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

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Find an estimate for the separation distance between two plates squeezing a long line of oil [on hold]

I was wondering if someone could give me some help on with this problem, I'm really struggling to get my head around it. A long line of oil is being squashed between two flat plates of length $L$ by ...
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29 views

How to scale a problem involving heat and the flow of a viscous fluid?

I recently got set this problem and I was wondering if anyone would be able to give me some help on the later parts. An incompressible thermal conducting fluid is contained between two infinite ...
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34 views

Chain Rule in Polar coordinates

I was looking for an intuitive explanation for the total derivative in polar coordinates. Let me be somewhat more specific: Take a standard line of reasoning that the gradient w.r.t. polar coordinates ...
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fluid dynamics in polar coordinates

On page 12 of Malham's fluid dynamics notes the following flow field is considered: $\boldsymbol u= (u,v) = (kx, -ky)$. It's easy to see in these Cartesian coordinates that this is solenoidal: ...
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Flow between two infinite horizontal plates

I recently got set this problem and I was wondering if anyone would be able to give me some hints/intuition on how to solve it. Thanks. An incompressible thermal conducting fluid is contained between ...
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2answers
56 views

Which of these 1-D representations of the Navier-Stokes equations is correct?

The incompressible Navier Stokes equations can be written as A. $$\frac{\partial (\rho \mathbf{v})}{\partial t} + \nabla \cdot (\rho \mathbf{v} \mathbf{v}) = S$$ or B. $$\frac{\partial ...
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26 views

Population balance model

I have some experimental data and I need to make a population balance model to compare the experimental results with. The experimental results are from the bubble size distribution in a bioreactor. I ...
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23 views

How can I modify this simple code to include the pressure term? (1-D Navier Stokes)

I have a mathematical model that involves a cylindrical container that is being modeled with a one dimensional simplification as the system is isotropic with respect to the z-axis. As part of the ...
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Fluid dynamics: mesh resolution close to the origin in spherical co-ord system

Suppose you have a spherical implosion calculation (e.g. ICF etc.) in which you have a material interface that you want to apply some sort of perturbation to. There are two possible configurations in ...
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recovering pressure term via Helmholtz-Hodge decomposition

Looking at the classical Navier-Stokes equations here. I wish to know how the pressure gradient can be recovered. In particular, I have trouble understanding how eq (1.7) can be stated as it is. ...
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Thermodynamics about turbines [migrated]

1.A turbine is rated at 650 hp when the flow of water through it is 0.85 m3/sec. Assuming an efficiency of 84%, what is the head acting on the turbine.
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Why does the pressure term complicate numerical methods for Navier-Stokes Equations?

I'm looking to code a solver for the Navier-Stokes equations. I will be using finite differences with the method of lines. Two questions: What is the significance of the pressure term in the full ...
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Helicity is Conserved

In fluid mechanics, the helicity is defined as $$\int_{R^3} u(x,t)\cdot \omega(x,t),$$ where $u(x,t)$ is a smooth solution of the Euler equations $$\partial_tu + (u \cdot \nabla) u = -\nabla p$$ ...
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Water explosion and Navier-Stokes global regularity problem

In this article, Terence Tao is talking about a water explosion thought experiment that can lead to the solution of Navier-­Stokes global regularity problem. Can anybody explain this in more ...
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35 views

Presentation of Navier-Stokes eqns

This may be a trivial point, perhaps it's a lack of understanding on my part? When I was first introduced to fluid mechanics I was instructed to write the continuity and (generalized) Navier-Stokes ...
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50 views

How much water could be stored?

We have two water storage tanks--- Tank A and Tank B --- on the roof of the upper storey of our two-storey house. The tanks are cylindrical in shape. Each of the two tanks have a circular opening ...
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1answer
46 views

Significance of 'faces' in Stress tensor components?

