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Fundamentals of Fluid Mechanics
Published in Ethirajan Rathakrishnan, Instrumentation, Measurements, and Experiments in Fluids, 2020
These are imaginary lines in the flow field such that the velocity at all points on these lines is always tangential to them. Flows are usually depicted graphically with the aid of streamlines. Streamlines proceeding through the periphery of an infinitesimal area at some time t form a tube called a stream tube, which is useful for the study of fluid flow phenomena. From the definition of streamlines, it can be inferred that Flow cannot cross a streamline, and the mass flow between two streamlines is confined.Based on the streamline concept, a function ψ called stream function can be defined. The velocity components of a flow field can be obtained by differentiating the stream function.
Wave, Diffusion and Potential Equations
Published in K.T. Chau, Theory of Differential Equations in Engineering and Mechanics, 2017
This is again the Laplace equation. Thus, both the potential function and stream function satisfy the Laplace equation. Physically the stream function indicates the flow line in the fluid. To see this, we can take the total differential of the function equal to a constant value, i.e., ψ = c: dψ=∂ψ∂xdx+∂ψ∂zdz=-vZdx+vXdz=0 $$ d\psi = \frac{{\partial \psi }}{{\partial x}}dx + \frac{{\partial \psi }}{{\partial z}}dz = - v_{Z} dx + v_{X} dz = 0 $$
Multidimensional Incompressible Laminar Flow
Published in M. Necati Özişik, Helcio R.B. Orlande, Marcelo José Colaço, Renato Machado Cotta, Finite Difference Methods in Heat Transfer, 2017
M. Necati Özişik, Helcio R.B. Orlande, Marcelo José Colaço, Renato Machado Cotta
For two-dimensional cases, the most successful numerical technique for solving such a system is based on the vorticity-stream function formulation that will be discussed next. However, a single-stream function does not exist for three-dimensional flow problems. For these cases, the numerical solution of the previous equations, subject to appropriate boundary conditions, is more adequate. This case will be discussed after the vorticity-stream function method.
Numerical simulations for heat transfer in peristalsis of Bingham fluid utilizing partial slip conditions
Published in Waves in Random and Complex Media, 2022
Sadia Hina, Meraj Mustafa, Zaheer Asghar, Sana Maryam Kayani
A better insight into the flow phenomenon under different controlling parameters can be obtained from detailed plots of stream function. To this end, contours of stream function are generated at different parameter values in Figures 8–11. These figures indicate the formation of closed streamlines in both right and left parts of the channel. Such closed streamlines are referred to as boluses or eddies. Level curves of stream function (surrounding the boundaries) acquire the shape of the wall. Moreover, the bolus shape appears symmetrical in left and right halves of the channel. In Figure 8(a–c), the consequences of yield stress on flow pattern can be studied. The results predict that bolus size would reduce whenever fluid with yield stress effect is employed. Such reduction in size becomes faster whenever the partial slip condition is invoked. Moreover, bolus size appears to shrink and the number of streamlines get reduced as one increases the slip coefficient (Figure 9(a–c)). Indeed, the bolus disappears for a sufficiently large choice of . Additionally, Figure 10(a–c) clarify that there is a marginal rise in bolus size whenever the buoyancy force term is retained.
Analysis of process efficiency: Role of flow and thermal characteristics on entropy production and heat transfer rates for thermal convection in porous beds confined within triangular configurations with hot slanted walls
Published in Numerical Heat Transfer, Part A: Applications, 2022
Leo Lukose, Pratibha Biswal, Tanmay Basak
Streamlines are used to represent fluid circulations or paths in two-dimensional flow field and each path corresponds to a value of streamfunction (ψ). Velocity components (UX and UY) at any point of the streamline are related with streamfunction (ψ) satisfying continuity equation (Eq. (2)) as The definition with sign convention dictates the direction of flow such that the clockwise or anticlockwise direction of flow can be visualized based on the negative and positive signs of ψ, respectively. Either derivative in Eq. (11) can be used to evaluate ψ at a node in computational domain. Alternatively, derivatives in Eq. (11) can be combined to Poisson equation as follows: Residual equations can be obtained for Eq. (12) at each element and final forms of residuals are obtained from assembly of elements. Employing following interpolation functions, the residuals would lead to set of algebraic equations. The no-slip condition at walls would dictate ψ = 0 leading to algebraic equation at all boundary nodes. The entire algebraic set can be solved following earlier method [35] to obtain ψk (k denotes the node of the domain).
Generation of waves due to bottom disturbances in a viscous fluid
Published in Geophysical & Astrophysical Fluid Dynamics, 2022
We consider the two-dimensional problem of wave generation in an incompressible homogeneous viscous fluid due to transient disturbances at the bottom. A Cartesian co-ordinate system is chosen in which the y-axis is taken vertically upwards and the free surface of the fluid region is initially horizontal, y = −h corresponds to the bottom of the fluid region where the transient ground disturbance takes place. The schematic diagram corresponding to the physical model is given in figure 1. The fluid motion can be described by a velocity potential and a stream function . The horizontal and vertical components of velocity are given by The continuity and linearised Navier-Stokes equations for a fluid, kinematic viscosity ν, are The linearised dynamic and kinematic conditions at the free surface, elevation , are and The free surface boundary conditions, namely the vanishing of tangential stress, , and continuity of normal stress, , are where and is the prescribed free surface pressure at time t. In terms of ϕ, ψ and η the free surface conditions (5), (6) reduce to while the last condition follows from (4).