Elementary flow

In the larger context of the Navier-Stokes equations (and especially in the context of potential theory), elementary flows are basic flows that can be combined, using various techniques, to construct more complex flows. In this article the term "flow" is used interchangeably with the term "solution" due to historical reasons. The techniques involved to create more complex solutions can be for example by superposition, by techniques such as topology or considering them as local solutions on a certain neighborhood, subdomain or boundary layer and to be patched together. Elementary flows can be considered the basic building blocks (fundamental solutions, local solutions and solitons) of the different types of equations derived from the Navier-Stokes equations. Some of the flows reflect specific constraints such as incompressible or irrotational flows, or both, as in the case of potential flow, and some of the flows may be limited to the case of two dimensions. Due to the relationship between fluid dynamics and field theory, elementary flows are relevant not only to aerodynamics but to all field theory in general.…

In the larger context of the Navier-Stokes equations (and especially in the context of potential theory), elementary flows are basic flows that can be combined, using various techniques, to construct more complex flows. In this article the term "flow" is used interchangeably with the term "solution" due to historical reasons. The techniques involved to create more complex solutions can be for example by superposition, by techniques such as topology or considering them as local solutions on a certain neighborhood, subdomain or boundary layer and to be patched together. Elementary flows can be considered the basic building blocks (fundamental solutions, local solutions and solitons) of the different types of equations derived from the Navier-Stokes equations. Some of the flows reflect specific constraints such as incompressible or irrotational flows, or both, as in the case of potential flow, and some of the flows may be limited to the case of two dimensions. Due to the relationship between fluid dynamics and field theory, elementary flows are relevant not only to aerodynamics but to all field theory in general. To put it in perspective boundary layers can be interpreted as topological defects on generic manifolds, and considering fluid dynamics analogies and limit cases in electromagnetism, quantum mechanics and general relativity one can see how all these solutions are at the core of recent developments in theoretical physics such as the ads/cft duality, the SYK model, the physics of nematic liquids, strongly correlated systems and even to quark gluon plasmas. \mathbf{v} = v_0 \cos(\theta_0)\, \mathbf{e}_x +v_0 \sin(\theta_0)\, \mathbf{e}_y where v_0 is the absolute magnitude of the velocity (i.e., v_0 = |\mathbf{v}| ); \theta_0 is the angle the velocity vector makes with the positive axis ( \theta_0 is positive for angles measured in a counterclockwise sense from the positive axis); and \mathbf{e}_x and \mathbf{e}_y are the unit basis vectors of the coordinate system. Because this flow is incompressible (i.e., \nabla \cdot \mathbf{v} = 0 ) and two-dimensional, its velocity can be expressed in terms of a stream function, \psi : v_x = \frac {\partial \psi} {\partial y} v_y = - \frac {\partial \psi} {\partial x} where \psi = \psi_0 - v_0 \sin (\theta_0)\, x + v_0 \cos (\theta_0)\, y and \psi_0 is a constant. In cylindrical coordinates: v_r = - \frac 1 r \frac{\partial \psi} {\partial \theta} v_\theta = \frac{\partial \psi} {\partial r} and \psi = \psi_0 + v_0\, r \sin (\theta - \theta_0) This flow is irrotational (i.e., \nabla \times \mathbf{v} = \mathbf{0} ) so its velocity can be expressed in terms of a potential function, \phi : v_x = - \frac{\partial \phi} {\partial x} v_y = - \frac {\partial \phi} {\partial y} where \phi = \phi_0 - v_0 \cos (\theta_0)\, x - v_0 \sin (\theta_0)\, y and \phi_0 is a constant. In cylindrical coordinates v_r = \frac {\partial \phi} {\partial r} v_\theta = \frac {1}{r} \frac {\partial \phi} {\partial \theta} \phi = \phi_0 - v_0\, r \cos(\theta - \theta_0) \mathbf{v} = v_r(r) \mathbf{e}_r Where the total outgoing flux is constant \int_S \mathbf{v} \cdot d \mathbf{S} = \int_{0}^{2 \pi} ( v_r(r) \, \mathbf{e}_r ) \cdot ( \mathbf{e}_r \, r \, d \theta ) = \! 