Relationship between shear stress and pressure gradient

Shear stress - Wikipedia

The aim of this study was to evaluate the mechanical triggers that may cause plaque rupture. Wall shear stress (WSS) and pressure gradient are the direct. In a non-Newtonian fluid, the relation between the shear stress and the .. are in the form of flow rate and pressure gradient measured in a. If I compare between the flow conditions in laminar with turbulent, there is With increasing Reynolds number the wall shear stress in a laminar flow The fluid floe to the wall is dragged up the adverse pressure gradient by the eddy viscosity! .. regarding smooth and rough pipes and subsequently friction factor relations.

The scope of the invention is intended to include the signal processor being a stand alone component or module, as well as the signal processor forming part a combined SONAR-based meter and signal processing device.

Viscosity – The Physics Hypertextbook

The one or more modules may also be implemented as apparatus taking the form of a computer-readable storage medium having computer-executable components for performing the steps of the aforementioned method.

Alternative Embodiment Alternatively, embodiments are also envisioned in which, in case the velocity profile zeta is not available, viscosity eta can be derived using a combination of flow rate Q and differential pressure dP. This method can use the Rabinowitsch equation from A control method based on the Rabinowitsch equation may be applied to any combination of gross volumetric flow rate and differential pressure.

The scope of the invention is intended to include such as alternative embodiment. For example, the applications may include other types or kind of pipe flows either now known or later developed in the future, including other types or kind of industrial processes either now known or later developed in the future.

relationship between shear stress and pressure gradient

The Scope of the Invention While the invention has been described with reference to an exemplary embodiment, it will be understood by those skilled in the art that various changes may be made and equivalents may be substituted for elements thereof without departing from the scope of the invention. In addition, modifications may be made to adapt a particular situation or material to the teachings of the invention without departing from the essential scope thereof.

Therefore, it is intended that the invention not be limited to the particular embodiment s disclosed herein as the best mode contemplated for carrying out this invention. Apparatus, including a signal processor, comprising: Apparatus according to claim 1wherein the one or more modules is configured to determine a zeta profile for a shear rate based at least partly on the information about the plurality of velocity profiles of the flow in the pipe.

Shear stress - Wikipedia

Apparatus according to claim 1wherein the one or more modules is configured to determine a Tau profile for a shear stress based at least partly on the information about the pressure gradient of the flow over the length of the pipe. Apparatus according to claim 1wherein the one or more modules is configured to: Apparatus according to claim 4wherein the one or more modules is configured to determine a viscosity or viscosity profile of the flow in the pipe based at least partly on a ratio of the Tau profile for the shear stress in relation to the zeta profile for the shear rate.

Apparatus according to claim 5wherein the one or more modules is configured to control or regulate the injection of the chemical into the flow in the pipe based at least partly on the viscosity or viscosity profile. Apparatus according to claim 1wherein the plurality of velocity profiles of the flow in the pipe includes a velocity profile of five velocities measured by a velocity profile meter, including a SONAR-based velocity profile meter.

Apparatus according to claim 7wherein each velocity profile is measured in a vertical plane through an axis of the pipe. Apparatus according to claim 7wherein the five velocity profiles are measured in five different vertical planes in relation to an axis of the pipe. Apparatus according towherein at least one velocity profile is based on a Chebyshev interpolation. Apparatus according to claim 10wherein the one or more modules is configured to use an nth order Chebyshev polynomial, where n is greater than 2, and to invoke a no slip condition, so that the flow velocity at the top and the bottom of the pipe is zero, and the signals includes information about n2 velocity values sampled at points distributed along a vertical axis between the top and bottom.

Apparatus according towherein the injection of the chemical includes flocculation chemicals. Apparatus, including a system, comprising: Apparatus according to claim 14wherein the one or more modules is configured to determine a zeta profile for a shear rate based at least partly on the information about the plurality of velocity profiles of the flow in the pipe. Apparatus according to claim 14wherein the one or more modules is configured to determine a Tau profile for a shear stress based at least partly on the information about the pressure gradient of the flow over the length of the pipe.

Apparatus according to claim 14wherein the one or more modules is configured to: During the pre-WWI era, the misguided notions of designers was compounded by the ever-increasing use of wind-tunnel tests. The wind tunnels used at the time were relatively small and ran at very low flow speeds.

