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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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Taylor expansion of characteristics

I am unable to follow the section of the solution I have underlined in green. Let us revisit the calculation in your notes that shows that a shock can form in finite time starting from appropriate initial conditions that are piecewise continuous.…
user144895
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Euler equation derivation

I am trying to follow the solution for i) , but i'm stuck on the parts I have underlined in green and orange.
user144895
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Point source derivation

I've just been reading this on a website to derive a point source. All makes sense, but I am not sure why it is $(2\pi r)v_r$. Thanks!
Mike Miller
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Find the constants in a 2D flow (incompressible, newtonian)

$$u_1=x_1^2x_2$$ $$u_2=A+Bx_1x_2^2$$ $$p=cosx_1$$ The fluid is an incompressible Newtonian fluid $\implies$ $u_i,i=0$ and $tor_ij=-p\delta_ij+\mu u_ij$ Fluid bounded by a stationary rigid plate at $x_2=0$ Find A and B So far, I have bounded by rigid…
James
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Streamlines tangent to velocity field

As from the title, I'm not too sure how they are related. Definition is that streamlines are instantaneously tangential to the velocity vector of the field. Why would a steamline that shows direction be a tangent to the velocity? Thanks!
Mike Miller
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Finding Stagnation Points

I am trying to find the stagnation point of a fluid flow from a complex potential. The complex potential is given by: $\Omega(z) = Uz + \cfrac{m}{2\pi}\ln z$. From this I found the streamfunction to be $\psi=Ur\sin\theta + \cfrac{m}{2\pi}\theta$ and…
Mike Miller
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Energy Density Conservation for Compressible Euler Equations

I have the following system of equations (for an ideal gas, from Exercise 2.1 from Fundamentals of Computational Fluid Dynamics): $$ \rho_t + \gamma p u_x + up_x = 0 $$ $$ p = (\gamma - 1) \left( e - \frac{1}{2} \rho u^2 \right) $$ I want to show…
Jacob Ivanov
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Why $\nabla \cdot(\vec \varphi \otimes \vec u)$ instead of $\nabla \cdot(\vec u \otimes \vec \varphi)$ in the continuity equation?

From the Wikipedia entry on the derivation of the Navier-Stokes equations, the continuity equations for scalar quantities has the following form： $$ \frac{\partial \varphi}{\partial t} + \nabla \cdot(\varphi \vec u) + \vec s = \vec…
ke zuo
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How to produce this graph in COMSOL?

I am trying to model generalised Couette flow in COMSOL. My goal is to reproduce this textbook graph in COMSOL. To cue in: Two parallel plates are in $h$ distance apart. The steady laminar flow of viscous incompressible fluid between them is along…
Kawrno
  • 131
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Total Time Derivative and partial time derivative

Hi guys i have some problems starting from $(1)$ deriving the $(2)$: $$\frac{d}{dt}\int_V \rho e dV=\int_V \rho \pmb{f}\cdot \textbf{u}dV+\int_{V}\rho~s~dV+\int_{\partial V}(\pmb{\sigma} \cdot \pmb{u}) \cdot \pmb{n} dA+\int_{\partial V}k \nabla T…
MatteoX
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Question regarding order of operations

(Fluid Mechanics: Fundamentals and Applications 4th Edition, Yunus Cengel and John Cimbala, problem 2-88) I am having some trouble with the order of operations. In the case of the below…
Kilian Olen
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Is D$_t\nabla f =\nabla$D$_tf$ where D$_t$ is the material derivative of a scalar function f?

I'm trying to prove that for $f$ being a scalar function with D$_tf=0$, where D$_t$ is the material derivative $D_t=\partial_t + v\cdot\nabla$, with $v$ solution of the incompressible Euler equations, then $D_t(\omega\cdot\nabla f)=0$, where…
Hapa
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Conservation of mass for fluids

My lecture notes contain the idea below, but I'm finding it difficult to understand one aspect. Consider a finite volume $V$, bounded by a surface $S$ and outward pointing normal $\hat{\bf{n}}$. A fluid occupies the space of which $V$ is a…
MHW
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Invariant form of material derivative

The material derivative in vector form is, $$\frac{DQ}{Dt}=\frac{\partial Q}{\partial t}+(V\cdot\nabla) Q$$ Where $Q$ is the fluid property. I didn't understand the following thing from here The form only works for Cartesian coordinates because it…
WhyMeasureTheory
  • 873
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Why is the streamfunction constant at the boundary of a pipe?

Consider a long cylindrical pipe rotating about its long axis with an imcompressible fluid inside. Why is the streamfunction constant at the boundary inside the pipe?
dannyybb
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