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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.

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Why does $\mathbf{v} \cdot \nabla H =0$ imply that $H$ is constant along a stream line?

Context I am trying to follow a derivation of Bernoulli's equation from Acheson's Elementary Fluid Dynamics 1990. After making some simplifying assumptions and doing some algebra I got to: $$( \nabla \times \mathbf{v}) \times \mathbf{v} = - \nabla…
Conor
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Finding the velocity field from the complex potential

The question i have is Consider two vortices with rotation Γ at positions $(a,0)$ and $(-a,0)$. Find the Complex potential and hence find the velocity field. Now my working is as follows $w =…
L G
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Complex velocity of line vortex, simplification

I am considering the potential flow where I have a uniform flow past wing and two line vortices, one at the origin and one at a position $(x_1,y_1)$ relative to a wing of chord $D$. I'm using the following…
WnGatRC456
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Linear Hydrodynamic Stability - Periodicity of Perturbations

I am currently taking a course in Hydrodynamic Stability, based around Drazin's 'Introduction to Hydrodynamic Stability'. Up until this point, we have considered linear stability, where we assume a basic background state of e.g $\mathbf{u, p,…
J. Jaksche
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Fluid Mechanics Questions

A steady two-dimensional flow (pure straining) is given by $u = k x$, $v = -k y$, for $k$ constant. Find the equation for a general streamline of the flow. At $t = 0$,the fluid on the curve $x^2 + y^2 = a^2$ is marked. Find the equation for this…
MathematicianP
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Fluid Mechanics - Honey flow inside pipe

Honey initially at rest in a tube starts flowing down at = 0 s due to gravity, eventually reaching steady state. The equation governing the transient velocity profile of the honey is equation of motion honey. The no-slip condition requires that the…
Pedro Reyes Ricardez
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How do you derive this boundary condition?

The Question: Suppose a circular cylinder of radius $a$ moves with constant velocity $U$ in the $x$-direction in a two-dimensional irrotational, incompressible flow whose velocity decays to zero at infinity. The circulation around the cylinder is…
glowstonetrees
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How would i show $H$ is constant

I was wondering if anyone could help me with the following problem. If $H=\frac{P}{\rho}+\frac{v^{2}}{2}+gz$, Then $H$ is constant along streamlines and if $\nabla \times v=0$, then $H$ is constant throughout the flow. I don't really understand how…
Gibberish
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Acceleration field from velocity field

The components of a flow $\mathbf{v=v}(x,y,x,t)$ are given in a Cartesian basis as: \begin{eqnarray} v_x &=& xy\\ v_y&=& -y^2 \\ v_z &=& zy\\ \end{eqnarray} What is the acceleration field? Would I be correct in thinking that in order to solve…
user549241
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Finding the Density Change of a Fluid

Consider the motion of a fluid with velocity field defined in Eulerian variables by the following equations $$u=kx,\,\,v=-ky,\,\,w=0$$ where $k$ is a constant. Also assume that the density is given by $$\rho = \rho_0 + Aye^{kt}$$ What is…
MRT
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Finding the equation of pressure in fluid dynamics

We did a simple example in lecture with sea pressure but for this question I'm kinda confused. It goes like this: We can model the atmosphere as a $\textbf{static fluid}$, with the air a $\textbf{compressible perfect gas}$ acting under a uniform…
MRT
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Material derivative

If I have a material derivative, lets say $$\frac{D \rho }{D t} = -c\rho$$ Is the solution in the lagrangian frame of reference $$\rho_L = \rho_{0L} \exp{(-ct)}$$ , or I is it wrong to treat a material derivative as a standard derivative? Thanks…
Petros
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Fluid dynamics problem of hydrostatic equilibrium

The problem is to calculate the force per unit width and the torque per unit width about the bottom of a lock gate which is produced when a fluid of depth $H$ and density $\rho$ lies at rest on one side of the gate, and fluid of depth $h
user395952
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Standing Wave problem-In deep water limit $h\to\infty$ show that $\omega^2=gk$

The equations I have are $\phi=(-ag/\omega)\cos(kx)\sin(\omega t)e^{kz}$ and $\eta=a\cos(kx)\cos(\omega t)$ I know that $d\phi/dz=d\eta/dt$ but when I partially differentiate and rearrange I get $gk e^{kz}= \omega^2$ and I don't know how to get rid…
Adam
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free boundary effect on Reynold equation

I was trying to derive Reynolds equation in case of free boundary and no slip conditions.I know that derivation of reynolds equation requires the following assumption 1-constant viscosity 2-thin film geometry 3-negligible body force 4-no slip…
Tien
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