Mathematics Stack Exchange
Mathematics Stack Exchange

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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Stream Function

Completely new to fluid dynamics so a bit unsure of what I'm doing here - could I have some guidance as to whether I'm doing this correctly and a little help in plotting please? I've been asked to plot the stream functions of the following complex…
Mike Miller
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Is $\textbf{u} \cdot \nabla (Det \ \textbf{J})=0$ where $\textbf{J}$ is the Jacobian matrix w.t.r the Lagrangian map?

The example 1.7 in "An introduction to Theoretical Fluid Mechanics" by Stephen Childress, he calculated the material derivative of the Jacobian determinant $Det \ \textbf{J}$ of the Lagrangian map \begin{align*} X:\Omega_0 \times [0,+\infty) \to…
Hapa
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existence of stream function

Show from $\frac{\partial}{\partial t}\int_{\Omega}\rho dV = -\int_{\partial \Omega} \rho v\cdot n dS$ that a stream function $\psi(x, y)$ exists, in two-dimensional steady flow such that the difference between the respective values of $\psi$ at any…
Math_Day
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Continuity equation and the square of density, velocity product

I have two questions related to the continuity equation. (1) In fluid mechanics, we have the continuity equation $$\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho v) = 0$$ I am interested in deriving an expression for the squared norm of $\rho…
hsiaomichiu
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Oseen's tensor derivation of Stokes' drag

I am going through some lecture notes and I have been absolutely stumped by the derivation of Stokes' drag (it is very short). The sphere is of radius $a$ and is instantaneously located at $\bf{x} = 0$ with velocity $\bf{U}$. We have Oseem's tensor…
FizzKicks
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Streamline functions for one-component velocity

I would like to plot the streamline function for the two-dimensional flow with only one non-zero velocity component ($v_x, v_y=0$). I have seen the comparable question, which has been asked here, but I want to get a more precise answer provided by…
John
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How to find curve which I have the tangent vector field to in polar coordinates?

I am having a fluid-dynamics problem. I am given the flow: $$u_{\theta}=a, u_{r}=b$$ in polar coordinates. I am interested in finding a set of curves which have tangets defined by the vector field $\vec{u} = (u_{\theta},u_r)$. How can I do…
zabop
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glass and a card

Take a cylindrical glass of height $L$ and radius $R$. Fill the glass to a height $h$. When you place a card that completely covers the opening of the glass, you can turn the glass over without the water running out. This is because the tension…
questmath
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Applying Bernoulli’s Equation to a Boat on a river?

I’m having difficulty getting my head around this scenario (we assume incompressibility): We have a boat on a river moving at speed $u$ which in turn causes a current of speed $v$. The area of the base of the boat is $A_B$. If we take a streamline…
maths54321
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Navier-Stokes: formal definition/intuition of "mean flow"?

In (computational) fluid dynamics there is a notion of mean flow that I don't quite understand: E.g. in the Wikipedia entry for turbulence modeling it says: Averaging the equations gives the Reynolds-averaged Navier–Stokes (RANS) equations, which…
don-joe
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determining a component of velocity vector of a fluid flow explicitly

If $(u(y),0,0)$ are the velocity components of an incompressible Newtonian fluid flow due to a pressure gradient in the $X$-direction, then $u(y)$ is a $(a)$ linear function of $y$. $(b)$ quadratic function of $y$. $(c)$ cubic function of…
am_11235...
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conservation of energy/mass/momentum related to a seemingly unknown equation

If we denote the density of a fluid media as $\rho$ and $\vec{q}$ as the flow velocity vector, then the equation $$\frac{1}{\rho}\bigg(\frac{\partial \rho}{\partial t}+\vec{q}.\vec{\nabla}\rho\bigg)+\vec{\nabla}.\vec{q}=0$$ is the equation…
am_11235...
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Deriving boundary condition on the free surface of a pool of water

I would like to know how one could derive the Bernoulli’s equation (and hence the boundary condition on the free surface of a pool of water), assuming that the fluid is inviscid and irrotational, and starting with the equations of the conservation…
Joker123
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Numerical Solution of Stoke's First Problem

$f''+2 \eta f-4mf=0$ where $f=1$ at $\eta=0$, $f \to 0$ as $\eta \to \infty$. The case of $m=0$ arises in the analysis of motion of fluid above a suddenly accelerated flat plate.
hellomecha
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Finding stream function from velocity fields.

Consider a velocity field $u = x\hat{i}+2y\hat{j}$. To find its stream function,$$\frac{\partial\psi}{\partial y}=x$$ $$\psi = xy + f(x)$$ and $$\frac{\partial\psi}{\partial x}=-2y$$ $$\psi=-2xy+g(y)$$ I am stuck at this point because both $\psi$ do…
Aniket Sharma
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