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Questions tagged [fluid-dynamics]

For questions about fluid dynamics which studies the flows of fluids and involves analysis and solution of partial differential equations like Euler equations, Navier-Stokes equations, etc. Tag with [tag:mathematical-physics] if necessary.

Fluid dynamics is a branch of physics that studies the the flows of fluids-liquids and gases, which involves analysis and solution of partial differential equations like Euler equations, Navier-Stokes equations, etc.

1103 questions
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Bernoulli, vorticity, stream function

I do not know how to demonstrate that \begin{equation} \omega(\psi) = \frac{d H(\psi)}{d\psi}, \end{equation} where $H$ stands for the Bernoulli's constant $H=\frac{u^2 +v^2}{2} + p/\rho$, $\psi$ for the stream function and $\omega$ for the…
user36390
  • 311
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When is a vector field the gradient of the pressure

In Frank White's book Fluid mechanics problem 4.27 (8th ed.) a 2D-velocity field is given as $$ \vec{v} = (2xy,-y^2) $$ Using Euler's equation (assuming stationary, incompressible flow and no gravitational field) $$ \rho\vec{v}\cdot\nabla\vec{v} =…
DiggingDeep
  • 117
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Material Derivative of a Line Integral?

I was wondering if there was an identity concerning the material derivative of a line integral, e.g., the material derivative of the zonal barotropic wind $u_{\tau}$ (which itself is the vertical-average of the total zonal wind $u$) $$\frac{\text{D}…
jltorchinsky
  • 163
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Air fan problem - fluid dynamic

For preparing an exam me and my classmates are unable to solve this question. Can anybody help us. Thanks. At a conference my colleagues had a room without air conditioning. At the reception they could get a fan to cool down the warm room. (a)…
notaMathqueen
  • 287
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Surface Pressure Over a Fluid Element

I came across this in Fluid Mechanics book by Prof F.M.White. Here he gave an expression for the Surface Pressure acting over a fluid element. I have great trouble understanding the expression. I would be grateful if somebody can help me…
Jenkins
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Fluid Mechanics - Sources/Sinks/Streamlines

My doubt is regarding the last part. How does one find the differential equation and then prove that they lie on the given surface? I am aware of the method, by writing dx/u = dy/v (or its equivalent in cylindrical). It's just that I am unable to…
AJ_
  • 273
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In fluid-dynamics: $2\nabla\cdot D = \nabla^2\textbf{v}+\nabla(\nabla\cdot\textbf{v})$, where $D$ is the deformation tensor.

Let $\textbf{v}\in C^2$ be a velocity field and $D$ the deformation tensor defined as $$d_{ij} = \dfrac{1}{2}\left(\dfrac{\partial v_i}{\partial x_j}+\dfrac{\partial v_j}{\partial x_i}\right).$$ Then $2\nabla\cdot D =…
Stijn D'hondt
  • 801
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For a fluid in equilibrium, the level surfaces of $\rho$ and $\Psi$ must coincide?

I am reading through An Introduction to Fluid Dynamics by G.K. Batchelor. Question background: Batchelor states that if a fluid is in equilibrium, then everywhere in the fluid we have $$ \rho\mathbf{F} = \nabla p$$ where $\rho = \rho(\mathbf{x},t)$,…
EssentialAnonymity
  • 1,399
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Weak Formulation of Navier-Stokes

I was looking into the Navier-Stokes Weak formulation Let $f\in L^2(\Omega_T)$, $u_0 \in H(\Omega)=\lbrace u\in L^2(\Omega):\text{div }u=0\text{ in }\Omega;u\cdot n|_{\partial \Omega}=0\rbrace$. A measurable function $u:\Omega_T\rightarrow…
user396656
  • 43
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potential flow question about sound speed

For isentropic $P = k\rho^\gamma$ steady irrotational $\vec{u} = \nabla\phi$ flow, the momentum equation implies the Bernoulli relation $$\tfrac{1}{2}|\nabla\phi|^2+\frac{k\gamma}{\gamma-1}\rho^{\gamma-1} = C$$ where $C$ is constant throughout the…
Bob Terrell
  • 3,812
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To find the velocity potential

The question is A simple source of strength $m$ is fixed at the origin $O$ in a uniform stream of incompressible fluid moving with velocity $U \vec{i}$. Show that the velocity potential $\phi$ at any point $P$ in the stream is given by $$ \phi…
Manoj Dash
  • 113
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Finding streamlines from complex potential.

I'm currently studying a Fluid Dynamics module and mock exam question has me completely stumped, I have been given the complex potential and shown it to be in the form given, however when trying to separate the complex potential into velocity…
user2662468
  • 69
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If a fluid has the complex potential $w(z)=\frac{-\Gamma i}{2 \pi}\operatorname{log}z$ what are its radial and transverse velocity components?

If a fluid has the complex potential $$w(z)=\frac{-\Gamma i}{2 \pi}\operatorname{log}z$$ Can anyone show me how to find its radial and transverse velocity components in polar coordinates? They are meant to be $u_r=0$ and $u_\theta=\frac{\Gamma}{2r…
Freeman
  • 5,399
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Velocity potential of flow under rigid disk

Determine velocity potential of the flow in this system: Rigid disk of radius R at a heigh h(t) above horizontal plane z=0 with incompressible, inviscid flow between them, and h< The flow is axisymmetric and has horizontal component independent of…
timni
  • 75
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Velocity of fluid in presence of Sphere

I am struggling with the following problem: A rigid sphere of radius $a$ is placed in a stream of fluid whose velocity in the undisturbed state is $V$. Determine the velocity of fluid at any point of the disturbed stream. I have just started to…
bhavesh
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