Question

Suppose a solid object in \(\mathbb{R}^{3}\) has a temperature distribution given by \(T(x, y, z) .\) The heat flow vector field in the object is \(\mathbf{F}=-k \nabla T,\) where the conductivity \(k>0\) is a property of the material. Note that the heat flow vector points in the direction opposite to that of the gradient, which is the direction of greatest temperature decrease. The divergence of the heat flow vector is \(\nabla \cdot \mathbf{F}=-k \nabla \cdot \nabla T=-k \nabla^{2} T(\text {the Laplacian of } T) .\) Compute the heat flow vector field and its divergence for the following temperature distributions. $$T(x, y, z)=100(1+\sqrt{x^{2}+y^{2}+z^{2}})$$

Short answer

Question: Compute the heat flow vector field \(\mathbf{F}\) and its divergence for the given temperature distribution \(T(x, y, z) = 100(1+\sqrt{x^{2}+y^{2}+z^{2}})\). Answer: The heat flow vector field \(\mathbf{F}\) is given by: $$ \mathbf{F} = \left\langle -k \frac{100x}{\sqrt{x^{2}+y^{2}+z^{2}}}, -k \frac{100y}{\sqrt{x^{2}+y^{2}+z^{2}}}, -k \frac{100z}{\sqrt{x^{2}+y^{2}+z^{2}}} \right\rangle. $$ The divergence of the heat flow vector field, \(\nabla \cdot \mathbf{F}\), is given by: $$ \nabla \cdot \mathbf{F} = -100k^2 \frac{x^{2}+y^{2}+z^{2}}{(x^{2}+y^{2}+z^{2})^{3/2}}. $$

Step-by-step solution

Find the gradient of \(T\)

To find the gradient of \(T\), we will compute the partial derivatives of \(T\) with respect to \(x, y, z\): $$ \frac{\partial T(x, y, z)}{\partial x} = \frac{100x}{\sqrt{x^{2}+y^{2}+z^{2}}}, \frac{\partial T(x, y, z)}{\partial y} = \frac{100y}{\sqrt{x^{2}+y^{2}+z^{2}}}, \frac{\partial T(x, y, z)}{\partial z} = \frac{100z}{\sqrt{x^{2}+y^{2}+z^{2}}}. $$ So the gradient of \(T\) is: $$ \nabla T = \left\langle \frac{100x}{\sqrt{x^{2}+y^{2}+z^{2}}}, \frac{100y}{\sqrt{x^{2}+y^{2}+z^{2}}}, \frac{100z}{\sqrt{x^{2}+y^{2}+z^{2}}} \right\rangle. $$

Compute the heat flow vector field \(\mathbf{F}\)

Now that we have \(\nabla T\), we can compute the heat flow vector field \(\mathbf{F}\) using the formula \(\mathbf{F} = -k \nabla T\). Multiply each component of \(\nabla T\) by \(-k\): $$ \mathbf{F} = \left\langle -k \frac{100x}{\sqrt{x^{2}+y^{2}+z^{2}}}, -k \frac{100y}{\sqrt{x^{2}+y^{2}+z^{2}}}, -k \frac{100z}{\sqrt{x^{2}+y^{2}+z^{2}}} \right\rangle. $$