The most popular design up to then was the biplane configuration held together by wires and struts, which introduced considerable amounts of parasitic drag and thereby limited the maximum speed of aircraft. Eliminating these supporting struts and wires meant that the flight loads needed to be carried by other means. The two spars were connected by the external wing skin to produce a closed box-section known as the wing box. Prandtl argued that the presence of a boundary layer supported the simplifying assumption that fluid flow can be split into two non-interacting portions; a thin layer close to the surface governed by viscosity the stickiness of the fluid and an inviscid mainstream.

This allowed Prandtl and his colleagues to make much more accurate predictions of the lift and drag performance of specific wing-shapes and greatly helped in the design of German WWI aircraft. Second, the thick aerofoil could be flown at a much higher angle of attack without stalling and hence improved the manoeuvrability of a plane during dog fighting.

Laminar flow is orderly and stratified without interchange of fluid particles between individual layers, whereas in turbulent flow there is significant exchange of fluid perpendicular to the flow direction. For example, due to the greater extent of mass interchange, a turbulent boundary layer is thicker than a laminar one and also features a steeper velocity gradient close to the surface, i. Velocity profile of laminar versus turbulent boundary layer.

Note how the turbulent flow increases velocity more rapidly away from the wall. Just like your hand experiences friction when sliding over a surface, so do layers of fluid in the boundary layer, i. This means that the velocity gradient throughout the boundary layer gives rise to internal shear stresses that are akin to friction acting on a surface.

As the velocity gradient at the surface is greater for turbulent than laminar flow, a streamlined body experiences more drag when the boundary layer flow over its surfaces is turbulent.

Shear stress

A typical example of a streamlined body is an aircraft wing at cruise, and hence it is no surprise that maintaining laminar flow over aircraft wings is an ongoing research topic. Over flat surfaces we can suitably ignore any changes in pressure in the flow direction. Under these conditions, the boundary layer remains stable but grows in thickness in the flow direction. Under these conditions the boundary layer can become unstable and separate from the surface. The boundary layer separation induces a second type of drag, known as pressure drag.

This type of drag is predominant for non-streamlined bodies, e. So why does the flow separate in the first place? To answer this question consider fluid flow over a cylinder. Right at the front of the cylinder fluid particles must come to rest.

This point is aptly called the stagnation point and is the point of maximum pressure to conserve energy the pressure needs to fall as fluid velocity increases, and vice versa. Hence, an area of accelerating flow and falling pressure occurs between the stagnation point and the poles of the cylinder. Hence, the curvature in the flow reduces and the flow slows down, turning the previously favourable pressure gradient into an adverse pressure gradient of rising pressure.

relationship between shear stress and pressure gradient

Boundary layer separation over a cylinder axis out out the page. To understand boundary layer separation we need to understand how these favourable and adverse pressure gradients influence the shape of the boundary layer.

relationship between shear stress and pressure gradient

From our discussion on boundary layers, we know that the fluid travels slower the closer we are to the surface due to the retarding action of the no-slip condition at the wall. As a result, the fluid is not decelerated as much close to the wall leading to a fuller U-shaped velocity profile, and the boundary layer grows more slowly.

relationship between shear stress and pressure gradient

By analogy, the opposite occurs for an adverse pressure gradient, i. So in the case of an adverse pressure gradient the pressure forces reinforce the retarding viscous friction forces close to the surface. As a result, the difference between the flow velocity close to the wall and the mainstream is more pronounced and the boundary layer grows more quickly.

If the adverse pressure gradient acts over a sufficiently extended distance, the deceleration in the flow will be sufficient to reverse the direction of flow in the boundary layer. Hence the boundary layer develops a point of inflection, known as the point of boundary layer separation, beyond which a circular flow pattern is established. For aircraft wings, boundary layer separation can lead to very significant consequences ranging from an increase in pressure drag to a dramatic loss of lift, known as aerodynamic stall.

Hence the airflow over the top convex surface of a wing follows the same basic principles outlined above: There is a point of stagnation at the leading edge. A region of accelerating mainstream flow favourable pressure gradient up to the point of maximum thickness.

A region of decelerating mainstream flow adverse pressure gradient beyond the point of maximum thickness. These three points are summarised in the schematic diagram below. Boundary layer separation over the top surface of a wing. Boundary layer separation is an important issue for aircraft wings as it induces a large wake that completely changes the flow downstream of the point of separation.

relationship between shear stress and pressure gradient