Steady Streamlines

For a two-dimensional velocity field in the form

\vec u(x,y) = u(x,y)\hat i + v(x,y)\hat j

the streamlines are:

    \[\int \frac{1}{v(x,y)}dy = \int \frac{1}{u(x,y)}dx\]

Unsteady Streamlines & Pathlines

For a two-dimensional unsteady field in the form

\vec u(x,y,t) = u(x,y,t)\hat i + v(x,y,t)\hat j

the parametrised pathlines are:

    \[x = \int u(x,y,t) dt\]

    \[y = \int v(x,y,t) dt\]

Forces in Fluids

Shear stress, \tau:

    \[\tau = \frac{F}{A}\]

    \[\tau = \mu \frac{du}{dy}\]

Pressure force, F_P:

    \[F_P = PA\]

Viscous Force, F_v:

    \[F_v = \tau A\]

    \[F_v = \mu \frac{du}{dy}A\]

Kinematic viscosity, v:

    \[v = \frac{\mu}{\rho}\]

Fluid Statics

Hydrostatic Equation:

    \[\frac{dP}{dz}=-\rho g\]

    \[\Delta P = -\rho g \Delta z\]

Archimedes’ Principle:

The magnitude of upthrust is equal to the weight of water displaced

Resultant Pressure Force, F_R:

    \[F_R = \int_A P dA\]

    \[F_R = \int_A \rho g y dA\]

Point of application, y':

    \[y'=\frac{\int_A Py dA}{\int_A P dA}\]

    \[y' = \frac{\int y^2 dy}{\int y dy}\]


    \[P_2-P_1 = gl(\rho_a-\rho_b)\]

Mass Flow Rate

Normal vectors, perpendicular component of velocity

    \[\dot m = \int_A \rho \vec u.\overrightarrow{dA}\]

When velocity is perpenduclar:

    \[\dot m = \rho uA\]

Reynolds Transport Theorem

    \[\frac{d}{dt} \int_{V_{sys(t)}} \eta \rho dV = \frac{d}{dt} \int_{CV} \eta \rho dV + \int_{CS} \eta \rho \vec u. \overrightarrow{dA}\]

  • The first term is the rate of change of N in the system
  • The second term is the rate of change of N in the control volume and system
  • The third term is the net flow rate of N out of the control volume

Where N is the property being conserved and N=\eta m. \rho\eta is the property of N per unit volume.

Conservation of Mass

    \[\frac{d}{dt} \int_{CV}\rho dV = -\int_{CS} \rho \vec u,\overrightarrow{dA}\]

Algebraic formulation:

\frac{dm}{dt}=\sum \dot m_{in} - \sum \dot m_{out}

Steady, uniform, incompressible flow:

\sum_{in} uA = \sum_{out} uA

Steady Flow:

\sum \dot m_{in} = \sum \dot m_{out}

\int_{CS} \rho \vec u.\overrightarrow{dA} =0

Conservation of Momentum

    \[\sum \vec F = \frac{d}{dt} \int_{CV} \vec u \rho dV + \int_{CS}\vec u \rho \vec u.\overrightarrow{dA}\]

For steady flow:

\sum \vec F = \int_{CS}\vec u \rho \vec u.\overrightarrow{dA}

For steady flow, resolved into components:

\sum F_x =  \int_{CS} u_x \rho \vec u. \overrightarrow{dA}

\sum F_y =  \int_{CS} v_y \rho \vec u. \overrightarrow{dA}

For steady, uniform, incompressible flow:

\sum F_x = \sum_{out} \dot mu - \sum_{in} \dot mu

\sum F_y = \sum_{out} \dot mv - \sum_{in} \dot mv

When mass is conserved for one inlet and one outlet:

\sum F_x = \dot m(u_{out} - u_{in})_x

\sum F_y = \dot m(u_{out}-u_{in})_y

The Bernoulli Equation


The four conditions for the Bernoulli Equation:

  • Steady flow
  • Inviscid flow
  • Incompressible flow
  • The two points are on the same streamline or have the same Bernoulli constant, c

Stagnation point flow:

Stagnation point flow, the Bernoulli Equation


Laminar Flow between Horizontal Plates

    \[u(y) = -\frac{h^2\Delta P}{2\mu L}(1-\frac{y^2}{h^2}\]

Where u_{max} =  \frac{h^2\Delta P}{2\mu L}

  • This is known as the Pouiselliu Law

Laminar Flow in a Circular Pipe

    \[u(r) = -\frac{R^2\Delta P}{4\mu L}(1-\frac{r^2}{R^2}\]

Where u_{max} = \frac{R^2\Delta P}{4\mu L}

  • This is known as the Hagen-Poiseuille Law

Conservation of Energy for Steady Flow

    \[\dot Q - \dot W = \int_{CS}(\frac{p}{\rho}+\frac{1}{2}u^2+gz+e)\rho\vec u.\overrightarrow{dA}\]

For a uniform velocity profile:


First law of thermodynamics:

    \[\frac{dE_{sys}}{dt}=\dot Q - \dot Q\]

Reynolds substitution:


The Pipe Flow Energy Equation (PFEE)

    \[\frac{P_1}{\rho}+\frac{1}{2}u_1^2+gz_1= \frac{P_2}{\rho}+\frac{1}{2}u_2^2+gz_2+w_L-w_P\]

  • w_L is the lost energy
  • w_P = w, the pump work

    \[ \frac{P_1}{\rho g}+\frac{1}{2g}u_1^2+z_1= \frac{P_2}{\rho g}+\frac{1}{2g}u_g^2+z_1+h_L-h_P\]

  • h_L is the lost head
  • h_P is the pump head

The pipe flow energy equation only applies to flow that is:

  • Steady
  • Adiabatic
  • Incompressible
  • Uniform velocity field
  • Between a single inlet and outlet

Turbulent Flow in Circular Pipes

Reynolds Number, Re:

    \[Re=\frac{\rho ud}{\mu}=\frac{ud}{v}\]

Mean velocity, u:


Friction factor, f:


Relative roughness, r:


Lost Head

Lost head, h_L:

    \[h_L = h_f+h_l\]

Major losses, h_f:


Minor losses, h_l:


Pump Head

Pump head, h_P:


    \[h_P=\frac{\dot W_P}{\dot mg}\]

    \[h_P = \frac{\Delta P_P}{\rho g}\]

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