Fluid Mechanics

Hagen-Poiseuille laminar pipe flow

Hagen-Poiseuille flow describes steady, fully developed, laminar flow of an incompressible Newtonian fluid in a circular pipe. The velocity profile is parabolic, u(r)=u_{max}\left(1-\frac{r^2}{R^2}\right), and the mean velocity is half of the centreline value, \bar V = u_{max}/2. The pressure drop over a length L at volumetric flow rate Q is \Delta P = \frac{128\mu L Q}{\pi D^4}, so the viscosity can be found from a measured \Delta P, Q, L and D, which is the basis of capillary viscometry.

At a fixed flow rate, the pressure drop is directly proportional to the viscosity and to the length, and it varies inversely with the fourth power of the diameter. Halving the diameter therefore raises the pressure drop sixteen-fold. A statement that \Delta P is independent of viscosity is always false for this flow.

The shear stress varies linearly across the radius, \tau(r)=\tau_w \frac{r}{R}, vanishing on the axis and reaching \tau_w at the wall. This result follows from a force balance on a cylindrical fluid element, so it holds for steady fully developed pipe flow of any fluid, not only Newtonian ones.

Mechanical energy balance for pump

For a steady flow of an incompressible fluid, the mechanical energy balance per unit mass is W_s = \frac{\Delta P}{\rho} + \frac{\Delta(\alpha \bar V^2)}{2} + g\Delta z + F, where W_s is the shaft work delivered to the fluid and F is the frictional loss. Without friction, with equal elevations at suction and discharge, the work reduces to \Delta P/\rho plus the change in kinetic energy per unit mass. The hydraulic power is \dot m W_s, and the shaft power is \dot m W_s/\eta, where \eta is the pump efficiency. Students often forget to divide by \eta when asked for the power drawn from the motor.

For a pump that moves liquid between two open reservoirs, the velocities at the free surfaces are negligible. The work per unit mass is then W_s = g\Delta z + F, and the power is \rho g Q H/\eta, where H = \Delta z + F/g is the total head that the pump must supply. Friction must be added to the static lift, and it is computed with the Darcy convention when the friction factor is given that way.

For an adiabatic pump, the work supplied is split into a reversible part v\Delta P, which is stored as pressure, and a dissipated part W_s - v\Delta P, which heats the liquid. The maximum temperature rise occurs when all of the dissipated work goes into the fluid, so \Delta T_{max} = \frac{W_s - v\Delta P}{c_p}. The flow rate does not change this specific temperature rise.

Continuity equation for incompressible flow

For an incompressible fluid the density is constant, and the continuity equation reduces to a zero-divergence condition, \nabla\cdot\vec V = \frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+\frac{\partial w}{\partial z}=0. A velocity field that makes the divergence a nonzero constant or a function of position is not admissible as an incompressible flow. The test is done by differentiating each component with respect to its own coordinate and summing the results.

The same condition is used to find a missing velocity component. If one component is known and the divergence is zero, the remaining component is obtained by integrating the relevant partial derivative. The integration constant is a function of the other coordinates and is fixed by a boundary condition, such as zero velocity at a wall. For example, if \partial u/\partial x = ay and \partial w/\partial z = 0, then \partial v/\partial y = -ay, giving v = -ay^2/2 + f(x,z) with f fixed by the boundary condition.

Frictional pressure drop in pipe flow

Two friction factor conventions are used. The Fanning factor f_F is defined by \tau_w = f_F \frac{1}{2}\rho \bar V^2, and the Darcy factor is f_D = 4f_F. In a fully developed horizontal pipe of constant diameter, a force balance between pressure and wall shear gives -\frac{dP}{dx} = \frac{4\tau_w}{D}, so the pressure falls linearly with distance. This linear region is the only one from which the friction factor should be found.

Near the pipe entrance the pressure falls non-linearly because the velocity profile is still developing. Measured pressures from the entrance region must be excluded when the fully developed slope is needed. The Darcy head loss in a straight pipe is h_f = f_D \frac{L}{D}\frac{\bar V^2}{2g}, or per unit mass F = f_D \frac{L}{D}\frac{\bar V^2}{2}. Minor losses are added separately with loss coefficients.

