Consider the flow of a newtonian fluid in a rectangular pipe. Consider a 3D coordinate system with the property that.that the flow of the fluid in the pipe is parallel to the x-axis. Let $V:\mathbb{R}^3\rightarrow \mathbb{R}^3$ return the velocity vector at each point of the 3D space. Assume that $V$ is differentiable with continuous partial derivatives. Consider a cuboid inside the fluid filling the pipe: $[x_0,x_0+\triangle x]\times [y_0,y_0+\triangle y]\times [z_0,z_0+\triangle z]$ . I am trying to compute the shear force on the face $\{(x_0,y,z)|y_0\leq y\leq y_0+\triangle y,z_0\leq z\leq z_0+\triangle z\}$ of the cuboid. My guess that it would be:
$$\int_{y_0}^{y_0+\triangle y}\int_{z_0}^{z_0+\triangle z} \mu[v_x(x_0,b,c)+v_y(x_0,b,c)]dc \,db$$
(Note about the notation: $V(x,y,z)=u(x,y,z)\mathbf{i}+v(x,y,z)\mathbf{j}+w(x,y,z)\mathbf{k}$)
Am I right? If I am not right could you please give the correct shear force on the face I am talking about.
Sorry if the question is naive. If the question is naive it would probably be because I only had an engineering course in fluid mechanics and the course considered only shear stress parallel to the direction of flow. I don't know if this is because shear stress only acts in the direction of shear flow or because the book was assuming (without mentioning) that some components of $V$ are zero or have zero partial derivatives. Thus the only equation for shear stress I saw in the course is $\tau=\mu \frac{\partial v}{\partial y}$. I am looking for the most general case.
Thank you
Edit: After thinking again, I modified my guess to:
$$\int_{y_0}^{y_0+\triangle y}\int_{z_0}^{z_0+\triangle z} \mu[v_x(x_0,b,c)+v_z(x_0,b,c)]dc \,db\,\mathbf{j}+\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\int_{y_0}^{y_0+\triangle y}\int_{z_0}^{z_0+\triangle z} \mu[w_x(x_0,b,c)+w_y(x_0,b,c)]dc \,db\,\mathbf{k}$$
However I am still unsure, and I would like someone to write the right answer for the net force acting on the face.
Reminder: The main question of this post is the following:
Is my second guess correct ? If not what is the right answer for the shear force ?
Best Answer
the stress tensor it $$\boldsymbol \tau = \left[ \begin{array}{ccc} \sigma_\text{xx} & \tau_\text{xy} & \tau_\text{xz}\\ \tau_\text{yx} & \sigma_\text{yy} & \tau_\text{yz}\\ \tau_\text{zx} & \tau_\text{zy} & \sigma_\text{zz}\\ \end{array} \right]$$ with the principal diagonal elements the normal stress and the off diagonal elements the shear stress. The shear stresses can be defined by many number of laws, simplest of which is the law from Newton. $$\tau = \mu \nabla v$$ in tensor form, shear stress can be expressed as the function of the viscosity $\mu$ and the gradient of the $i$th velocity in the $j$th direction, as given below.
$$\tau_{ij} = \mu \left(\frac{\partial v_i}{\partial x_j} + \frac{\partial v_j}{\partial x_i}\right)$$
that is the most general form of the viscosity equation. For more detailed info, please refer
Truckenbrodt, E. Fluidmechanik 1: Grundlagen und elementare Stromungsvorgange dichtebestandiger Fluide . Springer, Berlin, 2 Auflage, 1996 (in german) or
Munson, B. R., Young, D. F., and Okiishi, T. H.: Fundamentals of Fluid Mechanics . John Wiley and Sons Inc., 2006