2 \pi \, r \, v_r(r) = Q Therefore, v_r = \frac {Q}{2 \pi r} This is derived from a stream function \psi(r,\theta) = -\frac{Q}{2 \pi } \theta or from a potential function \phi(r,\theta) = -\frac{Q}{2 \pi } \ln r v_r = - \frac {Q}{2 \pi r} This is derived from a stream function \psi(r,\theta) = \frac{Q}{2 \pi } \theta or from a potential function \phi(r,\theta) = \frac{Q}{2 \pi } \ln r Given that the two results are the same a part from a minus sign we can treat transparently both line sources and line sinks with the same stream and potential functions permitting Q to assume both positive and negative values and absorbing the minus sign into the definition of Q. \psi(\mathbf{r}) = \psi_Q(\mathbf{r} - \mathbf{d}/2) - \psi_Q(\mathbf{r} + \mathbf{d}/2) \ \simeq \mathbf{d} \cdot \nabla \psi_Q(\mathbf{r}) The last approximation is to the first order in d. Given \mathbf{d} = d [ \cos (\theta_0) \mathbf{e}_x + \sin (\theta_0) \mathbf{e}_y] = d [ \cos (\theta-\theta_0) \mathbf{e}_r + \sin (\theta-\theta_0) \mathbf{e}_\theta] It remains \psi(r,\theta) = - \frac{Q d}{2 \pi} \frac{\sin(\theta-\theta_0)}{r} The velocity is then v_r(r,\theta) = \frac{Q d}{2 \pi} \frac{\cos(\theta-\theta_0)}{r^2} v_\theta(r,\theta) = \frac{Q d}{2 \pi} \frac{\sin(\theta-\theta_0)}{r^2} And the potential instead \phi(r,\theta) = \frac{Q d}{2 \pi} \frac{\cos(\theta-\theta_0)}{r} \mathbf{v} = v_\theta(r) \, \mathbf{e}_\theta Where the total circulation is constant for every closed line around the central vortex \oint \mathbf{v} \cdot d \mathbf{s} = \int_{0}^{2 \pi} (v_\theta(r) \, \mathbf{e}_\theta) \cdot (\mathbf{e}_\theta \, r \, d\theta) = \! 2 \pi \, r\, v_\theta(r) = \Gamma and is zero for any line not including the vortex. Therefore, v_\theta = \frac {\Gamma}{2 \pi r} This is derived from a stream function \psi(r,\theta) = \frac{\Gamma}{2 \pi } \ln r or from a potential function \phi(r,\theta) = - \frac{\Gamma}{2 \pi } \theta Which is dual to the previous case of a line source \nabla^2 \psi = 0 Which is in cylindrical coordinates \frac{1}{r} \frac{\partial}{\partial r} \left(r \frac{\partial \psi}{\partial r}\right) + \frac{1}{r^2} \frac{\partial^2 \psi}{\partial \theta^2}= 0 We look for a solution with separated variables: \psi(r,\theta) = R(r) \Theta(\theta) which gives \frac{r}{R(r)} \frac{d}{dr} \left(r \frac{d R(r)}{dr}\right) = -\frac{1}{\Theta(\theta)} \frac{d^2 \Theta(\theta)}{d\theta^2} Given the left part depends only on r and the right parts depends only on \theta , the two parts must be equal to a constant independent from r and \theta . The constant shall be positive . Therefore, r \frac{d}{dr} \left(r \frac{d}{dr} R(r)\right) = m^2 R(r) \frac{d^2 \Theta(\theta)}{d\theta^2} = - m^2 \Theta(\theta) The solution to the second equation is a linear combination of e^{i m \theta} and e^{-i m \theta} In order to have a single-valued velocity (and also a single-valued stream function) m shall be a positive integer. therefore the most generic solution is given by \psi = \alpha_0 + \beta_0 \ln r + \sum_{m > 0}{\left(\alpha_m r^m + \beta_m r^{-m}\right)\sin {[m(\theta - \theta_m)]}} The potential is instead given by \phi = \alpha_0 - \beta_0 \theta + \sum_{m \mathop > 0}{(\alpha_m r^m - \beta_m r^{-m})\cos {[m(\theta - \theta_m)]}} ==References== Specific ==Further reading== ==External links==

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