Shear stress distribution in turbulent pipe flow

In steady fully developed pipe flow the total shear stress is linear in r and equals zero on the axis. In turbulent flow the total stress is the sum of the viscous stress and the Reynolds stress -\rho\overline{u'v'}. The Reynolds stress is zero at the wall, where the no-slip condition damps the fluctuations, and it is zero at the centreline by symmetry. Between these two limits it rises to a maximum in the buffer region, so on moving from the wall toward the centre it first increases and then decreases.

The time-averaged Reynolds stress is not zero in the turbulent core, since it is what carries the momentum transfer. For a fully developed flow in a pipe of constant diameter, the mean pressure gradient along the flow direction is uniform. At high Reynolds number the mean velocity is about 0.8 times the centreline velocity, whereas the ratio of 0.5 applies only to laminar Poiseuille flow.

Venturi meter flow rate

A Venturi meter is a converging-diverging section in which the pressure drop between the upstream section and the throat is related to the flow rate. Applying Bernoulli's equation and continuity between the two sections gives Q = C_v A_t \sqrt{\frac{2\Delta P}{\rho(1-\beta^4)}}, where \beta = d_t/D is the diameter ratio and C_v is the discharge or Venturi coefficient. The factor 1-\beta^4 corrects for the approach velocity, and C_v accounts for friction and non-ideal profiles. Forgetting either correction gives a systematic error in the computed flow rate.

For a fixed geometry and a constant coefficient, Q is proportional to \sqrt{\Delta P}, so \Delta P \propto Q^2. If the pressure difference is read on a manometer, the reading is proportional to \Delta P for a given manometer fluid, since \Delta P = (\rho_m-\rho)g\Delta h. Doubling the flow rate therefore quadruples the manometer reading.

Torricelli tank draining

For a free surface open to the atmosphere, Torricelli's result gives the exit velocity through a small orifice as v = \sqrt{2gh}, where h is the height of the liquid surface above the orifice. If the surface is pressurized, the gauge pressure above the liquid adds to the head, and the result becomes v = \sqrt{2\left(\frac{\Delta P_{g}}{\rho}+g h\right)}, with h measured from the centreline of the nozzle. The surface kinetic energy is negligible when the tank is much larger than the outlet.

The draining rate follows from the mass balance A_T \frac{dh}{dt} = -a v, where A_T is the tank cross-section and a the outlet area, so \frac{dh}{dt} = -\frac{a}{A_T}v. The discharge coefficient of an orifice is C_d = C_c C_v, where C_c = A_{vc}/A_o is the contraction coefficient. For a circular orifice the contraction coefficient is the square of the ratio of the vena contracta diameter to the orifice diameter.

Plane Couette-Poiseuille flow between plates

In a channel of half-width H with the axis at y=0, a pressure gradient and the motion of one wall combine to produce a superposed velocity field. The pressure-driven part is parabolic, -\frac{1}{2\mu}\frac{dP}{dx}(H^2-y^2), and it vanishes at both walls. The wall-driven part is linear, and for a bottom wall moving at speed U and a fixed top wall it is U\frac{H-y}{2H}. The governing equation is \mu\frac{d^2u}{dy^2}=\frac{dP}{dx}, and the boundary conditions decide the signs of both terms.

Pressure falls in the flow direction, so dP/dx is negative for a pressure-driven flow in the +x direction. For a plane Poiseuille profile, the mean velocity over the gap is two-thirds of the centreline velocity, \bar u = \frac{2}{3}u_{max}. This relation is useful for converting a given velocity profile into its average value without integrating the full expression.

Kinetic energy correction factor in pipe flow

The kinetic energy correction factor is defined as \alpha = \frac{1}{A\bar V^3}\int v^3\,dA. It corrects the kinetic energy term in the energy balance, which uses the mean velocity, when the actual velocity profile is not uniform. For fully developed laminar Poiseuille flow \alpha = 2. A flatter turbulent profile gives a value much closer to unity.

For a power-law profile u/u_0 = (1-r/R)^{1/n}, the mean velocity satisfies \bar V/u_0 = \frac{2n^2}{(n+1)(2n+1)}, and the correction factor is \alpha = \left(\frac{u_0}{\bar V}\right)^3\frac{2n^2}{(n+3)(2n+3)}. Taking the ratio of the turbulent and laminar values is a common way of comparing the two regimes in exam problems.