Boring stuff

The previous equation is still too general, and a connection between stress and strain is still needed. Here we consider the case in which there is a linear relationship between both, which involves the coefficient of viscosity.

To begin with, let us consider a simple case in which a fluid is confined between two planes. One of them moves sideways with a certain speed u_0, while the other is kept fixed. After a certain transient, some force is needed in order to keep this shearing. The simplest expression is

F= \mu A \frac{u_0}{L}.

The force is proportional to the area and to the velocity difference between the planes. It is also inversely proportional to their separation, L (this fact being the least obvious). Finally, a constant of proportionality is given by \mu, the viscosity coefficient, or

simply “the viscosity”. This constant may vary with temperature, density, pressure, but the point with Newtonian fluids is that it does not vary with the velocity field (or its derivatives).

Later, in section …, this flow will be solved as a solution of the Navier-Stokes equations, the Couette flow. There, it will be shown that the velocity is everywhere in the direction of the force exerted on the upper plane, let us call it $x$, and varies linearly between the planes, in the y direction. Therefore, the only components of the strain rate tensor are \epsilon_{xy} = \epsilon_{yx} = u_0 / ( 2 L ). We therefore have

\tau_{xy} = \mu \epsilon_{xy}.

With this in mind, let us look for a general relationship between \tau and    \epsilon. This is much easier if we go to the principal strain axes. These are the coordinates on which the strain rate is diagonal. Such coordinate system always exist, since the strain rate tensor is symmetric. Notice that in these system strains are not due to shear, only to dilations.

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Better with stock pictures

pexels-photo-164531.jpeg

“A connection is needed”. Photo by Pixabay on Pexels.com

 

The previous equation is still too general, and a connection between stress and strain is still needed. Here we consider the case in which there is a linear relationship between both, which involves the coefficient of viscosity.

 

 

 

 

honey on white bowl

Holy cow, is honey viscous or what. Photo by Pixabay on Pexels.com

To begin with, let us consider a simple case in which a fluid is confined between two planes. One of them moves sideways with a certain speed u_0, while the other is kept fixed. After a certain transient, some force is needed in order to keep this shearing. The simplest expression is

F= \mu A \frac{u_0}{L}.

 

 

 

The force is proportional to the area and to the velocity difference between the planes. It is also inversely proportional to their separation, L (this fact being the least obvious). Finally, a constant of proportionality is given by \mu, the viscosity coefficient, or

action balls black and white illustration

Newton’s craddle. Another of this guy’s creations. Photo by Pixabay on Pexels.com

simply  “the viscosity”. This constant may vary with temperature, density, pressure, but the point with Newtonian fluids is that it does not vary with the velocity field (or its derivatives).

 

 

 

 

 

 

 

ocean water wave photo

What a wave. Its strain tensors must be on fire. Photo by Emiliano Arano on Pexels.com

Later, in section …, this flow will be solved as a solution of the Navier-Stokes equations, the Couette flow. There, it will be shown that the velocity is everywhere in the direction of the force exerted on the upper plane, let us call it $x$, and varies linearly between the planes, in the y direction. Therefore, the only components of the strain rate tensor are \epsilon_{xy} = \epsilon_{yx} = u_0 / ( 2 L ). We therefore have

\tau_{xy} = \mu \epsilon_{xy}.

 

 

With this in mind, let us look for a general relationship between \tau and $latex

black click pen on white paper

What a bunch of math. This is so hard. Photo by Lum3n.com on Pexels.com

\epsilon$. This is much easier if we go to the principal strain axes. These are the coordinates on which the strain rate is diagonal. Such coordinate system always exist, since the strain rate tensor is symmetric. Notice that in these system strains are not due to shear, only to dilations.

The o-nigiri project. Películas para ver con niños.

Una lista de pelis que ver con los niños que se salgan de la línea Disney.

Friends, la isla de los monstruos. En el enlace la tenéis enterita. Subtítulos en inglés. Producción japonesa de animación computerizada 3D (la primera que se ha hecho que sea largometraje, por lo visto).

Enest et célestine. Francesa, nominada al óscar 2014. Con moraleja, pero parece que es muy bonita visualmente.

Panique au Village. Colección de cortos belgas de stop-motion. Muy locos. Aquí tenéis uno de ellos.

Del Estudio Ghibli:

Las peores letras

Una colección de letras de canciones de llorar.

Escuela de Muñecos, Cantajuego 5.

Hay una escuela en algún lugar para muñecos
Es una escuela muy especial, es de muñecos
No me preguntes dónde está
Pues nadie nunca la pudo hallar
Porque esta escuela es solamente para muñecos

No Hay Marcha En Nueva York, Mecano

No hay marcha en Nueva York
Ni aunque lo jure Henry Ford
No hay marcha en Nueva York
Y los jamones son de York