Bernoulli's Theorem

In this book we look at an alternative way of deriving the governing equations of fluid flow using conservation of energy techniques on a differential element undergoing shear stress or viscous forces as it moves along a pipe and we use the expression for friction coefficient for laminar flow to derive the equations.We also derive other friction factors to explain observations. We also derive the equations that work for Torricelli flow and there conditions. We derive the turbulent flow equations too. We derive the general equation for all regimes laminar, transition and turbulent flow

P6.1 An engineer claims that flow of SAE 30W oil, at 20°C, through a 5-cm-diameter smooth pipe at 1 million N/h, is laminar. Do you agree? A million newtons is a lot, so this sounds like an awfully high flow rate. Solution: For SAE 30W oil at 20°C (Table A.3), take ρ = 891 kg/m 3 and μ = 0.29 kg/m-s. Convert the weight flow rate to volume flow rate in SI units:) D 2500 ≈ − = = = = = = = μ ρ π ρ

We are to define a fluid and how it differs between a solid and a gas.

We study the exact solutions for fully developed channel flow of model fluids obeying two typical constitutive equations: the Phan-Thien/Tanner and the Giesekus models. Particular attention is paid to the profiles of stress components across the channel and a peculiar behaviour is observed for the normal stresses: as the Deborah number increases up to unity, the level of stresses also increases, but for higher De the trend is reversed. We find that if stresses are scaled with the wall shear stress then the variation becomes monotonic.

In this book we look at deriving the governing equations of fluid flow using conservation of energy techniques on a differential element undergoing shear stress or viscous forces as it moves along a pipe and we use the expression for friction coefficient for laminar flow to derive the equations. We also derive a friction coefficient to work for Torricelli flow. We look at laminar, and turbulent flow. We look at cases where there is a pipe on a tank or an orifice and we develop the governing equations. We then develop a universal formula or equation that works for all types of flow i.e., laminar, transition and turbulent flow in one equation. We go ahead and demonstrate Pouiselle flow and the conditions under which it will be observed. We explain other phenomena too. [email protected]

Acta Mech, 2004

The effects of the curvature of the side walls on the flow of an incompressible viscous fluid in a duct of uniform cross-section due to the motion of the top wall and an imposed pressure gradient along the duct are considered. In order to show the effect of the wall curvature, two illustrative examples are given. One of them is the flow in a duct of semicircular cross-section and the other is the flow in a duct of rectangular cross-section. The results obtained for the flow in a duct of semicircular cross-section for zero flux and for separation at the bottom wall are compared with those of the flow in a duct of rectangular cross-section. It is a remarkable effect of the wall curvature on the flow in a duct that the separation at the bottom wall for the flow in a rectangular duct starts earlier than that in a semicircular duct.

American Journal of Physics, 2003

The way of teaching the basic principles of physics to first year engineering students is a matter of great interest to educationalists. These first year students already have a solid preparation in calculus, which means they can easily follow relatively complex mathematical interpretations; however, they are only just becoming familiar with differential equations in the calculus course corresponding to the first year of their degree. Differential equations are present in every branch of physics, so avoiding differential equations in a rigorous formulation of physics is not realistic. Moreover, the association of a differential equation with a particular principle of physics places this differential equation and its solution in a context, thereby providing the student with a background which could help him in future situations. Nevertheless, a non-rigorous

Journal of the American Helicopter Society

Several rotorcraft applications such as circulation control and tip jet-driven rotors involve internal spanwise flow along a ducted rotor blade. The primary goal of this work was to study a self-pumping pneumatically driven duct flow by both generating a quasi one-dimensional model for such flows and providing a validation data set for rotorcraft applications. The flow behavior inside a 1.32-m-long cylindrical duct, with a duct cross-sectional diameter of 52 mm, and rotating at speeds up to 1050 RPM was studied. Spanwise pressure distribution, duct velocity, hub forces, and moments from the numerical model showed good correlation with experiments. A considerable internal mass flow rate (∼0.3 kg/s) was also observed for a steadily rotating duct. In the presence of a time-varying valve at the inlet, transient spanwise pressure variations showed periodic fluctuations in pressure that diminished once the valve was fully open. The experimental results were compared with results of two computational models-a quasi one-dimensional finite volume Euler equation solver and a full threedimensional computational fluid dynamics solver. The ability to model a range of boundary conditions, time-varying duct cross-sectional area to simulate a flow control valve, frictional losses, duct sweep, and centrifugal as well as Coriolis effects on the flow is included. The experiments revealed key information about pressure at the duct's outlet. It was observed that when the duct's inlet is closed, the duct's outlet pressure is less than its ambient value. The knowledge of these boundary conditions is key in modeling flow through rotating ducts.

Communications in Numerical Methods in Engineering, 1993

The integral transform method is used to compute numerically the development of the velocity profile in the hydrodynamic entrance region for laminar flow inside a parallel plate channel. The results have user-prescribed accuracy and can be used to test other numerical methods.

In order to complete this tutorial you should already have completed level 1 or have a good basic knowledge of fluid mechanics equivalent to the Engineering Council part 1 examination 103.

International Communications in Heat and Mass Transfer, 1998

In this paper, a theoretical study is conducted, calculating the temperature distn'bution in the cross-section of a rectangular duct, under the conditions of newtonian and incompressa'ble fluid, fully developed laminar flow and steady-state regime. The governing equations are solved resorting to the finite Fourier transfornl The temperature distn'butions are obtained. The results concerning the temperature dism'bution in a square duct are shown by tables and figures, and a comparison between the present solution and some literature conm'butions is also presented. The viscous dissipation is responsible for a power generation that, for a particular Brinkman number (Brq = 1/~* or Br~=~), allows the wall heat flux to vanish. At last, the effects of viscous dissipation and wall heat flux are presented in some graphs, as a function of the duct aspect ratio.

This paper presents the observations of an experimental work dealing with the measurement of normalized mean velocity and wall static pressure of a 90° C-shaped curved duct. The test duct is made up of transparent perspexsheet to facilitate the flow visualization study. The duct has a centerline distance of 750mm. The Inlet and Outlet aspect ratio of the test duct used is 1.0. Wall pressures are measured with the help of an inclined manometer withthe inclination of 35°. The manometer had two tubes emanating from it: one left open to the atmosphere and the other connected to the steel pipes attached to the four walls of the curved duct. The difference in the readings helped calculate the static pressure and thereby the normalized pressure. Wall pressure distribution along the curved and parallel walls of the duct at 0°, 22.5°, 45°, 67.5° and 90° measuring sections was measured. The investigation for wall static pressure distribution is carried out at the velocity of 40m/s. This paper also presents an experimental work carried out with measurement of normalized mean velocity of the mentioned curved square C shaped duct, taken at the same velocity. The trend of the normalized mean velocity contour development inside the C duct shows continuous decrease of the normalized velocity as we move from the Inlet Section towards the Outlet Section. The distribution of wall static pressure and normalized mean velocity contours are mapped by using SURFER software package. The trend of wall static pressure development on the walls of C shaped duct shows that as the flow proceeds towards the curvature, there exists a high pressure gradient between the Outside Face and Inside Face due the centrifugal force acting along the curvature. This shows the bulk shifting of flow towards the Inside Face. This is due to the generation of secondary motion in a plane perpendicular to the primary flow. The main purpose of this investigation is to show the development of secondary flow which happens when the flow takes place through the bend in the curvature. This secondary flow arises as a result of a centrifugal force acting when the flow moves through the bend.