# [[Introduction to Fluid Mechanics - Fox, McDonald.pdf#page=41&selection=272,0,274,12|Introduction to Fluid Mechanics, 25 - 31]] ## Stress Field - **Surface force** $F_{S}$ is generated by contact with other particles or a solid surface (pressure, shear, resultant,...) ^11d9d9 - **Body force** $F_{B}$ is experienced throughout the particle (gravity, electromagnetic) ^195fa6 - Gravitational force acting on an element of volume is given by $\rho \vec{g}dV$ - Surface forces on a fluid particle lead to stresses, mostly generated by motion rather than by deflection - Contact force being generated between two particles creates a normal and shear stress $ \sigma_{n}=\lim_{ \delta A_{n} \to 0 } \frac{\delta F_{n}}{\delta A_{n}} \qquad \tau_{n}=\lim_{ \delta A_{n} \to 0 } \frac{\delta F_{t}}{\delta A_{n}} $ $ \tau_{xy}=\lim_{ \delta A_{x} \to 0 } \frac{\delta F_{y}}{\delta A_{x}}\quad \tau_{xz}=\lim_{ \delta A_{x} \to 0 } \frac{\delta F_{z}}{\delta A_{x}} $ - First subscript indicates the plane on which the stress acts; second subscript indicates the direction in which the stress acts - Stress component is positive when the direction of the stress component and the plane on which it acts are both positive or both negative ## Viscosity - For a fluid at rest, there will be no shear stresses - A fluid's viscosity is its resistance to shear stress $ \text{deformation rate }=\lim_{ \delta t \to 0 } \frac{\delta\alpha}{\delta t}=\frac{d\alpha}{dt}=\frac{du}{dy} $ where $\alpha$ is the angle formed from shear stress between the $y$ and $x$ axis - Fluids in which shear stress is directly proportional to the rate of deformation are **Newtonian fluids**. All other fluids are **Non-Newtonian** ### Newtonian Fluids If a fluid is Newtonian, then $ \tau_{yx}\propto \frac{du}{dy} $ The constant of proportionality is the **absolute (or dynamic) viscosity** $\mu$. Newton's law of viscosity is given for one-dimensional flow by $ \tau_{yx}=\mu \frac{du}{dy} $ In fluid mechanics, the ratio of absolute viscosity $\mu$ to density $\rho$ is called *kinematic viscosity* $\nu$ ### Non-Newtonian Fluids ![[Pasted image 20260908195721.png|center|600]] - Examples of Non-Newtonian fluids include toothpaste and Lucite paint - Toothpaste behaves as a "fluid" when squeezed from the tube, but it does not run out by itself when the cap is removed. There is a threshold or yield stress below which toothpaste behaves as a solid $ \tau_{yx}=k\left( \frac{du}{dy} \right)^n $ where $n$ is called the *flow behavior index* and $k$ is *consistency index* $ \tau_{yx}=k \left| \frac{du}{dy} \right|^{n-1}=\eta \frac{du}{dy} $ where $\eta=k|du/dy|^{n-1}$ is the *apparent viscosity* - Fluids in which the apparent viscosity decreases w/ increasing deformation rate ($n<1$) are called *pseudoplastic* - If the apparent viscosity increases w/ increasing deformation rate ($n>1$), the fluid is *dilatant* - A fluid that behaves as a solid until a minimum yield stress $\tau_{y}$ is exceeded and subsequently exhibits a linear relationship between stress & rate of deformation is called a *Bingham plastic:* $ \tau_{yx}=\tau_{y}+\mu_{{p}} \frac{du}{dy} $ # Lecture - **Viscosity:** measure of a fluid's internal friction resisting its movement - $\tau_{yx}$ is a function of $\frac{du_{x}}{dy}$ - **Newtonian fluids** follow $\tau_{yx}\propto \frac{du_{x}}{dy}\implies \tau_{yx}=\mu \frac{du_{x}}{dy}$ - **Kinematic viscosity** $\nu=\frac{\mu}{\rho}$, common quantity ![[Pasted image 20260913142028.png|center|500]] - No-slip boundary condition: fluid sticks to boundary so the fluid at the boundary has the same velocity as it $ u(y)=\left( \frac{u_{p}}{H} \right) $ $ \tau_{yx}=\mu \frac{du_{x}}{dy}=\mu\left( \frac{u_{p}}{H} \right)\to \text{ constant shear stress} $ - Inviscid flow has no viscous stresses, $\tau=0$ - Possible for C.V with no height variance (see p. 11) 1. Shear stress can vary from point to point 2. $\tau_{ij}=\mu \left(\frac{\partial u_{i}}{\partial x_{j}}+\frac{\partial u_{j}}{\partial x_{i}}\right)$ where $i,j=x,y,z$ 3. $dF_{s}=\tau dA$ 4.