I am trying to understand what the significance is of the face for which a force is acting on when talking about a stress tensor. Say we consider the components $T_{xx}$ and $T_{zx}$ of the stress ...
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1answer
48 views

Math Basis for Partial Fluid Change in Car Transmission

I have a fluid mixing problem with my car and I can't seem to find the answer: I have to change my transmission oil. The transmission has a 7 liter total capacity, but due to the torque converter, I ...
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55 views

Inverse Laplace Transform using Hetnarski's Algorithm

I'm trying to find the velocity component of an MHD flow using Laplace transforms. R.B. Hetnarski's algorithm for inverting the laplace transforms of some exponential functions was recommended to me ...
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1answer
69 views

Vector Calculus Operator $\vec{u} \cdot \nabla$

I just want to double check on this operator and it's properties. It pops up in fluid mechanics often and I just want to be sure about my understanding: 1) $$(\vec u \cdot \nabla)\vec u$$ Is this ...
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using complex potential to calculate stream function

The answer is $ \psi = Br^{12} sin12 \theta $ I dont know how to get this? say the complex potential equals $ Bz^{n} $ = $ Bre^{i\theta} $ how do i get the above answer? thank you
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1answer
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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 ...
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1answer
24 views

fluid dynamics Bernoulli's equation

I really cant seem to prove this, using A1u1 = A2u2 I get (r_a)^2/(r_b)^{2} = 1 using Bernoulli's equation, i cant figure out what the other terms should be? 1/2ro u_a^{2} + rogz1 + pa = ...
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Why we have to take the vector $\overrightarrow{e}$ ?

The differential equation of the balance of the momentum is $$\rho \frac{\partial{\overrightarrow{u}}}{\partial{t}}=-\rho (\overrightarrow{u} \cdot \nabla )\overrightarrow{u}-\nabla p+\rho ...
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27 views

Law of hydrostatic pressure

For a calm fluid of uniform density $\rho_0$, that occupies the space $W \subset \mathbb{R}^3$, and is subject to massive forces (per unit of mass) $\overrightarrow{b}(\overrightarrow{x})$, write the ...
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39 views

Stress vector - Stress tensor

Is the definition of the stress vector the following? The stress vector is the force per unit surface. The stress tensor is the matrix $\{\sigma_{ij}(x,t)\}$ and its $(i,j)$-component is the ...
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26 views

Find the streamlines of the velocity field

I have to find the streamlines of the following velocity fields: $$u=x(1+2t), v=y$$ $$u=xy, v=0$$ I have done the following: $$\frac{dx}{u}=\frac{dy}{v} \Rightarrow ...
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Streamlines - Pathlines

Construct and draw the streamlines of the velocity field $u=az-bt, v=\frac{b}{4}z-cy, w=2(a-1)$. Calculate $c$ (as a function of the constants $a$, $b$) such that the flow field ...
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22 views

Show the relation $W$ is constant

If the space $W$ is constant (doesn't move with the flow), show that $$\frac{d}{dt}\int_{W}\left (\frac{1}{2}\rho |\overrightarrow{u}|^2+\rho \epsilon\right )dV=-\int_{\partial{W}}\rho \left ...
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Visco-elastic fluid reference

What is a good book on visco-elastic fluids for self-teaching after one has studied Gurtin's Intro to Continuum mechanics? Thanks!
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42 views

Concept of a continuum

I have to explain the concept of a continuum that is used for the description of the dynamic behaviour of the fluids, and to explain how this concept is related on the one side with the laboratory ...
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32 views

How to prove the relation $\tan \theta = \hat{\vec{n}} \cdot \nabla h$

The relation for finding the contact angle is often given as $\tan \theta = - \hat{\vec{n}} \cdot \nabla h$ in papers such as in Sequential deposition of overlapping droplets to form a liquid line ...
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53 views

Find the mass flow rate, given a surface, density and velocity field

I have a confusion, I hope you can help me (I'd like that if you will respond, please read all my post). They ask me to find the mass flow rate passing through a surface, where the velocity field is ...
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38 views

How to find the complex potential for the following flow under certain conditions?