Compute the divergence of the heat flow vector field

To compute the divergence of \(\mathbf{F}\), we will find the partial derivatives of \(\mathbf{F}\) with respect to \(x, y, z\) and sum them up: $$ \begin{aligned} \nabla \cdot \mathbf{F} &= -k \nabla \cdot \nabla T \\ &= -k \left(\frac{\partial}{\partial x}\left(-k\frac{100x}{\sqrt{x^{2}+y^{2}+z^{2}}}\right) + \frac{\partial}{\partial y}\left(-k\frac{100y}{\sqrt{x^{2}+y^{2}+z^{2}}}\right) + \frac{\partial}{\partial z}\left(-k\frac{100z}{\sqrt{x^{2}+y^{2}+z^{2}}}\right)\right) \end{aligned} $$ Now let \(r=\sqrt{x^{2}+y^{2}+z^{2}}\). Then $$ \begin{aligned} \nabla \cdot \mathbf{F} &= -k\left(\frac{\partial}{\partial x}\left(-k\frac{100x}{r}\right) + \frac{\partial}{\partial y}\left(-k\frac{100y}{r}\right) + \frac{\partial}{\partial z}\left(-k\frac{100z}{r}\right)\right). \end{aligned} $$ Computing the partial derivatives and summing them up, we have: $$ \begin{aligned} \nabla \cdot \mathbf{F} &= -k(100k) \left( \frac{x^{2}+y^{2}+z^{2}-x^{2}}{(x^{2}+y^{2}+z^{2})^{3/2}} \right) \\ &= -100k^2 \frac{x^{2}+y^{2}+z^{2}}{(x^{2}+y^{2}+z^{2})^{3/2}}. \end{aligned} $$ Now, we found the heat flow vector field \(\mathbf{F}\) and its divergence \(\nabla\cdot \mathbf{F}\) for the temperature distribution \(T(x, y, z) = 100(1+\sqrt{x^{2}+y^{2}+z^{2}})\).

Key concepts

Temperature Distribution

Temperature distribution refers to how temperature varies in space within a given solid object. In the context of the given problem, the temperature across the object in three-dimensional space is defined by the function \(T(x, y, z) = 100(1 + \sqrt{x^2 + y^2 + z^2})\). This formula implies that temperature not only depends on the coordinates \((x, y, z)\) but also increases as the distance from the origin increases.

• "100" is a constant multiplier that determines the scale of temperature.
• "1 + \sqrt{x^2 + y^2 + z^2}" represents a radial temperature distribution that increases linearly with the distance from the origin.

Understanding how temperature varies is crucial since it affects how heat flows through the material. As temperature gradients arise, heat moves from regions of higher temperature to lower temperature.

Gradient Vector Field

The gradient of a temperature field is a vector field indicating the direction and rate of maximum temperature increase. For a scalar field \(T(x, y, z)\), its gradient \(abla T\) is computed by taking the partial derivatives with respect to each spatial variable \(x\), \(y\), and \(z\).
In this problem:

• The gradient \(abla T\) is \( \left\langle \frac{100x}{\sqrt{x^2+y^2+z^2}}, \frac{100y}{\sqrt{x^2+y^2+z^2}}, \frac{100z}{\sqrt{x^2+y^2+z^2}} \right\rangle \).
• This conveys that the gradient points radially outward from the origin.
• The magnitude of each component is proportional to their corresponding coordinate.

The gradient vector provides essential insights into how temperatures change across space, and it serves as a key element in determining the direction of heat flow.

Divergence of Vector Field

Divergence is a scalar measure that describes how much a vector field spreads out or converges at a point, providing insight into the local behavior of the field. Mathematically, divergence of a vector field \(\mathbf{F}\) is expressed as \(abla \cdot \mathbf{F}\).

In the exercise, the heat flow vector field \(\mathbf{F}\) is derived as \(-kabla T\). To find its divergence, we compute:

• Partial derivatives for each vector component: \(\frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z} \).
• Substituting these into the formula gives us the divergence \(abla \cdot \mathbf{F} = -100k^2 \frac{x^2 + y^2 + z^2}{(x^2+y^2+z^2)^{3/2}} \).

The negative sign signifies a decrease, highlighting that heat flows from higher to lower temperature regions. It reinforces the concept that divergence in the heat flow field is intrinsically connected to temperature gradients and material properties.

Laplacian

The Laplacian is an operator that describes how a function diverges from its average value at any point. For scalar fields, particularly in contexts involving temperature, the Laplacian quantifies the rate at which surrounding temperatures deviate from the point of interest.

In this exercise, the Laplacian of the temperature function \(T\) is denoted by \(abla^2 T\) and is inherently related to the divergence of the gradient \(abla \cdot abla T\).

Key points include:

• The divergence of the heat flow \(abla \cdot \mathbf{F} = -k abla^2 T\) directly links to the Laplacian.
• It represents how much the temperature field \(T\) diverges, impacting how heat distribution impacts the material.
• The term \(-k abla^2 T\) highlights how material properties (conductivity \(k\)) factor into the spread of temperature.

Understanding the Laplacian is crucial in heat transfer problems as it provides insight into the efficiency and dynamics of thermal conduction within materials.