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shell - Shell Balances R Shankar Subramanian When fluid...

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Shell Balances R. Shankar Subramanian When fluid flow occurs in a single direction everywhere in a system, shell balances are useful devices for applying the principle of conservation of momentum. An example is incompressible laminar flow of fluid in a straight circular pipe. Other examples include flow between two wide parallel plates or flow of a liquid film down an inclined plane. In the above situations, fluid velocity varies across the cross-section only in one coordinate direction and is uniform in the other direction normal to the flow direction. For flow through a straight circular tube, there is variation with the radial coordinate, but not with the polar angle. Similarly, for flow between wide parallel plates, the velocity varies with the distance coordinate between the two plates. If the plates are sufficiently wide, we can ignore variations in the other direction normal to the flow which runs parallel to the surfaces of the plates. If we neglect entrance and exit effects, the velocity does not vary with distance in the flow direction in both cases; this is the definition of fully-developed flow. A momentum balance can be written for a control volume called a shell, which is constructed by translating a differential cross-sectional area (normal to the flow) in the direction of the flow over a finite distance. The differential area itself is formed by taking a differential distance in the direction in which the velocity varies and translating it in the other cross-sectional coordinate over its full extent. We shall see by example how these shells are formed. The key idea is that we use a differential distance in the direction in which velocity varies. Later, we consider the limit as this distance approaches zero and obtain a differential equation. Typically this is an equation for the shear stress. By inserting a suitable rheological model connecting the shear stress to the velocity gradient, we can obtain a differential equation for the velocity distribution. This is then integrated with the boundary conditions relevant to the problem to obtain the velocity profile. Once the profile is known, we can calculate the volumetric flow rate and the average velocity as well as the maximum velocity. If desired, the shear stress distribution across the cross- section can be written as well. In the case of pipe flow, we shall see how this yields the well-known Hagen-Poiseuille equation connecting the pressure drop and the volumetric flow rate. Regarding boundary conditions, the most common condition we use is the “no slip” boundary condition. This states that the fluid adjacent to a solid surface assumes the velocity of the solid. Also, when necessary we use symmetry considerations to write boundary conditions. At a free liquid surface when flow is not driven by the adjoining gas dragging the liquid, we can set the shear stress to zero. In general at a fluid-fluid interface the velocity and the shear stress are continuous across the interface. Even though the interfacial region is made up of atoms and molecules and has a non-zero thickness, we treat is as a plane of zero thickness for this purpose. This means that the 1
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