We've used $z=i(Z+4/Z)$ as a conformal mapping to map the exterior of a circle $|Z|=2$ to the exterior of the line segment $(-4i,4i)$. We now want to write the complex potential of the uniform flow ...
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Help solving a first order non-linear differential equation derived from the navier-stokes equation

I am an engineer studying an unsteady-state flow through a pipe. The transient Bernoulli equation of this system, which I picked up from here ...
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10 views

Representation of polynomial order in CFD codes

I currently working on a CFD code over a cubic grid. Now, the number of elements used in the simulation is decomposed among the number of processors. Each of those processors (a section of the cube) ...
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1answer
25 views

Does the Laplace operator include the second derivative with respect to time variable?

Does the Laplacian of a function $f(x,z,t)$ equal $f_{xx} + f_{yy} + f_{tt}$? We aren't sure whether or not time is included in it or not.
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Is a 3D or 2D Poisson's equation separable or non sperable?

Can someone please explain to me if a 2/3D Poisson's equation is separable or non separable? Thank you
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1answer
39 views

When is shear useful?

I'd never heard of the shear of a vector field until reading this article. Shear is the symmetric, tracefree part of the gradient of a vector field. If you were to decompose the gradient of a vector ...
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Von Karman similarity solution of Navier-Stokes equations

I was wondering if anyone could help me; I've been looking at Von Karman's similarity solutions of the Nav-Stokes equations for a rotating disk in a fluid $u = r \Omega U(n)$ $v = r \Omega V(n)$ ...
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Solving a system of two second order ODEs using Runge-Kutta method (ode45) in MATLAB

I'm trying to solve the system of differential equations outlined in Von Karman's rotating disk flow. I got them into a system of ordinary differential equations: F(n), G(n), H(n) $$F'' = -G^2 + F^2 ...
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Linearisation of instability of flames

I am not sure it is a maths or physics question, kinda in between. Large-scale disturbances of a plane flame $x=\eta(y,t)$ are describe by Euler's equation and the continuity equation: $$ ...
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Problem on Majda's Vorticity and Incompressible Flow

I'm reading Majda's book, on page 110, I cannot understand how to get $v\in C_W (0,T;H^{m})$ from the above. And he wrote "$[\phi,v^{\epsilon}]\to[\phi,v]$ uniformly on $[0,T]$ " two times, do the ...
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Evaluating vorticity as a function of velocity components.

So i have the following question.. Consider the axisymmetric flow of a viscous fluid u = ($ \frac{-\alpha r}{2} $, v(r), $\alpha z$) in cylindrical polar coordinates, where $\alpha$ is a positive ...
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28 views

Two methods of calculating a Jacobian determinant

Suppose you have two fluid bodies, one described by a set of vectors $V$, and a perturbation of $V$ given by $V+\Delta V$. Suppose that the two regions are related by the transformation $\mathbf ...
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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 ...
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Linearising equations about a base state.

Consider a shallow-water system with mean depth H, where the base state consists of the flow (u,v)=($u_{0},$0), with a sloped water surface $\eta_{0}$(x,y) = - $\gamma y$, where u$_{0}$ and $\gamma$ ...
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Vorticity of a rigid body.

Consider a fluid in solid body roation about the z-axis with angular speed $\varOmega$ Derive an expression for the velocity field (u(x,y), v(x,y)) and show the vorticity field is the same at every ...
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The free surface of the wave is a material surface

If we define the free surface by: $F(x,y,t)=y-h(x,t)=0$ Then for this to be a material surface $\frac{DF}{Dt}=0$ on $y=h(x,t)$ However on $y=h(x,t)$, $F=0$, so doesn't this just imply ...
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Particle paths and standing waves

Here $x_0$ is the $x$ coordinate of a point in $x-y$ space Here $x_0$ is the $x$ coordinate of a point in $x-y$ space. I understand where the nodes and crests are on the figure. However, I don't ...