{U}_{x}}{dt}dm} = \frac{d}{dt}\int_{mass\ (system)}{\vec{U}_{x}dm}$$

$$ \displaystyle \vec{F} - \int_{V\ (system)}{\vec{a}_{rf}\rho\ dV} = \frac{d\vec{P}_{x}}{dt}|_{system}$$

For a moving CV we know that

$$ \displaystyle \frac{d\vec{P}_{x}}{dt}|_{system} = \frac{\partial}{\partial t}\int_{cv}{\vec{U}_{x}\rho\ dV} + \int_{cs}{\vec{U}_{x}\rho\vec{U}_{x}\cdot\vec{n}dA}$$

Let system and CV coincides at an instant $$ \displaystyle t_0$$:

$$ \displaystyle \vec{F}_{on\ system} -\int_{V\ system}{\vec{a}_{rf}\rho dV} = \vec{F}_{on\ cv} - \int_{cv}{\vec{a}_{rf}\rho dV}$$

$$ \displaystyle \vec{F}_{on\ cv} - \int_{cv}{\vec{a}_{rf}\rho dV} = \frac{\partial}{\partial t}\int_{cv}{\vec{U}_{x}\rho\ dV} + \int_{cs}{\vec{U}_{x}\rho \vec{U}_{x}\cdot \vec{n}dA}$$

Extension of Energy equation for CV
$$ \displaystyle \left[\dot{Q} + \dot{W}\right]_{on\ system} = \left[\dot{Q} + \dot{W}\right]_{on\ cv} = \frac{\partial}{\partial t}\int{e\rho\ dV} + \int_{cs}{e\rho \vec{U}\cdot\vec{n}dA} $$

where $$ \displaystyle e = U + \frac{\left|\vec{U}\right|^{2}}{2} + g\ z$$

$$ \displaystyle \dot{W}_{on\ cv} = \dot{W}_{body} + \dot{W}_{surface}$$

here $$ \displaystyle z = x_{2}$$

$$ \displaystyle \dot{W}_{body} = \dot{W}_{shaft} + \dot{W}_{elec} + \dot{W}_{other} $$

$$ \displaystyle \dot{W}_{surface} = \dot{W}_{normal} + \dot{W}_{shear} $$

$$ \displaystyle \vec{\tau}_{s}$$ is the stress in the plane of dA.

Special form of the Energy equation
For a CV with one inlet 1, one outlet 2 and steady uniform flow through it.

$$ \displaystyle \underbrace{\dot{Q} + \dot{W}_{shaft} + \dot{W}_{shear}}_{K} = \int_{cs}{\left(u + \frac{p}{\rho} + \frac{\left|\vec{U}^{2}\right|}{2} + gz\right)\rho\vec{U}\cdot\vec{n}dA}$$

For uniform flow properties at the inlet and outlet.

$$ \displaystyle K = \left(u_{1} + \frac{p_{1}}{\rho_{1}} + \frac{\left|\vec{U}_{1}^{2}\right|}{2} + gz_{1}\right)\underbrace{\int_{1}{\rho_{1}\vec{U}_{1}}\cdot\vec{n}dA}_{-\dot{m}} + \left(u_{2} + \frac{p_{2}}{\rho_{2}} + \frac{\left|\vec{U}_{2}^{2}\right|}{2} + gz_{2}\right)\underbrace{\int_{2}{\rho_{2}\vec{U}_{2}}\cdot\vec{n}dA}_{\dot{m}}$$

$$ \displaystyle K = \dot{m}\left[(u_{2} - u_{1}) + \left(\frac{p_{2}}{\rho_{2}} - \frac{p_{1}}{\rho_{1}}\right) + \left(\frac{\vec{U}_{2}^{2}}{2} - \frac{\vec{U}_{1}^{2}}{2}\right) + g(z_{2} - z_{1})\right]$$

Reform:

$$ \displaystyle \frac{p_{1}}{\rho_{1}} + \frac{\left|\vec{U}_{1}^{2}\right|}{2} + gz_{1} = \frac{p_{2}}{\rho_{2}} + \frac{\left|\vec{U}_{2}^{2}\right|}{2} + gz_{2} + (u_{2} - u_{1}) - \frac{\dot{Q}}{\dot{m}} - \frac{\dot{W}_{shaft}}{\dot{m}} - \frac{\dot{W}_{shear}}{\dot{m}}$$

For $$ \displaystyle \dot{W}_{shaft} = 0$$, $$ \displaystyle \dot{W}_{shear} = 0$$ and incompressible flow:

$$ \displaystyle \underbrace{\frac{p_{1}}{\rho} + \frac{\left|\vec{U}_{1}^{2}\right|}{2} + gz_{1}}_{mechanical\ evergy\ per\ unit\ mass\ at\ flow\ cross\ section} = \frac{p_{2}}{\rho} + \frac{\left|\vec{U}_{2}^{2}\right|}{2} + gz_{2} + (u_{2} - u_{1}) - \frac{\dot{Q}}{\dot{m}}$$

$$ \displaystyle \frac{p}{\rho} + \frac{U^{2}}{2} + gz$$: Mechanical energy per unit mass.

$$ \displaystyle u_{2} - u_{1} - \frac{\dot{Q}}{\dot{m}}$$: Irreversible conversion of mechanical energy to unwanted thermal energy $$ \displaystyle (u_{2} - u_{1})$$ and loss of energy via heat transfer $$ \displaystyle \left(-\frac{\dot{Q}}{\dot{m}}\right)$$.

Thus with this equation I can calculate the loss of energy through a device.

$$ \displaystyle h_{loss} = \left[u_{2} - u_{1} -\frac{\dot{Q}}{\dot{m}}\right]\frac{1}{g}$$

i.e.

$$ \displaystyle \frac{p_{1}}{\rho g} + \frac{U_{1}^{2}}{2g} + z_{1} = \frac{p_{2}}{\rho g} + \frac{U_{2}^{2}}{2g} + z_{2} + h_{loss}$$

One can add the work done by a pump or a turbine.

$$ \displaystyle \frac{p_{1}}{\rho g} + \frac{U_{1}^{2}}{2g} + z_{1} = \frac{p_{2}}{\rho g} + \frac{U_{2}^{2}}{2g} + z_{2} + h_{loss} - h_{pump} + h_{turbine}$$

Differential Control Volume Analysis:Bernoulli Equation
Integrate between 1 and 2 along a streamline:

$$ \displaystyle \frac{p_{1}}{\rho} + \frac{U_{1}^{2}}{2} + gz_{1} = \frac{p_{2}}{\rho} + \frac{U_{2}^{2}}{2} + gz_{2} = constant$$

Bernoulli equation is clearly related to the steady flow energy equation for a stream line. This form of the Bernoulli equation, when the following conditions are satisfied:

1. Steady flow. Note that theres is also an unsteady Bernoulli equation.

2. Incompressible flow. For example, in aerodynamics, flow can be accepted to be incompressible for Mach number $$ \displaystyle (M =\frac{\text{speed of flow}}{\text{speed of sound}}) $$ less than 0.3.

3. Frictionless flow, e.g. in the absence of solid walls.

4. Flow along a single streamline. Different streamline has a different constant.

5. No shaft work between 1 and 2.

6. No heat transfer between 1 and 2.

Example 3
 Find a relationship between the flow rate and the pressure difference inside a pipe which could be measured by venturimeter